Skip to content

Instantly share code, notes, and snippets.

@genkuroki

genkuroki/11 Kullback-Leibler information.ipynb

Last active April 3, 2019 11:04
Show Gist options
  • Star 0 You must be signed in to star a gist
  • Fork 0 You must be signed in to fork a gist
  • Save genkuroki/f6746b8a903fabe24d89d771b6fe23f2 to your computer and use it in GitHub Desktop.
Save genkuroki/f6746b8a903fabe24d89d771b6fe23f2 to your computer and use it in GitHub Desktop.
Download ZIP
11 Kullback-Leibler information
Display the source blob
Display the rendered blob
Raw
11 Kullback-Leibler information.ipynb
{
"cells": [
{
"metadata": {},
"cell_type": "markdown",
"source": "# 11 Kullback-Leibler情報量\n\n黒木玄\n\n2018-06-24～2019-04-03\n\n* Copyright 2018 Gen Kuroki\n* License: MIT https://opensource.org/licenses/MIT\n* Repository: https://github.com/genkuroki/Calculus\n\nこのファイルは次の場所できれいに閲覧できる:\n\n* http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/11+Kullback-Leibler+information.ipynb\n\n* https://genkuroki.github.io/documents/Calculus/11+Kullback-Leibler+information.pdf\n\nこのファイルは <a href=\"https://juliabox.com\">Julia Box</a> で利用できる.\n\n自分のパソコンに<a href=\"https://julialang.org/\">Julia言語</a>をインストールしたい場合には\n\n* [WindowsへのJulia言語のインストール](http://nbviewer.jupyter.org/gist/genkuroki/81de23edcae631a995e19a2ecf946a4f)\n\n* [Julia v1.1.0 の Windows 8.1 へのインストール](https://nbviewer.jupyter.org/github/genkuroki/msfd28/blob/master/install.ipynb)\n\nを参照せよ. 前者は古く, 後者の方が新しい.\n\n論理的に完璧な説明をするつもりはない. 細部のいい加減な部分は自分で訂正・修正せよ.\n\n$\n\\newcommand\\eps{\\varepsilon}\n\\newcommand\\ds{\\displaystyle}\n\\newcommand\\Z{{\\mathbb Z}}\n\\newcommand\\R{{\\mathbb R}}\n\\newcommand\\C{{\\mathbb C}}\n\\newcommand\\QED{\\text{□}}\n\\newcommand\\root{\\sqrt}\n\\newcommand\\bra{\\langle}\n\\newcommand\\ket{\\rangle}\n\\newcommand\\d{\\partial}\n\\newcommand\\sech{\\operatorname{sech}}\n$"
},
{
"metadata": {
"toc": true
},
"cell_type": "markdown",
"source": "<h1>目次<span class=\"tocSkip\"></span></h1>\n<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Kullback-Leibler情報量の基本性質\" data-toc-modified-id=\"Kullback-Leibler情報量の基本性質-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Kullback-Leibler情報量の基本性質</a></span><ul class=\"toc-item\"><li><span><a href=\"#KL情報量の定義\" data-toc-modified-id=\"KL情報量の定義-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>KL情報量の定義</a></span></li><li><span><a href=\"#KL情報量の正値性\" data-toc-modified-id=\"KL情報量の正値性-1.2\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>KL情報量の正値性</a></span></li><li><span><a href=\"#KL情報量の凸性\" data-toc-modified-id=\"KL情報量の凸性-1.3\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>KL情報量の凸性</a></span></li><li><span><a href=\"#KL情報量が-$L^1$-ノルムの2乗を上から押さえること\" data-toc-modified-id=\"KL情報量が-$L^1$-ノルムの2乗を上から押さえること-1.4\"><span class=\"toc-item-num\">1.4&nbsp;&nbsp;</span>KL情報量が $L^1$ ノルムの2乗を上から押さえること</a></span></li><li><span><a href=\"#KL情報量の使われ方\" data-toc-modified-id=\"KL情報量の使われ方-1.5\"><span class=\"toc-item-num\">1.5&nbsp;&nbsp;</span>KL情報量の使われ方</a></span></li></ul></li><li><span><a href=\"#多項分布のSanovの定理\" data-toc-modified-id=\"多項分布のSanovの定理-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>多項分布のSanovの定理</a></span><ul class=\"toc-item\"><li><span><a href=\"#Sanovの定理とその証明\" data-toc-modified-id=\"Sanovの定理とその証明-2.1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>Sanovの定理とその証明</a></span></li><li><span><a href=\"#Sanovの定理の使い方1\" data-toc-modified-id=\"Sanovの定理の使い方1-2.2\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>Sanovの定理の使い方1</a></span></li><li><span><a href=\"#二項分布の場合\" data-toc-modified-id=\"二項分布の場合-2.3\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>二項分布の場合</a></span></li><li><span><a href=\"#Sanovの定理の使い方2\" data-toc-modified-id=\"Sanovの定理の使い方2-2.4\"><span class=\"toc-item-num\">2.4&nbsp;&nbsp;</span>Sanovの定理の使い方2</a></span></li><li><span><a href=\"#連続的な確率分布の場合\" data-toc-modified-id=\"連続的な確率分布の場合-2.5\"><span class=\"toc-item-num\">2.5&nbsp;&nbsp;</span>連続的な確率分布の場合</a></span></li><li><span><a href=\"#正規分布による近似\" data-toc-modified-id=\"正規分布による近似-2.6\"><span class=\"toc-item-num\">2.6&nbsp;&nbsp;</span>正規分布による近似</a></span></li><li><span><a href=\"#多項分布の中心極限定理\" data-toc-modified-id=\"多項分布の中心極限定理-2.7\"><span class=\"toc-item-num\">2.7&nbsp;&nbsp;</span>多項分布の中心極限定理</a></span></li></ul></li></ul></div>"
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "using Base.MathConstants\nusing Base64\nusing Printf\nusing Statistics\nconst e = ℯ\nendof(a) = lastindex(a)\nlinspace(start, stop, length) = range(start, stop, length=length)\n\nusing Plots\ngr(); ENV[\"PLOTS_TEST\"] = \"true\"\n#clibrary(:colorcet)\nclibrary(:misc)\n\nfunction pngplot(P...; kwargs...)\n sleep(0.1)\n pngfile = tempname() * \".png\"\n savefig(plot(P...; kwargs...), pngfile)\n showimg(\"image/png\", pngfile)\nend\npngplot(; kwargs...) = pngplot(plot!(; kwargs...))\n\nshowimg(mime, fn) = open(fn) do f\n base64 = base64encode(f)\n display(\"text/html\", \"\"\"<img src=\"data:$mime;base64,$base64\">\"\"\")\nend\n\nusing SymPy\n#sympy.init_printing(order=\"lex\") # default\n#sympy.init_printing(order=\"rev-lex\")\n\nusing SpecialFunctions\nusing QuadGK",
"execution_count": 1,
"outputs": []
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## Kullback-Leibler情報量の基本性質"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### KL情報量の定義\n\n一般の確率分布についても定義可能だが, このノートでは簡単のため, 有限集合\n\n$$\n\\Omega=\\{1,2,\\ldots,r\\}\n$$\n\nの上の確率分布のみを扱う. $p=(p_1,\\ldots,p_r)$ が $\\Omega=\\{1,2,\\ldots,r\\}$ 上の**確率分布**であるとは\n\n$$\np_i\\geqq 0 \\quad (i=1,2,\\ldots,r) \\qquad \\sum_{i=1}^r p_i = 1\n$$\n\nが成立していることであると定める. \n\n$\\Omega=\\{1,2,\\ldots,r\\}$ 上の確率分布 $q,p$ に対して, $(q,p)$ の**Kullback-Leibler情報量** $D(q||p)$ を\n\n$$\nD(q||p) = \\sum_{i=1}^r q_i\\log\\frac{q_i}{p_i} = \n\\left(-\\sum_{i=1}^r q_i\\log p_i\\right) - \\left(-\\sum_{i=1}^r q_i\\log q_i\\right)\n$$\n\nと定める. ただし, $x\\log y$ は $x=0$ のとき $0$ になり, $\\log 0 = -\\infty$ であると約束しておく. Kullback-Leibler情報量の $-1$ 倍を**相対エントロピー**と呼ぶ. \n\n記述の簡単のため以下ではKullback-Leiblerを**KL**と略記することにする. さらに, $q,p$ は $\\Omega=\\{1,2,\\ldots,r\\}$ 上の確率分布であると仮定する."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### KL情報量の正値性\n\n**KL情報量の正値性:** $D(q||p)\\geqq 0$.\n\n**証明:** $x\\geqq0$ に対して $f(x)=x\\log x$ とおく. ただし, $f(0)=\\lim_{x\\searrow 0}f(x)=0$ と約束しておく. $f'(x)=\\log x + 1$, $f''(x)=1/x>0$ なので $f(x)$ は下に凸な函数である. 期待値汎函数 $g(x)\\mapsto E[g(x)]$ を\n\n$$\nE[g(x)] = \\sum_{i=1}^r p_i g(q_i/p_i)\n$$\n\nと定めることができる. ゆえにJensenの不等式より,\n\n$$\nE[f(x)] \\geqq f(E[x]).\n$$\n\nそして,\n\n$$\nE[f(x)] = \\sum_{i=1}^r p_i\\frac{q_i}{p_i}\\log\\frac{q_i}{p_i} = D(q||p), \\quad\nE[x] = \\sum_{i=1}^r p_i\\frac{q_i}{p_i} = \\sum_{i=1}^r q_i = 1, \\quad\nf(E[x]) = f(1) = 0\n$$\n\nなので $D(q||p)\\geqq 0$ を得る. $\\QED$"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### KL情報量の凸性\n\n**KL情報量の凸性:** $D(q||p)$ は $q$ の函数として下に凸である.\n\n**証明:** $\\Omega$ 上の確率分布 $q,q'$ と $0<t<1$ が任意に与えられているとする. $f(x)=x\\log x$ とおくと, $f(x)$ は下に凸な函数である. ゆえに\n\n$$\n\\begin{aligned}\nD((1-t)q+tq'||p) &= \\sum_{i=1}^r p_i f\\left(\\frac{(1-t)q_i+tq'_i}{p_i}\\right)\n\\\\ &\\leqq\n\\sum_{i=1}^r p_i\\left((1-t)f\\left(\\frac{q_i}{p_i}\\right)+tf\\left(\\frac{q'_i}{p_i}\\right)\\right)\n\\\\ &=\n(1-t)D(q||p)+tD(q'||p).\n\\qquad \\QED\n\\end{aligned}\n$$\n\n**注意:** $f(x)$ は強い意味で下に凸なので $D(q||p)$ も $q$ の函数として強い意味で下に凸になる. このことより, $q$ の函数としての $D(q||p)$ の最小値を与える $q=p$ とは異なる $q\\ne p$ において $D(q||p)>0$ となることがわかる. より精密な結果については次の節を見よ. $\\QED$"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### KL情報量が $L^1$ ノルムの2乗を上から押さえること\n\n$q=p$ ならば $q_i\\log(q_i/p_i)=0$ より, $D(q||p)=0$ となる.\n\n* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20160616KullbackLeibler.pdf\">Kullback-Leibler情報量とSanovの定理</a>\n\nの第5.3節で次の不等式が証明されている:\n\n$$\nD(q||p) \\geqq \\frac{1}{2}\\left(\\sum_{i=1}^r |q_i-p_i|\\right)^2.\n$$\n\nこれより, $q\\ne p$ ならば $D(q||p)>0$ となることがわかる. 特に $D(q||p)=0$ と $p=q$ が同値であることがわかる.\n\nさらに, 確率分布の列 $q^{(1)},q^{(2)},\\ldots$ について, $k\\to\\infty$ で $D(q^{(k)}||p)\\to 0$ ならば $q^{(k)}\\to p$ となり, 確率分布の列 $p^{(1)},p^{(2)},\\ldots$ について, $k\\to\\infty$ で $D(q||p^{(k)})\\to 0$ ならば $p^{(k)}\\to q$ となることもわかる. $\\QED$ "
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### KL情報量の使われ方\n\nKL情報量 $D(q||p)$ は確率分布 $q$ と確率分布 $p$ の違いの大きさを表す指標としてよく使われる. 一般に $D(q||p)\\ne D(p||q)$ であるが, そうなることには必然性がある. 「KL情報量の意味」は下の方で説明する**Sanovの定理**を見ればわかる. Sanovの定理はStirlingの公式を知っていれば容易に証明可能である. Stirlingの公式を理解しておけば「情報理論」「統計力学」「統計学」などなどで基本的なSanovの定理をすぐに理解できるようになる. \n\n* 階乗に関するStirlingの近似公式 $\\longrightarrow$ KL情報量に関するSanovの定理\n\nという経路をできるだけ早く踏破しておいた方が数学の応用について楽に考えることができるようになると思う. \n\n統計学へのSanovの定理の応用については赤池弘次氏による以下の2つの論説を参照せよ:\n\n* 赤池弘次, <a href=\"https://www.jstage.jst.go.jp/article/butsuri1946/35/7/35_7_608/_article/-char/ja/\">エントロピーとモデルの尤度(〈講座〉物理学周辺の確率統計)\n</a>, 日本物理学会誌, 35巻 (1980) 7号, 608-614\n\n* 赤池弘次, <a href=\"https://ismrepo.ism.ac.jp/?action=pages_view_main&active_action=repository_view_main_item_detail&item_id=32568&item_no=1&page_id=13&block_id=21\">統計的推論のパラダイムの変遷について</a>, 統計数理研究所彙報, 27巻, 1号, 5-12, 1980-03\n\nさらに\n\n* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20160616KullbackLeibler.pdf\">Kullback-Leibler情報量とSanovの定理</a>\n\nも参照して欲しい. 「<a href=\"https://genkuroki.github.io/documents/20160616KullbackLeibler.pdf\">Kullback-Leibler情報量とSanovの定理</a>」の第10節には平均符号長とKL情報量の関係も解説してある. 「情報量」という呼び方の由来はそちらを見た方が分かり易いかもしれない. $\\QED$"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## 多項分布のSanovの定理\n\n1回の試行で $\\Omega=\\{1,2,\\ldots,r\\}$ から $i$ を取り出す確率は $p_i$ であるとし, $k_i$ は非負の整数であり, $k_1+k_2+\\cdots+k_r=n$ を満たしているとする. このとき $n$ 回の独立試行で $1,2,\\ldots,r$ のそれぞれが $k_1,k_2,\\ldots,k_r$ 回取り出される確率は\n\n$$\n\\frac{n!}{k_1!k_2!\\cdots k_r!} p_1^{k_1}p_2^{k_2}\\cdots p_r^{k_r}\n\\tag{$*$}\n$$\n\nになる. この式における階乗の分数の因子はそれぞれ $k_i$ 個の $i$ 達を全部で $n$ 個順番に並べる方法の総数に等しいので, **多項係数**と呼ばれる. $(k_1,\\ldots,k_r)$ 全体の集合上の確率分布 ($*$) は**多項分布**と呼ばれる."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### Sanovの定理とその証明\n\n**Sanovの定理:** $n\\to\\infty$ で $k_i$ 達が $k_i/n\\sim q_i$ ($i=1,\\ldots,r$) のように振る舞うならば, $n\\to\\infty$ で\n\n$$\n\\frac{n!}{k_1!k_2!\\cdots k_r!} p_1^{k_1}p_2^{k_2}\\cdots p_r^{k_r} = \n\\exp\\left(-nD(q||p) + o(n)\\right).\n$$\n\nすなわち\n\n$$\n-\\log\\left(\\frac{n!}{k_1!k_2!\\cdots k_r!} p_1^{k_1}p_2^{k_2}\\cdots p_r^{k_r}\\right)\n= nD(q||p) + o(n).\n$$\n\n**証明:** $k_i/n\\sim q_i$ という仮定より, $k_i=nq_i+o(n)$ である. $q_i$ は固定されているので, $n$ が大きくなると $k_i$ も $O(n)$ のオーダーで大きくなる. だから, Stirlingの公式\n\n$$\nn! \\sim n^n e^{-n} \\sqrt{2\\pi n}, \\quad\nk_i! \\sim k_i^{k_i} e^{-k_i} \\sqrt{2\\pi k_i}\n$$\n\nを使用できる. ゆえに, $k_1+\\cdots+k_r=n$ を使うと, \n\n$$\n\\frac{n!}{k_1!k_2!\\cdots k_r!}\n\\sim\n\\frac{n^n e^{-n} \\sqrt{2\\pi n}}\n{k_1^{k_1} e^{-k_1} \\sqrt{2\\pi k_1}\\cdots k_r^{k_r} e^{-k_r} \\sqrt{2\\pi k_r}} =\n\\left(\n\\prod_{i=1}^r\\left(\\frac{k_i}{n}\\right)^{k_r}\n\\sqrt{(2\\pi n)^{r-1}\\prod_{i=1}^r\\frac{k_i}{n}}\n\\right)^{-1}.\n$$\nしたがって, \n$$\n\\begin{aligned}\n%&\n\\frac{n!}{k_1!k_2!\\cdots k_r!} p_1^{k_1}p_2^{k_2}\\cdots p_r^{k_i}\n&\\sim\n\\left(\n\\prod_{i=1}^r \\left(\\frac{k_i/n}{p_i}\\right)^{k_i}\n\\sqrt{(2\\pi n)^{r-1}\\prod_{i=1}^r\\frac{k_i}{n}}\n\\right)^{-1}\n\\\\ &=\n\\exp\\left(\n-\\sum_{i=1}^r k_i\\log\\frac{k_i/n}{p_i}\n-(r-1)\\log n-\\log\\sqrt{(2\\pi)^{r-1}\\prod_{i=1}^r\\frac{k_i}{n}}\n\\right).\n\\end{aligned}\n$$\n\nこれに $k_i/n=q_i+o(1)$, $k_i = nq_i + o(n)$ を代入すれば示したい結果が得られる. $\\log n$ や定数の項は $n$ で割って $n\\to\\infty$ とすると $0$ に収束するので, それらは $o(n)$ の項にまとめられることに注意せよ. さらにKL情報量の定義が\n\n$$\nD(q||p) = \\sum_{i=1}^r q_i\\log\\frac{q_i}{p_i}\n$$\n\nであったことも思い出して計算せよ. $\\QED$\n\n要するに多項分布にStirlingの近似公式と $k_i=nq_i+o(n)$ を単純に代入するだけで, Sanovの定理が得られる."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### Sanovの定理の使い方1 \n\n有限集合 $\\Omega=\\{1,2,\\ldots,r\\}$ から確率 $p_i$ で $i$ を選ぶ独立試行を $n$ 回行った結果, $i$ が $k_i$ 回取り出されたとき, $\\Omega=\\{1,2,\\ldots,r\\}$ 上の確率分布 $(k_1/n,\\ldots,k_r/n)$ を**経験分布**と呼ぶ. Sanovの定理における $k_i/n\\sim q_i$ ($n\\to\\infty$) という条件は, $n$ を大きくしたとき**経験分布が確率分布 $q$ に近いこと**を意味している. だから, Sanovの定理は $n$ が大きなとき\n\n* (確率分布 $p$ のもとでの独立試行の経験分布が確率分布 $q$ に近い確率) $\\sim\\exp\\left(-nD(q||p) + o(n)\\right)$.\n\nが成立することを意味している. \n\n$o(n)$ は $n$ で割ると $0$ に収束する量なので, $n$ が十分に大きなとき $-nD(q||p)$ と比較すると無視できるほど小さな量になる. ゆえに, $n$ を十分に大きくすると, 異なる $q$ ごとの「経験分布が $q$ に近くなる確率」の大小関係をKL情報量 $D(q||p)$ の大小関係で測って良い状況になる. $D(q||p)$ が小さな $q$ ほど相対的に経験分布が近くなる確率は高くなる. \n\n経験分布が確率分布 $q^*$ に近くなる確率で経験分布が確率分布 $q$ に近くなる確率を割った結果は\n\n$$\n\\exp\\left(-n\\left(D(q||p)-D(q^*||p)\\right) + o(n)\\right)\n$$\n\nのように振る舞う. $D(q||p) > D(q^*||p)$ ならばこの確率の商は $n\\to\\infty$ で $0$ に収束する. そのとき, 経験分布が $q$ に近くなる確率は経験分布が $q^*$ に近くなる確率と比較すると相対的に無視できるくらい小さくなる.\n\n確率分布の集合 $A$ が与えられているとする. ただし, $A$ の内部の閉包が $A$ 閉包に一致するような「良い集合」のみを考える. 例えば $r=2$ (二項分布のとき)で $0\\leqq a<b\\leqq 1$ のとき $A=\\{\\,(q_1,1-q_1)\\mid a\\leqq q_1\\leqq b\\,\\}$ はそのような「良い集合」の例になっている. (「内部」「閉包」の意味を知らない人はこの例を思い浮かべておけばよい.)\n\n$q$ が $A$ 上を動くとき, $D(q||p)$ を最小にする $q\\in A$ を $q^*$ と書く. \n\nこのとき, 経験分布が $A$ に入るという条件のもとでの条件付き確率分布を考えると, $q^*$ 以外の $q\\in A$ に経験分布が近くなる確率は $n\\to\\infty$ で $0$ に近付く. これは、経験分布が $A$ に入るという条件のもとでの条件付き確率分布が $n\\to\\infty$ で $q^*$ に台を持つデルタ分布に近付くことを意味している. "
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 二項分布の場合\n\n$r=2$ の二項分布の場合を考える. このとき, $q_1=q$, $q_2=1-q$, $p_1=p$, $p_2=1-p$ と書くと,\n\n$$\nD(q||p) = q\\log\\frac{q}{p} + (1-q)\\log\\frac{1-q}{1-p}.\n$$\n\n**問題:** この函数は $q=p$ で最小値 $0$ になることを自分で確認してみよ. $\\QED$\n\n次のセルのプロットを見ればわかるように $D(q||p)$ は $q$ の函数として $q=p$ で最小になり, 下に凸な函数になる."
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "D(q,p) = q*log(q/p) + (1-q)*log((1-q)/(1-p))\nq = 0:0.01:1\nPP = []\nfor p in [0.2, 0.4, 0.6, 0.8]\n P = plot(legend=false)\n plot!(title=\"y = D(q, $(round(p,digits=3)))\", titlefontsize=10)\n plot!(xlabel=\"q_1\", ylabel=\"y\")\n plot!(q, D.(q, p))\n push!(PP, P)\nend\nplot(PP..., size=(700, 500))",
"execution_count": 2,
"outputs": [
{
"data": {
"image/svg+xml": "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"700\" height=\"500\" viewBox=\"0 0 2800 2000\">\n<defs>\n <clipPath id=\"clip1600\">\n <rect x=\"0\" y=\"0\" width=\"2000\" height=\"2000\"/>\n </clipPath>\n</defs>\n<defs>\n <clipPath id=\"clip1601\">\n <rect x=\"0\" y=\"0\" width=\"2800\" height=\"2000\"/>\n </clipPath>\n</defs>\n<polygon clip-path=\"url(#clip1601)\" points=\"\n0,2000 2800,2000 2800,0 0,0 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip1602\">\n <rect x=\"560\" y=\"0\" width=\"1961\" height=\"1961\"/>\n </clipPath>\n</defs>\n<polygon clip-path=\"url(#clip1601)\" points=\"\n224.386,840.483 1321.26,840.483 1321.26,125.984 224.386,125.984 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip1603\">\n <rect x=\"224\" y=\"125\" width=\"1098\" height=\"715\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip1603)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 255.43,840.483 255.43,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1603)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 514.126,840.483 514.126,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1603)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 772.823,840.483 772.823,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1603)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1031.52,840.483 1031.52,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1603)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1290.22,840.483 1290.22,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1603)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,820.261 1321.26,820.261 \n \"/>\n<polyline clip-path=\"url(#clip1603)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,601.351 1321.26,601.351 \n \"/>\n<polyline clip-path=\"url(#clip1603)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,382.442 1321.26,382.442 \n \"/>\n<polyline clip-path=\"url(#clip1603)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,163.532 1321.26,163.532 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,840.483 1321.26,840.483 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,840.483 224.386,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 255.43,840.483 255.43,829.765 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 514.126,840.483 514.126,829.765 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 772.823,840.483 772.823,829.765 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1031.52,840.483 1031.52,829.765 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1290.22,840.483 1290.22,829.765 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,820.261 240.839,820.261 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,601.351 240.839,601.351 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,382.442 240.839,382.442 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,163.532 240.839,163.532 \n \"/>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 255.43, 896.483)\" x=\"255.43\" y=\"896.483\">0.00</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 514.126, 896.483)\" x=\"514.126\" y=\"896.483\">0.25</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 896.483)\" x=\"772.823\" y=\"896.483\">0.50</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1031.52, 896.483)\" x=\"1031.52\" y=\"896.483\">0.75</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1290.22, 896.483)\" x=\"1290.22\" y=\"896.483\">1.00</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 837.761)\" x=\"196.386\" y=\"837.761\">0.0</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 618.851)\" x=\"196.386\" y=\"618.851\">0.5</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 399.942)\" x=\"196.386\" y=\"399.942\">1.0</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 181.032)\" x=\"196.386\" y=\"181.032\">1.5</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 51.6)\" x=\"772.823\" y=\"51.6\">y = D(q, 0.2)</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 990.4)\" x=\"772.823\" y=\"990.4\">q_1</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 57.6, 483.233)\" x=\"57.6\" y=\"483.233\">y</text>\n</g>\n<polyline clip-path=\"url(#clip1603)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 265.778,741.013 276.125,753.349 286.473,763.349 296.821,771.816 307.169,779.131 317.517,785.518 327.865,791.126 338.213,796.059 348.561,800.396 358.908,804.197 \n 369.256,807.511 379.604,810.378 389.952,812.829 400.3,814.893 410.648,816.593 420.996,817.949 431.343,818.979 441.691,819.699 452.039,820.122 462.387,820.261 \n 472.735,820.126 483.083,819.727 493.431,819.072 503.779,818.171 514.126,817.029 524.474,815.654 534.822,814.051 545.17,812.226 555.518,810.184 565.866,807.929 \n 576.214,805.465 586.561,802.797 596.909,799.928 607.257,796.86 617.605,793.598 627.953,790.143 638.301,786.498 648.649,782.665 658.996,778.646 669.344,774.443 \n 679.692,770.058 690.04,765.492 700.388,760.746 710.736,755.821 721.084,750.719 731.432,745.44 741.779,739.984 752.127,734.353 762.475,728.546 772.823,722.564 \n 783.171,716.407 793.519,710.075 803.867,703.567 814.214,696.884 824.562,690.024 834.91,682.988 845.258,675.773 855.606,668.38 865.954,660.808 876.302,653.054 \n 886.65,645.118 896.997,636.998 907.345,628.692 917.693,620.198 928.041,611.514 938.389,602.637 948.737,593.566 959.085,584.296 969.432,574.826 979.78,565.15 \n 990.128,555.266 1000.48,545.169 1010.82,534.855 1021.17,524.319 1031.52,513.556 1041.87,502.559 1052.22,491.321 1062.56,479.837 1072.91,468.097 1083.26,456.093 \n 1093.61,443.816 1103.95,431.253 1114.3,418.395 1124.65,405.225 1135,391.73 1145.35,377.891 1155.69,363.689 1166.04,349.098 1176.39,334.093 1186.74,318.64 \n 1197.09,302.7 1207.43,286.224 1217.78,269.152 1228.13,251.405 1238.48,232.879 1248.82,213.425 1259.17,192.819 1269.52,170.68 1279.87,146.206 \n \"/>\n<polygon clip-path=\"url(#clip1601)\" points=\"\n1624.39,840.483 2721.26,840.483 2721.26,125.984 1624.39,125.984 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip1604\">\n <rect x=\"1624\" y=\"125\" width=\"1098\" height=\"715\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1655.43,840.483 1655.43,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1914.13,840.483 1914.13,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2172.82,840.483 2172.82,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2431.52,840.483 2431.52,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2690.22,840.483 2690.22,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,820.261 2721.26,820.261 \n \"/>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,662.815 2721.26,662.815 \n \"/>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,505.368 2721.26,505.368 \n \"/>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,347.922 2721.26,347.922 \n \"/>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,190.476 2721.26,190.476 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,840.483 2721.26,840.483 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,840.483 1624.39,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1655.43,840.483 1655.43,829.765 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1914.13,840.483 1914.13,829.765 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2172.82,840.483 2172.82,829.765 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2431.52,840.483 2431.52,829.765 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2690.22,840.483 2690.22,829.765 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,820.261 1640.84,820.261 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,662.815 1640.84,662.815 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,505.368 1640.84,505.368 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,347.922 1640.84,347.922 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,190.476 1640.84,190.476 \n \"/>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1655.43, 896.483)\" x=\"1655.43\" y=\"896.483\">0.00</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1914.13, 896.483)\" x=\"1914.13\" y=\"896.483\">0.25</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2172.82, 896.483)\" x=\"2172.82\" y=\"896.483\">0.50</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2431.52, 896.483)\" x=\"2431.52\" y=\"896.483\">0.75</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2690.22, 896.483)\" x=\"2690.22\" y=\"896.483\">1.00</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 837.761)\" x=\"1596.39\" y=\"837.761\">0.0</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 680.315)\" x=\"1596.39\" y=\"680.315\">0.2</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 522.868)\" x=\"1596.39\" y=\"522.868\">0.4</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 365.422)\" x=\"1596.39\" y=\"365.422\">0.6</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 207.976)\" x=\"1596.39\" y=\"207.976\">0.8</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 2172.82, 51.6)\" x=\"2172.82\" y=\"51.6\">y = D(q, 0.4)</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 2172.82, 990.4)\" x=\"2172.82\" y=\"990.4\">q_1</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 1457.6, 483.233)\" x=\"1457.6\" y=\"483.233\">y</text>\n</g>\n<polyline clip-path=\"url(#clip1604)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1665.78,459.017 1676.13,488.918 1686.47,514.62 1696.82,537.566 1707.17,558.441 1717.52,577.647 1727.86,595.452 1738.21,612.043 1748.56,627.563 1758.91,642.119 \n 1769.26,655.799 1779.6,668.674 1789.95,680.803 1800.3,692.236 1810.65,703.014 1821,713.174 1831.34,722.748 1841.69,731.764 1852.04,740.246 1862.39,748.216 \n 1872.73,755.695 1883.08,762.699 1893.43,769.243 1903.78,775.344 1914.13,781.012 1924.47,786.261 1934.82,791.1 1945.17,795.54 1955.52,799.589 1965.87,803.256 \n 1976.21,806.548 1986.56,809.472 1996.91,812.034 2007.26,814.24 2017.61,816.095 2027.95,817.604 2038.3,818.771 2048.65,819.601 2059,820.096 2069.34,820.261 \n 2079.69,820.097 2090.04,819.608 2100.39,818.796 2110.74,817.663 2121.08,816.21 2131.43,814.439 2141.78,812.351 2152.13,809.947 2162.48,807.227 2172.82,804.193 \n 2183.17,800.843 2193.52,797.179 2203.87,793.199 2214.21,788.903 2224.56,784.29 2234.91,779.359 2245.26,774.109 2255.61,768.537 2265.95,762.642 2276.3,756.422 \n 2286.65,749.874 2297,742.994 2307.35,735.781 2317.69,728.23 2328.04,720.337 2338.39,712.098 2348.74,703.508 2359.08,694.562 2369.43,685.254 2379.78,675.578 \n 2390.13,665.527 2400.48,655.094 2410.82,644.27 2421.17,633.047 2431.52,621.415 2441.87,609.362 2452.22,596.878 2462.56,583.949 2472.91,570.562 2483.26,556.699 \n 2493.61,542.345 2503.95,527.479 2514.3,512.079 2524.65,496.121 2535,479.577 2545.35,462.415 2555.69,444.599 2566.04,426.086 2576.39,406.827 2586.74,386.763 \n 2597.09,365.823 2607.43,343.919 2617.78,320.944 2628.13,296.756 2638.48,271.165 2648.82,243.907 2659.17,214.577 2669.52,182.491 2679.87,146.206 \n \"/>\n<polygon clip-path=\"url(#clip1601)\" points=\"\n224.386,1840.48 1321.26,1840.48 1321.26,1125.98 224.386,1125.98 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip1605\">\n <rect x=\"224\" y=\"1125\" width=\"1098\" height=\"715\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 255.43,1840.48 255.43,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 514.126,1840.48 514.126,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 772.823,1840.48 772.823,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1031.52,1840.48 1031.52,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1290.22,1840.48 1290.22,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,1820.26 1321.26,1820.26 \n \"/>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,1662.81 1321.26,1662.81 \n \"/>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,1505.37 1321.26,1505.37 \n \"/>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,1347.92 1321.26,1347.92 \n \"/>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,1190.48 1321.26,1190.48 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1840.48 1321.26,1840.48 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1840.48 224.386,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 255.43,1840.48 255.43,1829.77 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 514.126,1840.48 514.126,1829.77 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 772.823,1840.48 772.823,1829.77 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1031.52,1840.48 1031.52,1829.77 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1290.22,1840.48 1290.22,1829.77 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1820.26 240.839,1820.26 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1662.81 240.839,1662.81 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1505.37 240.839,1505.37 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1347.92 240.839,1347.92 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1190.48 240.839,1190.48 \n \"/>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 255.43, 1896.48)\" x=\"255.43\" y=\"1896.48\">0.00</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 514.126, 1896.48)\" x=\"514.126\" y=\"1896.48\">0.25</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 1896.48)\" x=\"772.823\" y=\"1896.48\">0.50</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1031.52, 1896.48)\" x=\"1031.52\" y=\"1896.48\">0.75</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1290.22, 1896.48)\" x=\"1290.22\" y=\"1896.48\">1.00</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 1837.76)\" x=\"196.386\" y=\"1837.76\">0.0</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 1680.31)\" x=\"196.386\" y=\"1680.31\">0.2</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 1522.87)\" x=\"196.386\" y=\"1522.87\">0.4</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 1365.42)\" x=\"196.386\" y=\"1365.42\">0.6</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 1207.98)\" x=\"196.386\" y=\"1207.98\">0.8</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 1051.6)\" x=\"772.823\" y=\"1051.6\">y = D(q, 0.6)</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 1990.4)\" x=\"772.823\" y=\"1990.4\">q_1</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 57.6, 1483.23)\" x=\"57.6\" y=\"1483.23\">y</text>\n</g>\n<polyline clip-path=\"url(#clip1605)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 265.778,1146.21 276.125,1182.49 286.473,1214.58 296.821,1243.91 307.169,1271.17 317.517,1296.76 327.865,1320.94 338.213,1343.92 348.561,1365.82 358.908,1386.76 \n 369.256,1406.83 379.604,1426.09 389.952,1444.6 400.3,1462.42 410.648,1479.58 420.996,1496.12 431.343,1512.08 441.691,1527.48 452.039,1542.34 462.387,1556.7 \n 472.735,1570.56 483.083,1583.95 493.431,1596.88 503.779,1609.36 514.126,1621.41 524.474,1633.05 534.822,1644.27 545.17,1655.09 555.518,1665.53 565.866,1675.58 \n 576.214,1685.25 586.561,1694.56 596.909,1703.51 607.257,1712.1 617.605,1720.34 627.953,1728.23 638.301,1735.78 648.649,1742.99 658.996,1749.87 669.344,1756.42 \n 679.692,1762.64 690.04,1768.54 700.388,1774.11 710.736,1779.36 721.084,1784.29 731.432,1788.9 741.779,1793.2 752.127,1797.18 762.475,1800.84 772.823,1804.19 \n 783.171,1807.23 793.519,1809.95 803.867,1812.35 814.214,1814.44 824.562,1816.21 834.91,1817.66 845.258,1818.8 855.606,1819.61 865.954,1820.1 876.302,1820.26 \n 886.65,1820.1 896.997,1819.6 907.345,1818.77 917.693,1817.6 928.041,1816.1 938.389,1814.24 948.737,1812.03 959.085,1809.47 969.432,1806.55 979.78,1803.26 \n 990.128,1799.59 1000.48,1795.54 1010.82,1791.1 1021.17,1786.26 1031.52,1781.01 1041.87,1775.34 1052.22,1769.24 1062.56,1762.7 1072.91,1755.69 1083.26,1748.22 \n 1093.61,1740.25 1103.95,1731.76 1114.3,1722.75 1124.65,1713.17 1135,1703.01 1145.35,1692.24 1155.69,1680.8 1166.04,1668.67 1176.39,1655.8 1186.74,1642.12 \n 1197.09,1627.56 1207.43,1612.04 1217.78,1595.45 1228.13,1577.65 1238.48,1558.44 1248.82,1537.57 1259.17,1514.62 1269.52,1488.92 1279.87,1459.02 \n \"/>\n<polygon clip-path=\"url(#clip1601)\" points=\"\n1624.39,1840.48 2721.26,1840.48 2721.26,1125.98 1624.39,1125.98 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip1606\">\n <rect x=\"1624\" y=\"1125\" width=\"1098\" height=\"715\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip1606)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1655.43,1840.48 1655.43,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1606)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1914.13,1840.48 1914.13,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1606)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2172.82,1840.48 2172.82,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1606)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2431.52,1840.48 2431.52,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1606)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2690.22,1840.48 2690.22,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1606)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,1820.26 2721.26,1820.26 \n \"/>\n<polyline clip-path=\"url(#clip1606)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,1601.35 2721.26,1601.35 \n \"/>\n<polyline clip-path=\"url(#clip1606)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,1382.44 2721.26,1382.44 \n \"/>\n<polyline clip-path=\"url(#clip1606)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,1163.53 2721.26,1163.53 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,1840.48 2721.26,1840.48 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,1840.48 1624.39,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1655.43,1840.48 1655.43,1829.77 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1914.13,1840.48 1914.13,1829.77 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2172.82,1840.48 2172.82,1829.77 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2431.52,1840.48 2431.52,1829.77 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2690.22,1840.48 2690.22,1829.77 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,1820.26 1640.84,1820.26 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,1601.35 1640.84,1601.35 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,1382.44 1640.84,1382.44 \n \"/>\n<polyline clip-path=\"url(#clip1601)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,1163.53 1640.84,1163.53 \n \"/>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1655.43, 1896.48)\" x=\"1655.43\" y=\"1896.48\">0.00</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1914.13, 1896.48)\" x=\"1914.13\" y=\"1896.48\">0.25</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2172.82, 1896.48)\" x=\"2172.82\" y=\"1896.48\">0.50</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2431.52, 1896.48)\" x=\"2431.52\" y=\"1896.48\">0.75</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2690.22, 1896.48)\" x=\"2690.22\" y=\"1896.48\">1.00</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 1837.76)\" x=\"1596.39\" y=\"1837.76\">0.0</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 1618.85)\" x=\"1596.39\" y=\"1618.85\">0.5</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 1399.94)\" x=\"1596.39\" y=\"1399.94\">1.0</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 1181.03)\" x=\"1596.39\" y=\"1181.03\">1.5</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 2172.82, 1051.6)\" x=\"2172.82\" y=\"1051.6\">y = D(q, 0.8)</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 2172.82, 1990.4)\" x=\"2172.82\" y=\"1990.4\">q_1</text>\n</g>\n<g clip-path=\"url(#clip1601)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 1457.6, 1483.23)\" x=\"1457.6\" y=\"1483.23\">y</text>\n</g>\n<polyline clip-path=\"url(#clip1606)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1665.78,1146.21 1676.13,1170.68 1686.47,1192.82 1696.82,1213.42 1707.17,1232.88 1717.52,1251.41 1727.86,1269.15 1738.21,1286.22 1748.56,1302.7 1758.91,1318.64 \n 1769.26,1334.09 1779.6,1349.1 1789.95,1363.69 1800.3,1377.89 1810.65,1391.73 1821,1405.23 1831.34,1418.39 1841.69,1431.25 1852.04,1443.82 1862.39,1456.09 \n 1872.73,1468.1 1883.08,1479.84 1893.43,1491.32 1903.78,1502.56 1914.13,1513.56 1924.47,1524.32 1934.82,1534.86 1945.17,1545.17 1955.52,1555.27 1965.87,1565.15 \n 1976.21,1574.83 1986.56,1584.3 1996.91,1593.57 2007.26,1602.64 2017.61,1611.51 2027.95,1620.2 2038.3,1628.69 2048.65,1637 2059,1645.12 2069.34,1653.05 \n 2079.69,1660.81 2090.04,1668.38 2100.39,1675.77 2110.74,1682.99 2121.08,1690.02 2131.43,1696.88 2141.78,1703.57 2152.13,1710.08 2162.48,1716.41 2172.82,1722.56 \n 2183.17,1728.55 2193.52,1734.35 2203.87,1739.98 2214.21,1745.44 2224.56,1750.72 2234.91,1755.82 2245.26,1760.75 2255.61,1765.49 2265.95,1770.06 2276.3,1774.44 \n 2286.65,1778.65 2297,1782.66 2307.35,1786.5 2317.69,1790.14 2328.04,1793.6 2338.39,1796.86 2348.74,1799.93 2359.08,1802.8 2369.43,1805.47 2379.78,1807.93 \n 2390.13,1810.18 2400.48,1812.23 2410.82,1814.05 2421.17,1815.65 2431.52,1817.03 2441.87,1818.17 2452.22,1819.07 2462.56,1819.73 2472.91,1820.13 2483.26,1820.26 \n 2493.61,1820.12 2503.95,1819.7 2514.3,1818.98 2524.65,1817.95 2535,1816.59 2545.35,1814.89 2555.69,1812.83 2566.04,1810.38 2576.39,1807.51 2586.74,1804.2 \n 2597.09,1800.4 2607.43,1796.06 2617.78,1791.13 2628.13,1785.52 2638.48,1779.13 2648.82,1771.82 2659.17,1763.35 2669.52,1753.35 2679.87,1741.01 \n \"/>\n</svg>\n"
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "D(q,p) = q*log(q/p) + (1-q)*log((1-q)/(1-p))\np = 0:0.01:1\nPP = []\nfor q in [0.2, 0.4, 0.6, 0.8]\n P = plot(legend=false)\n plot!(title=\"y = D($(round(q,digits=3)), p)\", titlefontsize=10)\n plot!(xlabel=\"p_1\", ylabel=\"y\")\n plot!(p, D.(q, p))\n push!(PP, P)\nend\nplot(PP..., size=(700, 500))",
"execution_count": 3,
"outputs": [
{
"data": {
"image/svg+xml": "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"700\" height=\"500\" viewBox=\"0 0 2800 2000\">\n<defs>\n <clipPath id=\"clip1800\">\n <rect x=\"0\" y=\"0\" width=\"2000\" height=\"2000\"/>\n </clipPath>\n</defs>\n<defs>\n <clipPath id=\"clip1801\">\n <rect x=\"0\" y=\"0\" width=\"2800\" height=\"2000\"/>\n </clipPath>\n</defs>\n<polygon clip-path=\"url(#clip1801)\" points=\"\n0,2000 2800,2000 2800,0 0,0 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip1802\">\n <rect x=\"560\" y=\"0\" width=\"1961\" height=\"1961\"/>\n </clipPath>\n</defs>\n<polygon clip-path=\"url(#clip1801)\" points=\"\n224.386,838.087 1321.26,838.087 1321.26,125.984 224.386,125.984 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip1803\">\n <rect x=\"224\" y=\"125\" width=\"1098\" height=\"713\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip1803)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 255.43,838.087 255.43,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1803)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 514.126,838.087 514.126,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1803)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 772.823,838.087 772.823,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1803)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1031.52,838.087 1031.52,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1803)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1290.22,838.087 1290.22,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1803)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,817.934 1321.26,817.934 \n \"/>\n<polyline clip-path=\"url(#clip1803)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,607.058 1321.26,607.058 \n \"/>\n<polyline clip-path=\"url(#clip1803)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,396.182 1321.26,396.182 \n \"/>\n<polyline clip-path=\"url(#clip1803)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,185.307 1321.26,185.307 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,838.087 1321.26,838.087 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,838.087 224.386,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 255.43,838.087 255.43,827.406 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 514.126,838.087 514.126,827.406 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 772.823,838.087 772.823,827.406 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1031.52,838.087 1031.52,827.406 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1290.22,838.087 1290.22,827.406 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,817.934 240.839,817.934 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,607.058 240.839,607.058 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,396.182 240.839,396.182 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,185.307 240.839,185.307 \n \"/>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 255.43, 894.087)\" x=\"255.43\" y=\"894.087\">0.00</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 514.126, 894.087)\" x=\"514.126\" y=\"894.087\">0.25</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 894.087)\" x=\"772.823\" y=\"894.087\">0.50</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1031.52, 894.087)\" x=\"1031.52\" y=\"894.087\">0.75</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1290.22, 894.087)\" x=\"1290.22\" y=\"894.087\">1.00</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 835.434)\" x=\"196.386\" y=\"835.434\">0</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 624.558)\" x=\"196.386\" y=\"624.558\">1</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 413.682)\" x=\"196.386\" y=\"413.682\">2</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 202.807)\" x=\"196.386\" y=\"202.807\">3</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 51.6)\" x=\"772.823\" y=\"51.6\">y = D(0.2, p)</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 990.4)\" x=\"772.823\" y=\"990.4\">p_1</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 57.6, 482.036)\" x=\"57.6\" y=\"482.036\">y</text>\n</g>\n<polyline clip-path=\"url(#clip1803)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 265.778,727.537 276.125,755.058 286.473,770.428 296.821,780.813 307.169,788.458 317.517,794.362 327.865,799.059 338.213,802.867 348.561,805.991 358.908,808.57 \n 369.256,810.705 379.604,812.468 389.952,813.916 400.3,815.091 410.648,816.028 420.996,816.753 431.343,817.29 441.691,817.656 452.039,817.866 462.387,817.934 \n 472.735,817.869 483.083,817.682 493.431,817.38 503.779,816.97 514.126,816.457 524.474,815.847 534.822,815.143 545.17,814.35 555.518,813.47 565.866,812.507 \n 576.214,811.463 586.561,810.339 596.909,809.137 607.257,807.86 617.605,806.507 627.953,805.079 638.301,803.578 648.649,802.003 658.996,800.356 669.344,798.635 \n 679.692,796.841 690.04,794.974 700.388,793.032 710.736,791.016 721.084,788.924 731.432,786.755 741.779,784.509 752.127,782.183 762.475,779.777 772.823,777.288 \n 783.171,774.715 793.519,772.056 803.867,769.307 814.214,766.468 824.562,763.534 834.91,760.502 845.258,757.371 855.606,754.135 865.954,750.79 876.302,747.333 \n 886.65,743.759 896.997,740.063 907.345,736.239 917.693,732.281 928.041,728.182 938.389,723.936 948.737,719.534 959.085,714.968 969.432,710.227 979.78,705.303 \n 990.128,700.182 1000.48,694.852 1010.82,689.298 1021.17,683.505 1031.52,677.455 1041.87,671.127 1052.22,664.498 1062.56,657.543 1072.91,650.233 1083.26,642.532 \n 1093.61,634.403 1103.95,625.799 1114.3,616.668 1124.65,606.946 1135,596.557 1145.35,585.411 1155.69,573.397 1166.04,560.375 1176.39,546.173 1186.74,530.565 \n 1197.09,513.257 1207.43,493.848 1217.78,471.777 1228.13,446.223 1238.48,415.912 1248.82,378.709 1259.17,330.614 1269.52,262.644 1279.87,146.138 \n \"/>\n<polygon clip-path=\"url(#clip1801)\" points=\"\n1624.39,838.087 2721.26,838.087 2721.26,125.984 1624.39,125.984 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip1804\">\n <rect x=\"1624\" y=\"125\" width=\"1098\" height=\"713\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1655.43,838.087 1655.43,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1914.13,838.087 1914.13,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2172.82,838.087 2172.82,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2431.52,838.087 2431.52,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2690.22,838.087 2690.22,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,817.934 2721.26,817.934 \n \"/>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,657.532 2721.26,657.532 \n \"/>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,497.131 2721.26,497.131 \n \"/>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,336.73 2721.26,336.73 \n \"/>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1624.39,176.329 2721.26,176.329 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,838.087 2721.26,838.087 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,838.087 1624.39,125.984 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1655.43,838.087 1655.43,827.406 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1914.13,838.087 1914.13,827.406 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2172.82,838.087 2172.82,827.406 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2431.52,838.087 2431.52,827.406 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2690.22,838.087 2690.22,827.406 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,817.934 1640.84,817.934 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,657.532 1640.84,657.532 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,497.131 1640.84,497.131 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,336.73 1640.84,336.73 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1624.39,176.329 1640.84,176.329 \n \"/>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1655.43, 894.087)\" x=\"1655.43\" y=\"894.087\">0.00</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1914.13, 894.087)\" x=\"1914.13\" y=\"894.087\">0.25</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2172.82, 894.087)\" x=\"2172.82\" y=\"894.087\">0.50</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2431.52, 894.087)\" x=\"2431.52\" y=\"894.087\">0.75</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2690.22, 894.087)\" x=\"2690.22\" y=\"894.087\">1.00</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 835.434)\" x=\"1596.39\" y=\"835.434\">0.0</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 675.032)\" x=\"1596.39\" y=\"675.032\">0.5</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 514.631)\" x=\"1596.39\" y=\"514.631\">1.0</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 354.23)\" x=\"1596.39\" y=\"354.23\">1.5</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1596.39, 193.829)\" x=\"1596.39\" y=\"193.829\">2.0</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 2172.82, 51.6)\" x=\"2172.82\" y=\"51.6\">y = D(0.4, p)</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 2172.82, 990.4)\" x=\"2172.82\" y=\"990.4\">p_1</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 1457.6, 482.036)\" x=\"1457.6\" y=\"482.036\">y</text>\n</g>\n<polyline clip-path=\"url(#clip1804)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1665.78,440.963 1676.13,527.954 1686.47,578.01 1696.82,612.931 1707.17,639.549 1717.52,660.908 1727.86,678.63 1738.21,693.684 1748.56,706.694 1758.91,718.087 \n 1769.26,728.167 1779.6,737.158 1789.95,745.229 1800.3,752.513 1810.65,759.115 1821,765.119 1831.34,770.593 1841.69,775.595 1852.04,780.171 1862.39,784.362 \n 1872.73,788.201 1883.08,791.719 1893.43,794.939 1903.78,797.884 1914.13,800.573 1924.47,803.022 1934.82,805.246 1945.17,807.258 1955.52,809.069 1965.87,810.689 \n 1976.21,812.127 1986.56,813.391 1996.91,814.488 2007.26,815.424 2017.61,816.205 2027.95,816.836 2038.3,817.321 2048.65,817.663 2059,817.866 2069.34,817.934 \n 2079.69,817.867 2090.04,817.669 2100.39,817.341 2110.74,816.884 2121.08,816.299 2131.43,815.588 2141.78,814.75 2152.13,813.785 2162.48,812.693 2172.82,811.474 \n 2183.17,810.126 2193.52,808.649 2203.87,807.041 2214.21,805.3 2224.56,803.424 2234.91,801.411 2245.26,799.257 2255.61,796.96 2265.95,794.515 2276.3,791.919 \n 2286.65,789.167 2297,786.253 2307.35,783.173 2317.69,779.92 2328.04,776.488 2338.39,772.867 2348.74,769.051 2359.08,765.029 2369.43,760.791 2379.78,756.326 \n 2390.13,751.621 2400.48,746.661 2410.82,741.431 2421.17,735.913 2431.52,730.086 2441.87,723.928 2452.22,717.413 2462.56,710.513 2472.91,703.193 2483.26,695.416 \n 2493.61,687.137 2503.95,678.305 2514.3,668.859 2524.65,658.726 2535,647.822 2545.35,636.043 2555.69,623.262 2566.04,609.322 2576.39,594.024 2586.74,577.113 \n 2597.09,558.251 2607.43,536.982 2617.78,512.667 2628.13,484.368 2638.48,450.633 2648.82,409.025 2659.17,354.982 2669.52,278.253 2679.87,146.138 \n \"/>\n<polygon clip-path=\"url(#clip1801)\" points=\"\n224.386,1838.09 1321.26,1838.09 1321.26,1125.98 224.386,1125.98 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip1805\">\n <rect x=\"224\" y=\"1125\" width=\"1098\" height=\"713\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 255.43,1838.09 255.43,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 514.126,1838.09 514.126,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 772.823,1838.09 772.823,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1031.52,1838.09 1031.52,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1290.22,1838.09 1290.22,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,1817.93 1321.26,1817.93 \n \"/>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,1657.53 1321.26,1657.53 \n \"/>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,1497.13 1321.26,1497.13 \n \"/>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,1336.73 1321.26,1336.73 \n \"/>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 224.386,1176.33 1321.26,1176.33 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1838.09 1321.26,1838.09 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1838.09 224.386,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 255.43,1838.09 255.43,1827.41 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 514.126,1838.09 514.126,1827.41 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 772.823,1838.09 772.823,1827.41 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1031.52,1838.09 1031.52,1827.41 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1290.22,1838.09 1290.22,1827.41 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1817.93 240.839,1817.93 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1657.53 240.839,1657.53 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1497.13 240.839,1497.13 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1336.73 240.839,1336.73 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 224.386,1176.33 240.839,1176.33 \n \"/>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 255.43, 1894.09)\" x=\"255.43\" y=\"1894.09\">0.00</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 514.126, 1894.09)\" x=\"514.126\" y=\"1894.09\">0.25</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 1894.09)\" x=\"772.823\" y=\"1894.09\">0.50</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1031.52, 1894.09)\" x=\"1031.52\" y=\"1894.09\">0.75</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1290.22, 1894.09)\" x=\"1290.22\" y=\"1894.09\">1.00</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 1835.43)\" x=\"196.386\" y=\"1835.43\">0.0</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 1675.03)\" x=\"196.386\" y=\"1675.03\">0.5</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 1514.63)\" x=\"196.386\" y=\"1514.63\">1.0</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 1354.23)\" x=\"196.386\" y=\"1354.23\">1.5</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 196.386, 1193.83)\" x=\"196.386\" y=\"1193.83\">2.0</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 1051.6)\" x=\"772.823\" y=\"1051.6\">y = D(0.6, p)</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 772.823, 1990.4)\" x=\"772.823\" y=\"1990.4\">p_1</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 57.6, 1482.04)\" x=\"57.6\" y=\"1482.04\">y</text>\n</g>\n<polyline clip-path=\"url(#clip1805)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 265.778,1146.14 276.125,1278.25 286.473,1354.98 296.821,1409.03 307.169,1450.63 317.517,1484.37 327.865,1512.67 338.213,1536.98 348.561,1558.25 358.908,1577.11 \n 369.256,1594.02 379.604,1609.32 389.952,1623.26 400.3,1636.04 410.648,1647.82 420.996,1658.73 431.343,1668.86 441.691,1678.31 452.039,1687.14 462.387,1695.42 \n 472.735,1703.19 483.083,1710.51 493.431,1717.41 503.779,1723.93 514.126,1730.09 524.474,1735.91 534.822,1741.43 545.17,1746.66 555.518,1751.62 565.866,1756.33 \n 576.214,1760.79 586.561,1765.03 596.909,1769.05 607.257,1772.87 617.605,1776.49 627.953,1779.92 638.301,1783.17 648.649,1786.25 658.996,1789.17 669.344,1791.92 \n 679.692,1794.51 690.04,1796.96 700.388,1799.26 710.736,1801.41 721.084,1803.42 731.432,1805.3 741.779,1807.04 752.127,1808.65 762.475,1810.13 772.823,1811.47 \n 783.171,1812.69 793.519,1813.78 803.867,1814.75 814.214,1815.59 824.562,1816.3 834.91,1816.88 845.258,1817.34 855.606,1817.67 865.954,1817.87 876.302,1817.93 \n 886.65,1817.87 896.997,1817.66 907.345,1817.32 917.693,1816.84 928.041,1816.21 938.389,1815.42 948.737,1814.49 959.085,1813.39 969.432,1812.13 979.78,1810.69 \n 990.128,1809.07 1000.48,1807.26 1010.82,1805.25 1021.17,1803.02 1031.52,1800.57 1041.87,1797.88 1052.22,1794.94 1062.56,1791.72 1072.91,1788.2 1083.26,1784.36 \n 1093.61,1780.17 1103.95,1775.59 1114.3,1770.59 1124.65,1765.12 1135,1759.12 1145.35,1752.51 1155.69,1745.23 1166.04,1737.16 1176.39,1728.17 1186.74,1718.09 \n 1197.09,1706.69 1207.43,1693.68 1217.78,1678.63 1228.13,1660.91 1238.48,1639.55 1248.82,1612.93 1259.17,1578.01 1269.52,1527.95 1279.87,1440.96 \n \"/>\n<polygon clip-path=\"url(#clip1801)\" points=\"\n1584.24,1838.09 2721.26,1838.09 2721.26,1125.98 1584.24,1125.98 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip1806\">\n <rect x=\"1584\" y=\"1125\" width=\"1138\" height=\"713\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip1806)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1616.42,1838.09 1616.42,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1806)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1884.59,1838.09 1884.59,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1806)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2152.75,1838.09 2152.75,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1806)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2420.92,1838.09 2420.92,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1806)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2689.08,1838.09 2689.08,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1806)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1584.24,1817.93 2721.26,1817.93 \n \"/>\n<polyline clip-path=\"url(#clip1806)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1584.24,1607.06 2721.26,1607.06 \n \"/>\n<polyline clip-path=\"url(#clip1806)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1584.24,1396.18 2721.26,1396.18 \n \"/>\n<polyline clip-path=\"url(#clip1806)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1584.24,1185.31 2721.26,1185.31 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1584.24,1838.09 2721.26,1838.09 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1584.24,1838.09 1584.24,1125.98 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1616.42,1838.09 1616.42,1827.41 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1884.59,1838.09 1884.59,1827.41 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2152.75,1838.09 2152.75,1827.41 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2420.92,1838.09 2420.92,1827.41 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2689.08,1838.09 2689.08,1827.41 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1584.24,1817.93 1601.3,1817.93 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1584.24,1607.06 1601.3,1607.06 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1584.24,1396.18 1601.3,1396.18 \n \"/>\n<polyline clip-path=\"url(#clip1801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1584.24,1185.31 1601.3,1185.31 \n \"/>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1616.42, 1894.09)\" x=\"1616.42\" y=\"1894.09\">0.00</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1884.59, 1894.09)\" x=\"1884.59\" y=\"1894.09\">0.25</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2152.75, 1894.09)\" x=\"2152.75\" y=\"1894.09\">0.50</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2420.92, 1894.09)\" x=\"2420.92\" y=\"1894.09\">0.75</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2689.08, 1894.09)\" x=\"2689.08\" y=\"1894.09\">1.00</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1556.24, 1835.43)\" x=\"1556.24\" y=\"1835.43\">0</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1556.24, 1624.56)\" x=\"1556.24\" y=\"1624.56\">1</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1556.24, 1413.68)\" x=\"1556.24\" y=\"1413.68\">2</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1556.24, 1202.81)\" x=\"1556.24\" y=\"1202.81\">3</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:59px; text-anchor:middle;\" transform=\"rotate(0, 2152.75, 1051.6)\" x=\"2152.75\" y=\"1051.6\">y = D(0.8, p)</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 2152.75, 1990.4)\" x=\"2152.75\" y=\"1990.4\">p_1</text>\n</g>\n<g clip-path=\"url(#clip1801)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 1457.6, 1482.04)\" x=\"1457.6\" y=\"1482.04\">y</text>\n</g>\n<polyline clip-path=\"url(#clip1806)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1627.15,1146.14 1637.88,1262.64 1648.6,1330.61 1659.33,1378.71 1670.06,1415.91 1680.78,1446.22 1691.51,1471.78 1702.24,1493.85 1712.96,1513.26 1723.69,1530.57 \n 1734.41,1546.17 1745.14,1560.38 1755.87,1573.4 1766.59,1585.41 1777.32,1596.56 1788.05,1606.95 1798.77,1616.67 1809.5,1625.8 1820.23,1634.4 1830.95,1642.53 \n 1841.68,1650.23 1852.41,1657.54 1863.13,1664.5 1873.86,1671.13 1884.59,1677.45 1895.31,1683.51 1906.04,1689.3 1916.77,1694.85 1927.49,1700.18 1938.22,1705.3 \n 1948.95,1710.23 1959.67,1714.97 1970.4,1719.53 1981.13,1723.94 1991.85,1728.18 2002.58,1732.28 2013.31,1736.24 2024.03,1740.06 2034.76,1743.76 2045.49,1747.33 \n 2056.21,1750.79 2066.94,1754.13 2077.67,1757.37 2088.39,1760.5 2099.12,1763.53 2109.84,1766.47 2120.57,1769.31 2131.3,1772.06 2142.02,1774.72 2152.75,1777.29 \n 2163.48,1779.78 2174.2,1782.18 2184.93,1784.51 2195.66,1786.76 2206.38,1788.92 2217.11,1791.02 2227.84,1793.03 2238.56,1794.97 2249.29,1796.84 2260.02,1798.63 \n 2270.74,1800.36 2281.47,1802 2292.2,1803.58 2302.92,1805.08 2313.65,1806.51 2324.38,1807.86 2335.1,1809.14 2345.83,1810.34 2356.56,1811.46 2367.28,1812.51 \n 2378.01,1813.47 2388.74,1814.35 2399.46,1815.14 2410.19,1815.85 2420.92,1816.46 2431.64,1816.97 2442.37,1817.38 2453.1,1817.68 2463.82,1817.87 2474.55,1817.93 \n 2485.28,1817.87 2496,1817.66 2506.73,1817.29 2517.45,1816.75 2528.18,1816.03 2538.91,1815.09 2549.63,1813.92 2560.36,1812.47 2571.09,1810.7 2581.81,1808.57 \n 2592.54,1805.99 2603.27,1802.87 2613.99,1799.06 2624.72,1794.36 2635.45,1788.46 2646.17,1780.81 2656.9,1770.43 2667.63,1755.06 2678.35,1727.54 \n \"/>\n</svg>\n"
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "**問題(Laplaceの方法の弱形の一種):** 実数 $E_1,\\ldots,E_N$ について, $n\\to\\infty$ のとき, \n\n$$\n-\\frac{1}{n}\\log\\sum_{i=1}^N e^{-nE_i} \\to \\min\\{E_1,\\cdots,E_N\\}\n\\tag{$*$}\n$$\n\nとなることを示せ. \n\n**解答例:** $E_1=\\cdots=E_k<E_{k+1}\\leqq\\cdots\\leqq E_N$ と仮定してよい. このとき,\n\n$$\n-\\frac{1}{n}\\log\\sum_{i=1}^N e^{-nE_i} =\n-\\frac{1}{n}\\log\\left(e^{-nE_1}\\left(k+\\sum_{i=k+1}^n e^{-n(E_i-E_1)}\\right)\\right) =\nE_1 - \\frac{1}{n}\\log\\left(k+\\sum_{i=k+1}^n e^{-n(E_i-E_1)}\\right).\n$$\n\nこれより, ($*$)が成立することがわかる. $\\QED$\n\n**注意:** 上の問題の結果の $\\max$ 版\n\n$$\n\\frac{1}{n}\\log\\sum_{i=1}^N e^{nE_i} \\to \\max\\{E_1,\\cdots,E_N\\}\n$$\n\nは<a href=\"https://www.google.co.jp/search?q=%E3%83%88%E3%83%AD%E3%83%94%E3%82%AB%E3%83%AB%E5%B9%BE%E4%BD%95\">トロピカル幾何</a>における <a href=\"https://www.google.co.jp/search?q=max-plus%E4%BB%A3%E6%95%B0\">max-plus 代数</a>の max が極限で出て来る仕組みとしても現われる. $\\QED$\n\n**注意:** コンピューターによる統計がらみの数値計算では<a href=\"https://www.google.co.jp/search?q=logsumexp\">logsumexp</a>という函数を適切に定義して使うことがある. $\\QED$"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "確率 $p$ で当たるくじを引く独立試行を $n$ 回繰り返したとき, 当たりが $k$ 回出る確率は\n\n$$\n\\binom{n}{k}p^k(1-p)^{n-k} = \\frac{n!}{k!(n-k)!}p^k(1-p)^{n-k}\n$$\n\nになる. $0\\leqq a<b\\leqq 1$ であるとする. このとき当たりが出た回数の割合 $k/n$ が $a$ 以上 $b$ 以下になる確率は\n\n$$\n\\sum_{a\\leqq k/n\\leqq b}\\binom{n}{k}p^k(1-p)^{n-k}\n$$\n\nになる. この確率の対数の $-1/n$ 倍\n\n$$\n-\\frac{1}{n}\\log \\sum_{a\\leqq k\\leqq b}\\binom{n}{k}p^k(1-p)^{n-k}\n$$\n\nが $n\\to\\infty$ でどうなるかについて考えよう.\n\nSanovの定理より, $k/n\\sim q$ のとき,\n\n$$\n\\binom{n}{k}p^k(1-p)^{n-k} = \\exp(-nD(q|p) + o(n)).\n$$\n\nしたがって, 上の問題の結果の一般化によって, $n\\to\\infty$ で\n\n$$\n-\\frac{1}{n}\\log \\sum_{a\\leqq k\\leqq b}\\binom{n}{k}p^k(1-p)^{n-k} \\to\n\\max_{a\\leqq q\\leqq b} D(q||p) =\n\\begin{cases}\n0 & (a\\leqq p\\leqq b) \\\\\nD(b||p) & (b < p) \\\\\nD(a||p) & (p < a) \\\\\n\\end{cases}\n$$\n\nとなると考えられる. 要するに $a$ 以上 $b$ 以下の $k/n$ で $p$ に最も近いものが生じる確率の寄与だけが, 効いて来るということである. (前節の説明も再読せよ.)\n\n以上の議論は少々厳密さに欠けた荒い議論だが, 結論は正しい. 正確な議論の仕方については\n\n* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20160616KullbackLeibler.pdf\">Kullback-Leibler情報量とSanovの定理</a>\n\nの第3節を参照せよ."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### Sanovの定理の使い方2\n\nKL情報量 $D(q||p)$ は, Sanovの定理によって, 確率分布 $p$ の試行を独立に $n$ 回繰り返したとき, 経験分布として $q$ に近い分布が出て来る確率が $\\exp(-nD(q||p)+o(n))$ になるという性質を持っているのであった.\n\nこれより, 確率分布 $p$ の独立試行によって確率分布 $q$ をシミュレートしたときの, 予測誤差の大きさが $D(q||p)$ で測られることを意味していると考えることができる. \n\n$q=p$ と $D(q||p)=0$ が同値であることも, この解釈をサポートしている. \n\nさらに, 最も簡単(で極端)な次の場合について考えてみよう. 毎日雨が確実に降る地域で, 毎日確率 $p$ で「明日は雨が降ります」と予報を出し, 確率 $1-p$ で「明日は雨が降りません」と予報を出すことを $n$ 回繰りかすとき, 予報の出し方が間違っていたことがばれずにすむ確率は $p^n = \\exp(-n(-\\log p))$ になる. そして, この場合のKL情報量は\n\n$$\nD(1||p) = 1\\log\\frac{1}{p} + (1-1)\\log\\frac{1-1}{1-p} = -\\log p\n$$\n\nとなる(二項分布の場合の記号法をそのまま使った). 予報の出し方が間違っていたことがばれずにすむ確率が下がる速さは $D(1||p)=-\\log p$ で決まっている. ゆえに, KL情報量 $D(1||p)$ は予報の出し方が間違っていたことが確率的にばれる速さを表していると考えられる. この極端で単純な場合も, KL情報量 $D(q||p)$ を確率分布 $p$ による確率分布 $q$ のシミュレーションの予測誤差を表しているという解釈をサポートしている.\n\nより一般の場合には, Sanovの定理は\n\n* (確率分布 $p$ の独立試行の経験分布が $q$ に近い確率) $= \\exp(-nD(q||p)+o(n))$\n\nと要約でき, 確率分布 $p$ の独立試行の経験分布が $q$ に近いことは, 確率分布 $p$ の独立試行により確率分布 $q$ の予測が間違っていることがばれずに済んでいることを意味していると解釈できるので, やはり $D(q||p)$ は予測が間違っていたことがばれるまでの速さになっていると解釈できる.\n\nKL情報量は以上の意味で予測誤差になっていると考えられる:\n\n* $D(q||p) =$ (確率分布 $p$ の独立試行による確率分布 $q$ のシミュレーションの予測誤差)."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 連続的な確率分布の場合\n\n連続的な確率分布の場合であっても, 確率密度函数 $q(x)$, $p(x)$ に対するKL情報量が\n\n$$\nD(q||p) = \\int q(x)\\log\\frac{q(x)}{p(x)}\\,dx\n$$\n\nと定義され($\\int$ は定積分を意味する), 適当な条件と適切な定式化のもとでSanovの定理が成立していることが知られている:\n\n* $D(q||p) =$ (確率分布 $p$ の独立試行による確率分布 $q$ のシミュレーションの予測誤差).\n\n正確な内容については「<a href=\"https://www.google.co.jp/search?q=%E5%A4%A7%E5%81%8F%E5%B7%AE%E5%8E%9F%E7%90%86+Sanov%E3%81%AE%E5%AE%9A%E7%90%86\">大偏差原理 Sanovの定理</a>」について検索して欲しい. (連続的な $x$ 軸を $r$ 個に区切れば, $x$ 軸上の確率分布は有限集合 $\\Omega=\\{1,\\ldots,r\\}$ 上の確率分布で近似できることに注意せよ.)\n\n確率分布 $p$ による独立試行で確率分布 $q$ をシミュレートする状況を考えよう. $\\int_{-\\infty}^\\infty$ を単に $\\int$ と書くことにすると, その場合の予測誤差は\n\n$$\nD(q|p) = \\int q(x)\\log\\frac{q(x)}{p(x)}\\,dx = G(q||p) - S(q).\n$$\n\nここで\n\n$$\nG(q||p) = -\\int q(x)\\log p(x)\\,dx, \\quad\nS(q)=-\\int q(x)\\log q(x)\\,dx\n$$\n\nといおいた. $S(q)$ は**Shannon情報量**と呼ばれ, $G(q||p)$ は**汎化誤差**と呼ばれる."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 正規分布による近似\n\n例として, 正規分布\n\n$$\np(x) = p_N(x|\\mu,\\sigma) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-(x-\\mu)^2/(2\\sigma^2)}\n$$\n\nによる独立試行で確率分布 $q(x)$ をシミュレートする場合について考えよう. 予測誤差を小さくするためには, パラメーター $\\mu,\\sigma$ を調節して汎化誤差 $G(q||p)$ を小さくすればよい. $p$ が正規分布の場合には\n\n$$\n\\begin{aligned}\nG(q||p) &= \n\\int q(x)\\left(\\frac{(x-\\mu)^2}{2\\sigma^2} + \\frac{1}{2}\\log\\sigma^2 + \\log\\sqrt{2\\pi}\\right)\\,dx\n\\\\ &=\n\\frac{1}{\\sigma^2}\\int q(x)(x-\\mu)^2\\,dx + \\frac{1}{2}\\log\\sigma^2 + \\log\\sqrt{2\\pi}.\n\\end{aligned}\n$$\n\n$q(x)$ の平均と分散を\n\n$$\n\\mu_q = \\int q(x)x\\,dx, \\quad\n\\sigma_q^2 = \\int q(x)(x-\\mu_q)^2\\,dx\n$$\n\nと書くと, \n\n$$\n\\int q(x)(x-\\mu)^2\\,dx =\\int q(x)((x-\\mu_q)-(\\mu-\\mu_q))^2\\,dx =\n\\sigma_q^2 + (\\mu-\\mu_q)^2\n$$\n\nなので\n\n$$\nG(q||p) =\n\\frac{(\\mu-\\mu_q)^2}{2\\sigma^2} + \\frac{\\sigma_q^2}{2\\sigma^2} + \\frac{1}{2}\\log\\sigma^2 + \\log\\sqrt{2\\pi}.\n$$\n\nこれより, $G(q||p)$ を最小にする $\\mu,\\sigma$ は $\\mu=\\mu_q$, $\\sigma=\\sigma_q$ となることがわかる. \n\nすなわち, 正規分布による確率分布 $q(x)$ のシミュレーションの予測誤差を最小にするためには, 正規分布の平均と分散を確率分布 $q(x)$ の平均と分散に一致させればよいことがわかった.\n\nこの結果は未知の確率分布 $q(x)$ のサンプルに基いてその平均と分散の両方を推定することは, 未知の確率分布 $q(x)$ を最も小さな予測誤差でシミュレートできる正規分布を推定することと同じことであることがわかる.\n\n平均と分散に関する統計学は, KL情報量のSanovの定理の立場から見ると, 本質的に正規分布の統計学であるということがわかった.\n\n以上のように, **KL情報量のSanovの定理は, 「パラメーターの推定」という考え方をより普遍的な「確率分布の推定」という考え方に高めるため役に立つ**."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "**問題:** 上の計算で省略した部分を埋めよ.\n\n**解答例:** $\\sigma$ を固定したときの $G(q||p)$ の最小値は $\\mu=\\mu_q$ で与えられることは明らか. \n\n$$\nf(s) = \\frac{a}{s} + \\log s\n$$\n\nとおくと, \n\n$$\nf'(s) = -\\frac{a}{s^2}+\\frac{1}{s} = \\frac{s-a}{s^2}\n$$\n\nなので $f(s)$ は $s=a$ で最小になることがわかる. これより, $G(q||p)$ は $\\mu=\\mu_q$, $\\sigma=\\sigma_q$ のときに最小になることがわかる. $\\QED$"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "**問題:** 次の確率密度函数で定義される分布をロジスティック分布と呼ぶ:\n\n$$\nq(x) = \\frac{1}{4s} \\sech^2\\left(\\frac{x-\\mu}{2s}\\right).\n$$\n\nここで\n\n$$\n\\sech x = \\frac{1}{\\cosh x}, \\quad \\cosh x = \\frac{e^x+e^{-x}}{2}.\n$$\n\nロジスティック分布の平均と分散はそれぞれ\n\n$$\n\\mu_q = \\mu, \\quad \\sigma_q^2 = \\frac{\\pi^2 s^2}{3}.\n$$\n\nロジスティック分布を最もよくシミュレートする正規分布の確率密度函数ともとのロジスティック分布の確率密度函数を同時にプロットしてみよ. $\\QED$"
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "pdflogistic(μ,s,x) = 1/(4s)*sech((x-μ)/(2s))^2\npdfnormal(μ,σ,x) = exp(-(x-μ)^2/(2*σ^2))/√(2π*σ^2)\n\nμ = 0.0\ns = 1.0\nσ = π*s/√3\nx = -10:0.02:10\n\nP = plot(legend=:topleft, ylims=(0, 0.26))\nplot!(x, pdflogistic.(μ,s,x), label=\"logistic\", lw=2)\nplot!(x, pdfnormal.(μ,σ,x), label=\"normal\", lw=2, ls=:dash)",
"execution_count": 4,
"outputs": [
{
"data": {
"image/svg+xml": "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n<defs>\n <clipPath id=\"clip2000\">\n <rect x=\"0\" y=\"0\" width=\"2000\" height=\"2000\"/>\n </clipPath>\n</defs>\n<defs>\n <clipPath id=\"clip2001\">\n <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"/>\n </clipPath>\n</defs>\n<polygon clip-path=\"url(#clip2001)\" points=\"\n0,1600 2400,1600 2400,0 0,0 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip2002\">\n <rect x=\"480\" y=\"0\" width=\"1681\" height=\"1600\"/>\n </clipPath>\n</defs>\n<polygon clip-path=\"url(#clip2001)\" points=\"\n188.156,1501.08 2321.26,1501.08 2321.26,47.2441 188.156,47.2441 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip2003\">\n <rect x=\"188\" y=\"47\" width=\"2134\" height=\"1455\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 248.527,1501.08 248.527,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 751.618,1501.08 751.618,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1254.71,1501.08 1254.71,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1757.8,1501.08 1757.8,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2260.89,1501.08 2260.89,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,1501.08 2321.26,1501.08 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,1221.5 2321.26,1221.5 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,941.912 2321.26,941.912 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,662.328 2321.26,662.328 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,382.745 2321.26,382.745 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,103.161 2321.26,103.161 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,1501.08 2321.26,1501.08 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,1501.08 188.156,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 248.527,1501.08 248.527,1479.27 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 751.618,1501.08 751.618,1479.27 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1254.71,1501.08 1254.71,1479.27 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1757.8,1501.08 1757.8,1479.27 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2260.89,1501.08 2260.89,1479.27 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,1501.08 220.153,1501.08 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,1221.5 220.153,1221.5 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,941.912 220.153,941.912 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,662.328 220.153,662.328 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,382.745 220.153,382.745 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,103.161 220.153,103.161 \n \"/>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 248.527, 1555.08)\" x=\"248.527\" y=\"1555.08\">-10</text>\n</g>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 751.618, 1555.08)\" x=\"751.618\" y=\"1555.08\">-5</text>\n</g>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1254.71, 1555.08)\" x=\"1254.71\" y=\"1555.08\">0</text>\n</g>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1757.8, 1555.08)\" x=\"1757.8\" y=\"1555.08\">5</text>\n</g>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2260.89, 1555.08)\" x=\"2260.89\" y=\"1555.08\">10</text>\n</g>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 164.156, 1518.58)\" x=\"164.156\" y=\"1518.58\">0.00</text>\n</g>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 164.156, 1239)\" x=\"164.156\" y=\"1239\">0.05</text>\n</g>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 164.156, 959.412)\" x=\"164.156\" y=\"959.412\">0.10</text>\n</g>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 164.156, 679.828)\" x=\"164.156\" y=\"679.828\">0.15</text>\n</g>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 164.156, 400.245)\" x=\"164.156\" y=\"400.245\">0.20</text>\n</g>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 164.156, 120.661)\" x=\"164.156\" y=\"120.661\">0.25</text>\n</g>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#009af9; stroke-width:8; stroke-opacity:1; fill:none\" points=\"\n 248.527,1500.83 250.54,1500.82 252.552,1500.82 254.564,1500.81 256.577,1500.8 258.589,1500.8 260.601,1500.79 262.614,1500.79 264.626,1500.78 266.638,1500.78 \n 268.651,1500.77 270.663,1500.76 272.676,1500.76 274.688,1500.75 276.7,1500.74 278.713,1500.74 280.725,1500.73 282.737,1500.72 284.75,1500.72 286.762,1500.71 \n 288.774,1500.7 290.787,1500.69 292.799,1500.69 294.812,1500.68 296.824,1500.67 298.836,1500.66 300.849,1500.65 302.861,1500.64 304.873,1500.64 306.886,1500.63 \n 308.898,1500.62 310.91,1500.61 312.923,1500.6 314.935,1500.59 316.948,1500.58 318.96,1500.57 320.972,1500.56 322.985,1500.55 324.997,1500.54 327.009,1500.53 \n 329.022,1500.51 331.034,1500.5 333.046,1500.49 335.059,1500.48 337.071,1500.47 339.084,1500.46 341.096,1500.44 343.108,1500.43 345.121,1500.42 347.133,1500.4 \n 349.145,1500.39 351.158,1500.38 353.17,1500.36 355.182,1500.35 357.195,1500.33 359.207,1500.32 361.219,1500.3 363.232,1500.29 365.244,1500.27 367.257,1500.25 \n 369.269,1500.24 371.281,1500.22 373.294,1500.2 375.306,1500.18 377.318,1500.17 379.331,1500.15 381.343,1500.13 383.355,1500.11 385.368,1500.09 387.38,1500.07 \n 389.393,1500.05 391.405,1500.03 393.417,1500.01 395.43,1499.99 397.442,1499.96 399.454,1499.94 401.467,1499.92 403.479,1499.9 405.491,1499.87 407.504,1499.85 \n 409.516,1499.82 411.529,1499.8 413.541,1499.77 415.553,1499.75 417.566,1499.72 419.578,1499.69 421.59,1499.66 423.603,1499.63 425.615,1499.6 427.627,1499.57 \n 429.64,1499.54 431.652,1499.51 433.665,1499.48 435.677,1499.45 437.689,1499.42 439.702,1499.38 441.714,1499.35 443.726,1499.31 445.739,1499.28 447.751,1499.24 \n 449.763,1499.2 451.776,1499.17 453.788,1499.13 455.8,1499.09 457.813,1499.05 459.825,1499.01 461.838,1498.97 463.85,1498.92 465.862,1498.88 467.875,1498.84 \n 469.887,1498.79 471.899,1498.74 473.912,1498.7 475.924,1498.65 477.936,1498.6 479.949,1498.55 481.961,1498.5 483.974,1498.45 485.986,1498.39 487.998,1498.34 \n 490.011,1498.28 492.023,1498.23 494.035,1498.17 496.048,1498.11 498.06,1498.05 500.072,1497.99 502.085,1497.93 504.097,1497.86 506.11,1497.8 508.122,1497.73 \n 510.134,1497.67 512.147,1497.6 514.159,1497.53 516.171,1497.45 518.184,1497.38 520.196,1497.31 522.208,1497.23 524.221,1497.15 526.233,1497.07 528.246,1496.99 \n 530.258,1496.91 532.27,1496.83 534.283,1496.74 536.295,1496.65 538.307,1496.56 540.32,1496.47 542.332,1496.38 544.344,1496.29 546.357,1496.19 548.369,1496.09 \n 550.381,1495.99 552.394,1495.89 554.406,1495.78 556.419,1495.68 558.431,1495.57 560.443,1495.46 562.456,1495.34 564.468,1495.23 566.48,1495.11 568.493,1494.99 \n 570.505,1494.87 572.517,1494.74 574.53,1494.61 576.542,1494.48 578.555,1494.35 580.567,1494.21 582.579,1494.08 584.592,1493.93 586.604,1493.79 588.616,1493.64 \n 590.629,1493.49 592.641,1493.34 594.653,1493.18 596.666,1493.03 598.678,1492.86 600.691,1492.7 602.703,1492.53 604.715,1492.36 606.728,1492.18 608.74,1492 \n 610.752,1491.82 612.765,1491.63 614.777,1491.44 616.789,1491.25 618.802,1491.05 620.814,1490.85 622.827,1490.64 624.839,1490.43 626.851,1490.22 628.864,1490 \n 630.876,1489.78 632.888,1489.55 634.901,1489.32 636.913,1489.08 638.925,1488.84 640.938,1488.59 642.95,1488.34 644.962,1488.09 646.975,1487.83 648.987,1487.56 \n 651,1487.29 653.012,1487.01 655.024,1486.73 657.037,1486.44 659.049,1486.15 661.061,1485.84 663.074,1485.54 665.086,1485.23 667.098,1484.91 669.111,1484.58 \n 671.123,1484.25 673.136,1483.91 675.148,1483.57 677.16,1483.22 679.173,1482.86 681.185,1482.49 683.197,1482.12 685.21,1481.74 687.222,1481.35 689.234,1480.96 \n 691.247,1480.55 693.259,1480.14 695.272,1479.72 697.284,1479.3 699.296,1478.86 701.309,1478.41 703.321,1477.96 705.333,1477.5 707.346,1477.02 709.358,1476.54 \n 711.37,1476.05 713.383,1475.55 715.395,1475.04 717.408,1474.52 719.42,1473.99 721.432,1473.44 723.445,1472.89 725.457,1472.33 727.469,1471.75 729.482,1471.17 \n 731.494,1470.57 733.506,1469.96 735.519,1469.34 737.531,1468.71 739.544,1468.06 741.556,1467.4 743.568,1466.73 745.581,1466.04 747.593,1465.34 749.605,1464.63 \n 751.618,1463.91 753.63,1463.17 755.642,1462.41 757.655,1461.64 759.667,1460.85 761.679,1460.05 763.692,1459.24 765.704,1458.41 767.717,1457.56 769.729,1456.69 \n 771.741,1455.81 773.754,1454.91 775.766,1453.99 777.778,1453.06 779.791,1452.11 781.803,1451.13 783.815,1450.14 785.828,1449.13 787.84,1448.1 789.853,1447.06 \n 791.865,1445.99 793.877,1444.9 795.89,1443.78 797.902,1442.65 799.914,1441.5 801.927,1440.32 803.939,1439.12 805.951,1437.9 807.964,1436.65 809.976,1435.38 \n 811.989,1434.08 814.001,1432.76 816.013,1431.42 818.026,1430.05 820.038,1428.65 822.05,1427.23 824.063,1425.78 826.075,1424.3 828.087,1422.79 830.1,1421.25 \n 832.112,1419.69 834.125,1418.09 836.137,1416.47 838.149,1414.81 840.162,1413.13 842.174,1411.41 844.186,1409.66 846.199,1407.87 848.211,1406.05 850.223,1404.2 \n 852.236,1402.32 854.248,1400.39 856.26,1398.44 858.273,1396.44 860.285,1394.41 862.298,1392.34 864.31,1390.23 866.322,1388.09 868.335,1385.9 870.347,1383.67 \n 872.359,1381.4 874.372,1379.09 876.384,1376.74 878.396,1374.35 880.409,1371.91 882.421,1369.42 884.434,1366.9 886.446,1364.32 888.458,1361.7 890.471,1359.03 \n 892.483,1356.31 894.495,1353.55 896.508,1350.73 898.52,1347.87 900.532,1344.95 902.545,1341.98 904.557,1338.96 906.57,1335.88 908.582,1332.75 910.594,1329.57 \n 912.607,1326.33 914.619,1323.03 916.631,1319.67 918.644,1316.26 920.656,1312.79 922.668,1309.25 924.681,1305.66 926.693,1302 928.706,1298.28 930.718,1294.5 \n 932.73,1290.65 934.743,1286.74 936.755,1282.77 938.767,1278.72 940.78,1274.61 942.792,1270.43 944.804,1266.18 946.817,1261.86 948.829,1257.47 950.841,1253 \n 952.854,1248.47 954.866,1243.86 956.879,1239.17 958.891,1234.41 960.903,1229.58 962.916,1224.66 964.928,1219.67 966.94,1214.6 968.953,1209.45 970.965,1204.23 \n 972.977,1198.92 974.99,1193.52 977.002,1188.05 979.015,1182.49 981.027,1176.85 983.039,1171.13 985.052,1165.32 987.064,1159.42 989.076,1153.44 991.089,1147.37 \n 993.101,1141.21 995.113,1134.96 997.126,1128.63 999.138,1122.2 1001.15,1115.69 1003.16,1109.08 1005.18,1102.39 1007.19,1095.6 1009.2,1088.72 1011.21,1081.75 \n 1013.22,1074.69 1015.24,1067.53 1017.25,1060.28 1019.26,1052.94 1021.27,1045.51 1023.29,1037.98 1025.3,1030.36 1027.31,1022.64 1029.32,1014.84 1031.34,1006.94 \n 1033.35,998.945 1035.36,990.86 1037.37,982.682 1039.39,974.412 1041.4,966.051 1043.41,957.598 1045.42,949.055 1047.43,940.422 1049.45,931.7 1051.46,922.888 \n 1053.47,913.99 1055.48,905.004 1057.5,895.932 1059.51,886.775 1061.52,877.535 1063.53,868.212 1065.55,858.808 1067.56,849.324 1069.57,839.762 1071.58,830.123 \n 1073.6,820.409 1075.61,810.621 1077.62,800.762 1079.63,790.833 1081.64,780.837 1083.66,770.775 1085.67,760.65 1087.68,750.464 1089.69,740.219 1091.71,729.919 \n 1093.72,719.566 1095.73,709.162 1097.74,698.711 1099.76,688.215 1101.77,677.677 1103.78,667.101 1105.79,656.49 1107.81,645.847 1109.82,635.177 1111.83,624.481 \n 1113.84,613.765 1115.86,603.032 1117.87,592.286 1119.88,581.531 1121.89,570.771 1123.9,560.01 1125.92,549.253 1127.93,538.504 1129.94,527.768 1131.95,517.049 \n 1133.97,506.352 1135.98,495.682 1137.99,485.043 1140,474.441 1142.02,463.881 1144.03,453.368 1146.04,442.907 1148.05,432.503 1150.07,422.162 1152.08,411.889 \n 1154.09,401.69 1156.1,391.569 1158.11,381.533 1160.13,371.587 1162.14,361.737 1164.15,351.989 1166.16,342.347 1168.18,332.818 1170.19,323.408 1172.2,314.122 \n 1174.21,304.966 1176.23,295.946 1178.24,287.066 1180.25,278.334 1182.26,269.755 1184.28,261.333 1186.29,253.076 1188.3,244.987 1190.31,237.074 1192.32,229.34 \n 1194.34,221.792 1196.35,214.435 1198.36,207.274 1200.37,200.313 1202.39,193.559 1204.4,187.015 1206.41,180.687 1208.42,174.579 1210.44,168.695 1212.45,163.041 \n 1214.46,157.62 1216.47,152.436 1218.49,147.493 1220.5,142.795 1222.51,138.346 1224.52,134.148 1226.54,130.206 1228.55,126.522 1230.56,123.099 1232.57,119.94 \n 1234.58,117.047 1236.6,114.423 1238.61,112.069 1240.62,109.988 1242.63,108.181 1244.65,106.65 1246.66,105.395 1248.67,104.418 1250.68,103.72 1252.7,103.301 \n 1254.71,103.161 1256.72,103.301 1258.73,103.72 1260.75,104.418 1262.76,105.395 1264.77,106.65 1266.78,108.181 1268.79,109.988 1270.81,112.069 1272.82,114.423 \n 1274.83,117.047 1276.84,119.94 1278.86,123.099 1280.87,126.522 1282.88,130.206 1284.89,134.148 1286.91,138.346 1288.92,142.795 1290.93,147.493 1292.94,152.436 \n 1294.96,157.62 1296.97,163.041 1298.98,168.695 1300.99,174.579 1303,180.687 1305.02,187.015 1307.03,193.559 1309.04,200.313 1311.05,207.274 1313.07,214.435 \n 1315.08,221.792 1317.09,229.34 1319.1,237.074 1321.12,244.987 1323.13,253.076 1325.14,261.333 1327.15,269.755 1329.17,278.334 1331.18,287.066 1333.19,295.946 \n 1335.2,304.966 1337.21,314.122 1339.23,323.408 1341.24,332.818 1343.25,342.347 1345.26,351.989 1347.28,361.737 1349.29,371.587 1351.3,381.533 1353.31,391.569 \n 1355.33,401.69 1357.34,411.889 1359.35,422.162 1361.36,432.503 1363.38,442.907 1365.39,453.368 1367.4,463.881 1369.41,474.441 1371.43,485.043 1373.44,495.682 \n 1375.45,506.352 1377.46,517.049 1379.47,527.768 1381.49,538.504 1383.5,549.253 1385.51,560.01 1387.52,570.771 1389.54,581.531 1391.55,592.286 1393.56,603.032 \n 1395.57,613.765 1397.59,624.481 1399.6,635.177 1401.61,645.847 1403.62,656.49 1405.64,667.101 1407.65,677.677 1409.66,688.215 1411.67,698.711 1413.68,709.162 \n 1415.7,719.566 1417.71,729.919 1419.72,740.219 1421.73,750.464 1423.75,760.65 1425.76,770.775 1427.77,780.837 1429.78,790.833 1431.8,800.762 1433.81,810.621 \n 1435.82,820.409 1437.83,830.123 1439.85,839.762 1441.86,849.324 1443.87,858.808 1445.88,868.212 1447.89,877.535 1449.91,886.775 1451.92,895.932 1453.93,905.004 \n 1455.94,913.99 1457.96,922.888 1459.97,931.7 1461.98,940.422 1463.99,949.055 1466.01,957.598 1468.02,966.051 1470.03,974.412 1472.04,982.682 1474.06,990.86 \n 1476.07,998.945 1478.08,1006.94 1480.09,1014.84 1482.1,1022.64 1484.12,1030.36 1486.13,1037.98 1488.14,1045.51 1490.15,1052.94 1492.17,1060.28 1494.18,1067.53 \n 1496.19,1074.69 1498.2,1081.75 1500.22,1088.72 1502.23,1095.6 1504.24,1102.39 1506.25,1109.08 1508.27,1115.69 1510.28,1122.2 1512.29,1128.63 1514.3,1134.96 \n 1516.32,1141.21 1518.33,1147.37 1520.34,1153.44 1522.35,1159.42 1524.36,1165.32 1526.38,1171.13 1528.39,1176.85 1530.4,1182.49 1532.41,1188.05 1534.43,1193.52 \n 1536.44,1198.92 1538.45,1204.23 1540.46,1209.45 1542.48,1214.6 1544.49,1219.67 1546.5,1224.66 1548.51,1229.58 1550.53,1234.41 1552.54,1239.17 1554.55,1243.86 \n 1556.56,1248.47 1558.57,1253 1560.59,1257.47 1562.6,1261.86 1564.61,1266.18 1566.62,1270.43 1568.64,1274.61 1570.65,1278.72 1572.66,1282.77 1574.67,1286.74 \n 1576.69,1290.65 1578.7,1294.5 1580.71,1298.28 1582.72,1302 1584.74,1305.66 1586.75,1309.25 1588.76,1312.79 1590.77,1316.26 1592.78,1319.67 1594.8,1323.03 \n 1596.81,1326.33 1598.82,1329.57 1600.83,1332.75 1602.85,1335.88 1604.86,1338.96 1606.87,1341.98 1608.88,1344.95 1610.9,1347.87 1612.91,1350.73 1614.92,1353.55 \n 1616.93,1356.31 1618.95,1359.03 1620.96,1361.7 1622.97,1364.32 1624.98,1366.9 1627,1369.42 1629.01,1371.91 1631.02,1374.35 1633.03,1376.74 1635.04,1379.09 \n 1637.06,1381.4 1639.07,1383.67 1641.08,1385.9 1643.09,1388.09 1645.11,1390.23 1647.12,1392.34 1649.13,1394.41 1651.14,1396.44 1653.16,1398.44 1655.17,1400.39 \n 1657.18,1402.32 1659.19,1404.2 1661.21,1406.05 1663.22,1407.87 1665.23,1409.66 1667.24,1411.41 1669.25,1413.13 1671.27,1414.81 1673.28,1416.47 1675.29,1418.09 \n 1677.3,1419.69 1679.32,1421.25 1681.33,1422.79 1683.34,1424.3 1685.35,1425.78 1687.37,1427.23 1689.38,1428.65 1691.39,1430.05 1693.4,1431.42 1695.42,1432.76 \n 1697.43,1434.08 1699.44,1435.38 1701.45,1436.65 1703.46,1437.9 1705.48,1439.12 1707.49,1440.32 1709.5,1441.5 1711.51,1442.65 1713.53,1443.78 1715.54,1444.9 \n 1717.55,1445.99 1719.56,1447.06 1721.58,1448.1 1723.59,1449.13 1725.6,1450.14 1727.61,1451.13 1729.63,1452.11 1731.64,1453.06 1733.65,1453.99 1735.66,1454.91 \n 1737.67,1455.81 1739.69,1456.69 1741.7,1457.56 1743.71,1458.41 1745.72,1459.24 1747.74,1460.05 1749.75,1460.85 1751.76,1461.64 1753.77,1462.41 1755.79,1463.17 \n 1757.8,1463.91 1759.81,1464.63 1761.82,1465.34 1763.84,1466.04 1765.85,1466.73 1767.86,1467.4 1769.87,1468.06 1771.89,1468.71 1773.9,1469.34 1775.91,1469.96 \n 1777.92,1470.57 1779.93,1471.17 1781.95,1471.75 1783.96,1472.33 1785.97,1472.89 1787.98,1473.44 1790,1473.99 1792.01,1474.52 1794.02,1475.04 1796.03,1475.55 \n 1798.05,1476.05 1800.06,1476.54 1802.07,1477.02 1804.08,1477.5 1806.1,1477.96 1808.11,1478.41 1810.12,1478.86 1812.13,1479.3 1814.14,1479.72 1816.16,1480.14 \n 1818.17,1480.55 1820.18,1480.96 1822.19,1481.35 1824.21,1481.74 1826.22,1482.12 1828.23,1482.49 1830.24,1482.86 1832.26,1483.22 1834.27,1483.57 1836.28,1483.91 \n 1838.29,1484.25 1840.31,1484.58 1842.32,1484.91 1844.33,1485.23 1846.34,1485.54 1848.35,1485.84 1850.37,1486.15 1852.38,1486.44 1854.39,1486.73 1856.4,1487.01 \n 1858.42,1487.29 1860.43,1487.56 1862.44,1487.83 1864.45,1488.09 1866.47,1488.34 1868.48,1488.59 1870.49,1488.84 1872.5,1489.08 1874.52,1489.32 1876.53,1489.55 \n 1878.54,1489.78 1880.55,1490 1882.56,1490.22 1884.58,1490.43 1886.59,1490.64 1888.6,1490.85 1890.61,1491.05 1892.63,1491.25 1894.64,1491.44 1896.65,1491.63 \n 1898.66,1491.82 1900.68,1492 1902.69,1492.18 1904.7,1492.36 1906.71,1492.53 1908.73,1492.7 1910.74,1492.86 1912.75,1493.03 1914.76,1493.18 1916.78,1493.34 \n 1918.79,1493.49 1920.8,1493.64 1922.81,1493.79 1924.82,1493.93 1926.84,1494.08 1928.85,1494.21 1930.86,1494.35 1932.87,1494.48 1934.89,1494.61 1936.9,1494.74 \n 1938.91,1494.87 1940.92,1494.99 1942.94,1495.11 1944.95,1495.23 1946.96,1495.34 1948.97,1495.46 1950.99,1495.57 1953,1495.68 1955.01,1495.78 1957.02,1495.89 \n 1959.03,1495.99 1961.05,1496.09 1963.06,1496.19 1965.07,1496.29 1967.08,1496.38 1969.1,1496.47 1971.11,1496.56 1973.12,1496.65 1975.13,1496.74 1977.15,1496.83 \n 1979.16,1496.91 1981.17,1496.99 1983.18,1497.07 1985.2,1497.15 1987.21,1497.23 1989.22,1497.31 1991.23,1497.38 1993.24,1497.45 1995.26,1497.53 1997.27,1497.6 \n 1999.28,1497.67 2001.29,1497.73 2003.31,1497.8 2005.32,1497.86 2007.33,1497.93 2009.34,1497.99 2011.36,1498.05 2013.37,1498.11 2015.38,1498.17 2017.39,1498.23 \n 2019.41,1498.28 2021.42,1498.34 2023.43,1498.39 2025.44,1498.45 2027.46,1498.5 2029.47,1498.55 2031.48,1498.6 2033.49,1498.65 2035.5,1498.7 2037.52,1498.74 \n 2039.53,1498.79 2041.54,1498.84 2043.55,1498.88 2045.57,1498.92 2047.58,1498.97 2049.59,1499.01 2051.6,1499.05 2053.62,1499.09 2055.63,1499.13 2057.64,1499.17 \n 2059.65,1499.2 2061.67,1499.24 2063.68,1499.28 2065.69,1499.31 2067.7,1499.35 2069.71,1499.38 2071.73,1499.42 2073.74,1499.45 2075.75,1499.48 2077.76,1499.51 \n 2079.78,1499.54 2081.79,1499.57 2083.8,1499.6 2085.81,1499.63 2087.83,1499.66 2089.84,1499.69 2091.85,1499.72 2093.86,1499.75 2095.88,1499.77 2097.89,1499.8 \n 2099.9,1499.82 2101.91,1499.85 2103.92,1499.87 2105.94,1499.9 2107.95,1499.92 2109.96,1499.94 2111.97,1499.96 2113.99,1499.99 2116,1500.01 2118.01,1500.03 \n 2120.02,1500.05 2122.04,1500.07 2124.05,1500.09 2126.06,1500.11 2128.07,1500.13 2130.09,1500.15 2132.1,1500.17 2134.11,1500.18 2136.12,1500.2 2138.13,1500.22 \n 2140.15,1500.24 2142.16,1500.25 2144.17,1500.27 2146.18,1500.29 2148.2,1500.3 2150.21,1500.32 2152.22,1500.33 2154.23,1500.35 2156.25,1500.36 2158.26,1500.38 \n 2160.27,1500.39 2162.28,1500.4 2164.3,1500.42 2166.31,1500.43 2168.32,1500.44 2170.33,1500.46 2172.35,1500.47 2174.36,1500.48 2176.37,1500.49 2178.38,1500.5 \n 2180.39,1500.51 2182.41,1500.53 2184.42,1500.54 2186.43,1500.55 2188.44,1500.56 2190.46,1500.57 2192.47,1500.58 2194.48,1500.59 2196.49,1500.6 2198.51,1500.61 \n 2200.52,1500.62 2202.53,1500.63 2204.54,1500.64 2206.56,1500.64 2208.57,1500.65 2210.58,1500.66 2212.59,1500.67 2214.6,1500.68 2216.62,1500.69 2218.63,1500.69 \n 2220.64,1500.7 2222.65,1500.71 2224.67,1500.72 2226.68,1500.72 2228.69,1500.73 2230.7,1500.74 2232.72,1500.74 2234.73,1500.75 2236.74,1500.76 2238.75,1500.76 \n 2240.77,1500.77 2242.78,1500.78 2244.79,1500.78 2246.8,1500.79 2248.81,1500.79 2250.83,1500.8 2252.84,1500.8 2254.85,1500.81 2256.86,1500.82 2258.88,1500.82 \n 2260.89,1500.83 \n \"/>\n<polyline clip-path=\"url(#clip2003)\" style=\"stroke:#e26f46; stroke-width:8; stroke-opacity:1; fill:none\" stroke-dasharray=\"32, 20\" points=\"\n 248.527,1501.08 250.54,1501.08 252.552,1501.08 254.564,1501.08 256.577,1501.08 258.589,1501.08 260.601,1501.08 262.614,1501.08 264.626,1501.08 266.638,1501.08 \n 268.651,1501.08 270.663,1501.08 272.676,1501.08 274.688,1501.08 276.7,1501.08 278.713,1501.08 280.725,1501.08 282.737,1501.08 284.75,1501.08 286.762,1501.08 \n 288.774,1501.08 290.787,1501.08 292.799,1501.08 294.812,1501.08 296.824,1501.08 298.836,1501.08 300.849,1501.08 302.861,1501.08 304.873,1501.08 306.886,1501.08 \n 308.898,1501.08 310.91,1501.08 312.923,1501.08 314.935,1501.08 316.948,1501.08 318.96,1501.08 320.972,1501.08 322.985,1501.08 324.997,1501.08 327.009,1501.08 \n 329.022,1501.08 331.034,1501.08 333.046,1501.08 335.059,1501.08 337.071,1501.08 339.084,1501.08 341.096,1501.08 343.108,1501.07 345.121,1501.07 347.133,1501.07 \n 349.145,1501.07 351.158,1501.07 353.17,1501.07 355.182,1501.07 357.195,1501.07 359.207,1501.07 361.219,1501.07 363.232,1501.07 365.244,1501.07 367.257,1501.07 \n 369.269,1501.07 371.281,1501.07 373.294,1501.07 375.306,1501.07 377.318,1501.07 379.331,1501.07 381.343,1501.07 383.355,1501.07 385.368,1501.06 387.38,1501.06 \n 389.393,1501.06 391.405,1501.06 393.417,1501.06 395.43,1501.06 397.442,1501.06 399.454,1501.06 401.467,1501.06 403.479,1501.06 405.491,1501.06 407.504,1501.05 \n 409.516,1501.05 411.529,1501.05 413.541,1501.05 415.553,1501.05 417.566,1501.05 419.578,1501.04 421.59,1501.04 423.603,1501.04 425.615,1501.04 427.627,1501.04 \n 429.64,1501.03 431.652,1501.03 433.665,1501.03 435.677,1501.03 437.689,1501.02 439.702,1501.02 441.714,1501.02 443.726,1501.02 445.739,1501.01 447.751,1501.01 \n 449.763,1501.01 451.776,1501 453.788,1501 455.8,1500.99 457.813,1500.99 459.825,1500.99 461.838,1500.98 463.85,1500.98 465.862,1500.97 467.875,1500.97 \n 469.887,1500.96 471.899,1500.96 473.912,1500.95 475.924,1500.94 477.936,1500.94 479.949,1500.93 481.961,1500.92 483.974,1500.91 485.986,1500.91 487.998,1500.9 \n 490.011,1500.89 492.023,1500.88 494.035,1500.87 496.048,1500.86 498.06,1500.85 500.072,1500.84 502.085,1500.83 504.097,1500.82 506.11,1500.81 508.122,1500.79 \n 510.134,1500.78 512.147,1500.77 514.159,1500.75 516.171,1500.74 518.184,1500.72 520.196,1500.71 522.208,1500.69 524.221,1500.67 526.233,1500.65 528.246,1500.63 \n 530.258,1500.61 532.27,1500.59 534.283,1500.57 536.295,1500.55 538.307,1500.53 540.32,1500.5 542.332,1500.48 544.344,1500.45 546.357,1500.42 548.369,1500.39 \n 550.381,1500.36 552.394,1500.33 554.406,1500.3 556.419,1500.27 558.431,1500.23 560.443,1500.19 562.456,1500.16 564.468,1500.12 566.48,1500.08 568.493,1500.03 \n 570.505,1499.99 572.517,1499.94 574.53,1499.89 576.542,1499.85 578.555,1499.79 580.567,1499.74 582.579,1499.68 584.592,1499.63 586.604,1499.57 588.616,1499.5 \n 590.629,1499.44 592.641,1499.37 594.653,1499.3 596.666,1499.23 598.678,1499.16 600.691,1499.08 602.703,1499 604.715,1498.91 606.728,1498.83 608.74,1498.74 \n 610.752,1498.65 612.765,1498.55 614.777,1498.45 616.789,1498.35 618.802,1498.24 620.814,1498.13 622.827,1498.01 624.839,1497.89 626.851,1497.77 628.864,1497.64 \n 630.876,1497.51 632.888,1497.37 634.901,1497.23 636.913,1497.08 638.925,1496.93 640.938,1496.78 642.95,1496.61 644.962,1496.45 646.975,1496.27 648.987,1496.09 \n 651,1495.91 653.012,1495.72 655.024,1495.52 657.037,1495.31 659.049,1495.1 661.061,1494.88 663.074,1494.66 665.086,1494.42 667.098,1494.18 669.111,1493.93 \n 671.123,1493.68 673.136,1493.41 675.148,1493.14 677.16,1492.85 679.173,1492.56 681.185,1492.26 683.197,1491.95 685.21,1491.63 687.222,1491.3 689.234,1490.96 \n 691.247,1490.61 693.259,1490.25 695.272,1489.87 697.284,1489.49 699.296,1489.09 701.309,1488.69 703.321,1488.26 705.333,1487.83 707.346,1487.39 709.358,1486.93 \n 711.37,1486.45 713.383,1485.97 715.395,1485.46 717.408,1484.95 719.42,1484.42 721.432,1483.87 723.445,1483.31 725.457,1482.73 727.469,1482.13 729.482,1481.52 \n 731.494,1480.89 733.506,1480.24 735.519,1479.58 737.531,1478.9 739.544,1478.19 741.556,1477.47 743.568,1476.73 745.581,1475.97 747.593,1475.18 749.605,1474.38 \n 751.618,1473.55 753.63,1472.71 755.642,1471.84 757.655,1470.94 759.667,1470.03 761.679,1469.08 763.692,1468.12 765.704,1467.13 767.717,1466.11 769.729,1465.07 \n 771.741,1464 773.754,1462.91 775.766,1461.78 777.778,1460.63 779.791,1459.45 781.803,1458.24 783.815,1457 785.828,1455.73 787.84,1454.43 789.853,1453.1 \n 791.865,1451.74 793.877,1450.34 795.89,1448.91 797.902,1447.45 799.914,1445.95 801.927,1444.42 803.939,1442.85 805.951,1441.25 807.964,1439.61 809.976,1437.93 \n 811.989,1436.21 814.001,1434.46 816.013,1432.67 818.026,1430.83 820.038,1428.96 822.05,1427.05 824.063,1425.09 826.075,1423.09 828.087,1421.05 830.1,1418.96 \n 832.112,1416.84 834.125,1414.66 836.137,1412.44 838.149,1410.18 840.162,1407.87 842.174,1405.51 844.186,1403.1 846.199,1400.65 848.211,1398.15 850.223,1395.59 \n 852.236,1392.99 854.248,1390.34 856.26,1387.63 858.273,1384.87 860.285,1382.06 862.298,1379.2 864.31,1376.28 866.322,1373.31 868.335,1370.29 870.347,1367.21 \n 872.359,1364.07 874.372,1360.88 876.384,1357.63 878.396,1354.32 880.409,1350.95 882.421,1347.53 884.434,1344.04 886.446,1340.5 888.458,1336.9 890.471,1333.23 \n 892.483,1329.51 894.495,1325.72 896.508,1321.88 898.52,1317.97 900.532,1314 902.545,1309.96 904.557,1305.86 906.57,1301.7 908.582,1297.47 910.594,1293.18 \n 912.607,1288.83 914.619,1284.41 916.631,1279.93 918.644,1275.38 920.656,1270.76 922.668,1266.08 924.681,1261.33 926.693,1256.52 928.706,1251.64 930.718,1246.69 \n 932.73,1241.68 934.743,1236.6 936.755,1231.45 938.767,1226.24 940.78,1220.96 942.792,1215.61 944.804,1210.2 946.817,1204.72 948.829,1199.17 950.841,1193.56 \n 952.854,1187.88 954.866,1182.14 956.879,1176.33 958.891,1170.45 960.903,1164.51 962.916,1158.5 964.928,1152.43 966.94,1146.29 968.953,1140.09 970.965,1133.83 \n 972.977,1127.5 974.99,1121.11 977.002,1114.66 979.015,1108.14 981.027,1101.57 983.039,1094.93 985.052,1088.23 987.064,1081.48 989.076,1074.66 991.089,1067.79 \n 993.101,1060.86 995.113,1053.87 997.126,1046.83 999.138,1039.73 1001.15,1032.58 1003.16,1025.38 1005.18,1018.12 1007.19,1010.82 1009.2,1003.46 1011.21,996.056 \n 1013.22,988.602 1015.24,981.102 1017.25,973.556 1019.26,965.965 1021.27,958.332 1023.29,950.656 1025.3,942.94 1027.31,935.184 1029.32,927.39 1031.34,919.56 \n 1033.35,911.694 1035.36,903.795 1037.37,895.863 1039.39,887.901 1041.4,879.909 1043.41,871.89 1045.42,863.845 1047.43,855.775 1049.45,847.683 1051.46,839.569 \n 1053.47,831.437 1055.48,823.286 1057.5,815.12 1059.51,806.94 1061.52,798.748 1063.53,790.545 1065.55,782.334 1067.56,774.116 1069.57,765.894 1071.58,757.67 \n 1073.6,749.444 1075.61,741.221 1077.62,733 1079.63,724.785 1081.64,716.578 1083.66,708.38 1085.67,700.194 1087.68,692.021 1089.69,683.865 1091.71,675.727 \n 1093.72,667.609 1095.73,659.513 1097.74,651.443 1099.76,643.399 1101.77,635.384 1103.78,627.401 1105.79,619.451 1107.81,611.537 1109.82,603.661 1111.83,595.825 \n 1113.84,588.032 1115.86,580.284 1117.87,572.583 1119.88,564.932 1121.89,557.332 1123.9,549.786 1125.92,542.296 1127.93,534.865 1129.94,527.495 1131.95,520.188 \n 1133.97,512.946 1135.98,505.771 1137.99,498.667 1140,491.634 1142.02,484.676 1144.03,477.794 1146.04,470.991 1148.05,464.268 1150.07,457.629 1152.08,451.075 \n 1154.09,444.608 1156.1,438.23 1158.11,431.944 1160.13,425.752 1162.14,419.655 1164.15,413.656 1166.16,407.756 1168.18,401.958 1170.19,396.264 1172.2,390.675 \n 1174.21,385.194 1176.23,379.822 1178.24,374.561 1180.25,369.413 1182.26,364.38 1184.28,359.463 1186.29,354.664 1188.3,349.985 1190.31,345.427 1192.32,340.993 \n 1194.34,336.683 1196.35,332.499 1198.36,328.442 1200.37,324.515 1202.39,320.718 1204.4,317.053 1206.41,313.52 1208.42,310.122 1210.44,306.86 1212.45,303.734 \n 1214.46,300.746 1216.47,297.897 1218.49,295.187 1220.5,292.619 1222.51,290.192 1224.52,287.908 1226.54,285.767 1228.55,283.771 1230.56,281.919 1232.57,280.213 \n 1234.58,278.654 1236.6,277.241 1238.61,275.975 1240.62,274.858 1242.63,273.888 1244.65,273.067 1246.66,272.395 1248.67,271.872 1250.68,271.499 1252.7,271.274 \n 1254.71,271.2 1256.72,271.274 1258.73,271.499 1260.75,271.872 1262.76,272.395 1264.77,273.067 1266.78,273.888 1268.79,274.858 1270.81,275.975 1272.82,277.241 \n 1274.83,278.654 1276.84,280.213 1278.86,281.919 1280.87,283.771 1282.88,285.767 1284.89,287.908 1286.91,290.192 1288.92,292.619 1290.93,295.187 1292.94,297.897 \n 1294.96,300.746 1296.97,303.734 1298.98,306.86 1300.99,310.122 1303,313.52 1305.02,317.053 1307.03,320.718 1309.04,324.515 1311.05,328.442 1313.07,332.499 \n 1315.08,336.683 1317.09,340.993 1319.1,345.427 1321.12,349.985 1323.13,354.664 1325.14,359.463 1327.15,364.38 1329.17,369.413 1331.18,374.561 1333.19,379.822 \n 1335.2,385.194 1337.21,390.675 1339.23,396.264 1341.24,401.958 1343.25,407.756 1345.26,413.656 1347.28,419.655 1349.29,425.752 1351.3,431.944 1353.31,438.23 \n 1355.33,444.608 1357.34,451.075 1359.35,457.629 1361.36,464.268 1363.38,470.991 1365.39,477.794 1367.4,484.676 1369.41,491.634 1371.43,498.667 1373.44,505.771 \n 1375.45,512.946 1377.46,520.188 1379.47,527.495 1381.49,534.865 1383.5,542.296 1385.51,549.786 1387.52,557.332 1389.54,564.932 1391.55,572.583 1393.56,580.284 \n 1395.57,588.032 1397.59,595.825 1399.6,603.661 1401.61,611.537 1403.62,619.451 1405.64,627.401 1407.65,635.384 1409.66,643.399 1411.67,651.443 1413.68,659.513 \n 1415.7,667.609 1417.71,675.727 1419.72,683.865 1421.73,692.021 1423.75,700.194 1425.76,708.38 1427.77,716.578 1429.78,724.785 1431.8,733 1433.81,741.221 \n 1435.82,749.444 1437.83,757.67 1439.85,765.894 1441.86,774.116 1443.87,782.334 1445.88,790.545 1447.89,798.748 1449.91,806.94 1451.92,815.12 1453.93,823.286 \n 1455.94,831.437 1457.96,839.569 1459.97,847.683 1461.98,855.775 1463.99,863.845 1466.01,871.89 1468.02,879.909 1470.03,887.901 1472.04,895.863 1474.06,903.795 \n 1476.07,911.694 1478.08,919.56 1480.09,927.39 1482.1,935.184 1484.12,942.94 1486.13,950.656 1488.14,958.332 1490.15,965.965 1492.17,973.556 1494.18,981.102 \n 1496.19,988.602 1498.2,996.056 1500.22,1003.46 1502.23,1010.82 1504.24,1018.12 1506.25,1025.38 1508.27,1032.58 1510.28,1039.73 1512.29,1046.83 1514.3,1053.87 \n 1516.32,1060.86 1518.33,1067.79 1520.34,1074.66 1522.35,1081.48 1524.36,1088.23 1526.38,1094.93 1528.39,1101.57 1530.4,1108.14 1532.41,1114.66 1534.43,1121.11 \n 1536.44,1127.5 1538.45,1133.83 1540.46,1140.09 1542.48,1146.29 1544.49,1152.43 1546.5,1158.5 1548.51,1164.51 1550.53,1170.45 1552.54,1176.33 1554.55,1182.14 \n 1556.56,1187.88 1558.57,1193.56 1560.59,1199.17 1562.6,1204.72 1564.61,1210.2 1566.62,1215.61 1568.64,1220.96 1570.65,1226.24 1572.66,1231.45 1574.67,1236.6 \n 1576.69,1241.68 1578.7,1246.69 1580.71,1251.64 1582.72,1256.52 1584.74,1261.33 1586.75,1266.08 1588.76,1270.76 1590.77,1275.38 1592.78,1279.93 1594.8,1284.41 \n 1596.81,1288.83 1598.82,1293.18 1600.83,1297.47 1602.85,1301.7 1604.86,1305.86 1606.87,1309.96 1608.88,1314 1610.9,1317.97 1612.91,1321.88 1614.92,1325.72 \n 1616.93,1329.51 1618.95,1333.23 1620.96,1336.9 1622.97,1340.5 1624.98,1344.04 1627,1347.53 1629.01,1350.95 1631.02,1354.32 1633.03,1357.63 1635.04,1360.88 \n 1637.06,1364.07 1639.07,1367.21 1641.08,1370.29 1643.09,1373.31 1645.11,1376.28 1647.12,1379.2 1649.13,1382.06 1651.14,1384.87 1653.16,1387.63 1655.17,1390.34 \n 1657.18,1392.99 1659.19,1395.59 1661.21,1398.15 1663.22,1400.65 1665.23,1403.1 1667.24,1405.51 1669.25,1407.87 1671.27,1410.18 1673.28,1412.44 1675.29,1414.66 \n 1677.3,1416.84 1679.32,1418.96 1681.33,1421.05 1683.34,1423.09 1685.35,1425.09 1687.37,1427.05 1689.38,1428.96 1691.39,1430.83 1693.4,1432.67 1695.42,1434.46 \n 1697.43,1436.21 1699.44,1437.93 1701.45,1439.61 1703.46,1441.25 1705.48,1442.85 1707.49,1444.42 1709.5,1445.95 1711.51,1447.45 1713.53,1448.91 1715.54,1450.34 \n 1717.55,1451.74 1719.56,1453.1 1721.58,1454.43 1723.59,1455.73 1725.6,1457 1727.61,1458.24 1729.63,1459.45 1731.64,1460.63 1733.65,1461.78 1735.66,1462.91 \n 1737.67,1464 1739.69,1465.07 1741.7,1466.11 1743.71,1467.13 1745.72,1468.12 1747.74,1469.08 1749.75,1470.03 1751.76,1470.94 1753.77,1471.84 1755.79,1472.71 \n 1757.8,1473.55 1759.81,1474.38 1761.82,1475.18 1763.84,1475.97 1765.85,1476.73 1767.86,1477.47 1769.87,1478.19 1771.89,1478.9 1773.9,1479.58 1775.91,1480.24 \n 1777.92,1480.89 1779.93,1481.52 1781.95,1482.13 1783.96,1482.73 1785.97,1483.31 1787.98,1483.87 1790,1484.42 1792.01,1484.95 1794.02,1485.46 1796.03,1485.97 \n 1798.05,1486.45 1800.06,1486.93 1802.07,1487.39 1804.08,1487.83 1806.1,1488.26 1808.11,1488.69 1810.12,1489.09 1812.13,1489.49 1814.14,1489.87 1816.16,1490.25 \n 1818.17,1490.61 1820.18,1490.96 1822.19,1491.3 1824.21,1491.63 1826.22,1491.95 1828.23,1492.26 1830.24,1492.56 1832.26,1492.85 1834.27,1493.14 1836.28,1493.41 \n 1838.29,1493.68 1840.31,1493.93 1842.32,1494.18 1844.33,1494.42 1846.34,1494.66 1848.35,1494.88 1850.37,1495.1 1852.38,1495.31 1854.39,1495.52 1856.4,1495.72 \n 1858.42,1495.91 1860.43,1496.09 1862.44,1496.27 1864.45,1496.45 1866.47,1496.61 1868.48,1496.78 1870.49,1496.93 1872.5,1497.08 1874.52,1497.23 1876.53,1497.37 \n 1878.54,1497.51 1880.55,1497.64 1882.56,1497.77 1884.58,1497.89 1886.59,1498.01 1888.6,1498.13 1890.61,1498.24 1892.63,1498.35 1894.64,1498.45 1896.65,1498.55 \n 1898.66,1498.65 1900.68,1498.74 1902.69,1498.83 1904.7,1498.91 1906.71,1499 1908.73,1499.08 1910.74,1499.16 1912.75,1499.23 1914.76,1499.3 1916.78,1499.37 \n 1918.79,1499.44 1920.8,1499.5 1922.81,1499.57 1924.82,1499.63 1926.84,1499.68 1928.85,1499.74 1930.86,1499.79 1932.87,1499.85 1934.89,1499.89 1936.9,1499.94 \n 1938.91,1499.99 1940.92,1500.03 1942.94,1500.08 1944.95,1500.12 1946.96,1500.16 1948.97,1500.19 1950.99,1500.23 1953,1500.27 1955.01,1500.3 1957.02,1500.33 \n 1959.03,1500.36 1961.05,1500.39 1963.06,1500.42 1965.07,1500.45 1967.08,1500.48 1969.1,1500.5 1971.11,1500.53 1973.12,1500.55 1975.13,1500.57 1977.15,1500.59 \n 1979.16,1500.61 1981.17,1500.63 1983.18,1500.65 1985.2,1500.67 1987.21,1500.69 1989.22,1500.71 1991.23,1500.72 1993.24,1500.74 1995.26,1500.75 1997.27,1500.77 \n 1999.28,1500.78 2001.29,1500.79 2003.31,1500.81 2005.32,1500.82 2007.33,1500.83 2009.34,1500.84 2011.36,1500.85 2013.37,1500.86 2015.38,1500.87 2017.39,1500.88 \n 2019.41,1500.89 2021.42,1500.9 2023.43,1500.91 2025.44,1500.91 2027.46,1500.92 2029.47,1500.93 2031.48,1500.94 2033.49,1500.94 2035.5,1500.95 2037.52,1500.96 \n 2039.53,1500.96 2041.54,1500.97 2043.55,1500.97 2045.57,1500.98 2047.58,1500.98 2049.59,1500.99 2051.6,1500.99 2053.62,1500.99 2055.63,1501 2057.64,1501 \n 2059.65,1501.01 2061.67,1501.01 2063.68,1501.01 2065.69,1501.02 2067.7,1501.02 2069.71,1501.02 2071.73,1501.02 2073.74,1501.03 2075.75,1501.03 2077.76,1501.03 \n 2079.78,1501.03 2081.79,1501.04 2083.8,1501.04 2085.81,1501.04 2087.83,1501.04 2089.84,1501.04 2091.85,1501.05 2093.86,1501.05 2095.88,1501.05 2097.89,1501.05 \n 2099.9,1501.05 2101.91,1501.05 2103.92,1501.06 2105.94,1501.06 2107.95,1501.06 2109.96,1501.06 2111.97,1501.06 2113.99,1501.06 2116,1501.06 2118.01,1501.06 \n 2120.02,1501.06 2122.04,1501.06 2124.05,1501.06 2126.06,1501.07 2128.07,1501.07 2130.09,1501.07 2132.1,1501.07 2134.11,1501.07 2136.12,1501.07 2138.13,1501.07 \n 2140.15,1501.07 2142.16,1501.07 2144.17,1501.07 2146.18,1501.07 2148.2,1501.07 2150.21,1501.07 2152.22,1501.07 2154.23,1501.07 2156.25,1501.07 2158.26,1501.07 \n 2160.27,1501.07 2162.28,1501.07 2164.3,1501.07 2166.31,1501.07 2168.32,1501.08 2170.33,1501.08 2172.35,1501.08 2174.36,1501.08 2176.37,1501.08 2178.38,1501.08 \n 2180.39,1501.08 2182.41,1501.08 2184.42,1501.08 2186.43,1501.08 2188.44,1501.08 2190.46,1501.08 2192.47,1501.08 2194.48,1501.08 2196.49,1501.08 2198.51,1501.08 \n 2200.52,1501.08 2202.53,1501.08 2204.54,1501.08 2206.56,1501.08 2208.57,1501.08 2210.58,1501.08 2212.59,1501.08 2214.6,1501.08 2216.62,1501.08 2218.63,1501.08 \n 2220.64,1501.08 2222.65,1501.08 2224.67,1501.08 2226.68,1501.08 2228.69,1501.08 2230.7,1501.08 2232.72,1501.08 2234.73,1501.08 2236.74,1501.08 2238.75,1501.08 \n 2240.77,1501.08 2242.78,1501.08 2244.79,1501.08 2246.8,1501.08 2248.81,1501.08 2250.83,1501.08 2252.84,1501.08 2254.85,1501.08 2256.86,1501.08 2258.88,1501.08 \n 2260.89,1501.08 \n \"/>\n<polygon clip-path=\"url(#clip2001)\" points=\"\n260.156,312.204 647.253,312.204 647.253,130.764 260.156,130.764 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 260.156,312.204 647.253,312.204 647.253,130.764 260.156,130.764 260.156,312.204 \n \"/>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#009af9; stroke-width:8; stroke-opacity:1; fill:none\" points=\"\n 284.156,191.244 428.156,191.244 \n \"/>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 452.156, 208.744)\" x=\"452.156\" y=\"208.744\">logistic</text>\n</g>\n<polyline clip-path=\"url(#clip2001)\" style=\"stroke:#e26f46; stroke-width:8; stroke-opacity:1; fill:none\" stroke-dasharray=\"32, 20\" points=\"\n 284.156,251.724 428.156,251.724 \n \"/>\n<g clip-path=\"url(#clip2001)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 452.156, 269.224)\" x=\"452.156\" y=\"269.224\">normal</text>\n</g>\n</svg>\n"
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "ロジスティック分布の正規分布による近似に関しては\n\n* 黒木玄, <a href=\"http://nbviewer.jupyter.org/gist/genkuroki/96b0508a5773035e3a5247beff1d4f99\">ロジスティック分布の2通りの正規分布近似の比較</a>\n\nも参照せよ."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 多項分布の中心極限定理\n\nSanovの定理の証明によって, $k_i/n\\sim q_i$ ($n\\to\\infty$) のとき, $n\\to\\infty$ で,\n$$\n\\frac{n!}{k_1!k_2!\\cdots k_r!} p_1^{k_1}p_2^{k_2}\\cdots p_r^{k_r}\n\\sim\n\\left(\n\\prod_{i=1}^r \\left(\\frac{k_i/n}{p_i}\\right)^{k_i}\n\\sqrt{(2\\pi n)^{r-1}\\prod_{i=1}^r\\frac{k_i}{n}}\n\\right)^{-1}.\n$$\n\nこれを\n\n$$\n\\frac{k_i}{n} = p_i + frac{x_i}{\\sqrt{n}} + \\eps_i, \\quad\n\\eps_i=o\\left(\\frac{1}{\\sqrt{n}}\\right)\n$$\n\nの場合に適用すると,\n\n$$\n\\begin{aligned}\n&\n\\sum_{i=1}^r\\left(\\frac{x_i}{\\sqrt{n}}+\\eps_i\\right) = \n\\sum_{i=1}^r \\frac{k_i}{n} - \\sum_{i=1}^r p_i = 0,\n\\\\ &\n\\log\\left(\\frac{k_i/n}{p_i}\\right)^{k_r} =\nnp_i\\left(1+\\frac{x_i}{\\sqrt{n}\\;p_i}+\\frac{\\eps_i}{p_i}\\right)\n\\log\\left(1+\\frac{x_i}{\\sqrt{n}\\;p_i}+\\frac{\\eps_i}{p_i}\\right) =\nn\\left(\\frac{x_i}{\\sqrt{n}} + \\eps_i\\right) + \\frac{1}{2}\\frac{x_i^2}{p_i} + o(1),\n\\\\ & \\therefore\\quad\n\\log\\prod_{i=1}^r \\left(\\frac{k_i/n}{p_i}\\right)^{k_i} =\n\\sum_{i=1}^r \\log\\left(\\frac{k_i/n}{p_i}\\right)^{k_i} = \\frac{1}{2}\\sum_{i=1}^r \\frac{x_i^2}{p_i} + o(1),\n\\\\ &\n\\prod_{i=1}^r \\frac{k_i}{n} = p_1\\cdots p_r(1+o(1))\n\\end{aligned}\n$$\n\nなので\n\n$$\n\\frac{n!}{k_1!k_2!\\cdots k_r!} p_1^{k_1}p_2^{k_2}\\cdots p_r^{k_r}\n\\sim\n\\frac{\\ds\\exp\\left(-\\frac{1}{2}\\sum_{i=1}^r\\frac{x_i^2}{p_i}\\right)}\n{\\sqrt{(2\\pi n)^{r-1}p_1\\cdots p_r}}.\n$$\n\n両辺に $dk_1\\cdots k_{r-1} = \\sqrt{n}\\;dx_1\\cdots\\sqrt{n}\\;dx_{r-1}$ をかけると,\n\n$$\n\\frac{n!}{k_1!k_2!\\cdots k_r!} p_1^{k_1}p_2^{k_2}\\cdots p_r^{k_r}\\,dk_1\\cdots dk_{r-1}\n\\sim\n\\frac{\\ds\\exp\\left(-\\frac{1}{2}\\sum_{i=1}^r\\frac{x_i^2}{p_i}\\right)}\n{\\sqrt{(2\\pi)^{r-1}p_1\\cdots p_r}}\n\\,dx_1\\cdots dx_{r-1}.\n$$\n\n$k_1+\\cdots+k_r=n$ より $x_1+\\cdots+x_r=o(1)$ であることに注意せよ. 右辺は $(x_1,\\ldots,x_{r-1})$ の多変量正規分布の確率密度函数の形になっている.\n\n以上の結果は多項分布が $n$ が大きなときに多変量正規分布で近似できることを意味している. この結果を**多項分布の中心極限定理**と呼ぶ."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "**二項分布の中心極限定理:** $r=2$ のとき, $0\\leqq p\\leqq 1$, $k/n = p + x/\\sqrt{n} + o(1/\\sqrt{n})$ のとき,\n\n$$\n\\binom{n}{k}p^k(1-p)^{n-k}\\,dk \\sim\n\\frac{\\ds\\exp\\left(-\\frac{1}{2}\\left(\\frac{x^2}{p}+\\frac{(-x)^2}{1-p}\\right)\\right)}\n{\\sqrt{2\\pi p(1-p)}}\\,dx =\n\\frac{\\ds\\exp\\left(-\\frac{x^2}{2p(1-p)}\\right)}\n{\\sqrt{2\\pi p(1-p)}}\\,dx.\n$$\n\nこの結果を**二項分布の中心極限定理**と呼ぶ. $\\QED$"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "より一般の場合の中心極限定理の証明については\n\n* 黒木玄, <a href=\"https://genkuroki.github.io/documents/IntroProbability.pdf\">確率論入門</a>\n\nの第2.4節を参照せよ. 本質的にTaylorの定理のみを使って中心極限定理は証明される. (積率母函数や特性函数の理論(本質的にFourier-Laplace解析の理論)を使わなくても証明できる.)"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "**問題:** 二項分布の中心極限定理を実際にグラフをプロットすることによって確認せよ. $\\QED$\n\n次のセルを見よ."
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "binomcoeff(n,k) = exp(lgamma(n+1)-lgamma(k+1)-lgamma(n-k+1))\npdfbinom(n,p,k) = 0 ≤ k ≤ n ? binomcoeff(n,k)*p^k*(1-p)^(n-k) : 0.0\npdfnormal(n,p,x) = exp(-x^2/(2p*(1-p)))/√(2π*n*p*(1-p))\n\nPP = []\nfor (n,p,l) in [(6, 0.4, :bottom), (12, 0.4, :topright), (25, 0.4, :topright), (50, 0.4, :topright)]\n k = 0:n\n kk = 0:n/400:n\n x = (kk .- n*p)./√n\n P = plot(legend=l, legendfontsize=7, xlabel=\"k\")\n scatter!(k, pdfbinom.(n,p,k), label=\"Bin($n, $p)\", markersize=2.5)\n plot!(kk, pdfnormal.(n,p,x), label=\"Normal dist\", lw=1.5, ls=:dash)\n push!(PP, P)\nend\nplot(PP..., size=(800, 600))",
"execution_count": 5,
"outputs": [
{
"data": {
"image/svg+xml": "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"800\" height=\"600\" viewBox=\"0 0 3200 2400\">\n<defs>\n <clipPath id=\"clip2200\">\n <rect x=\"0\" y=\"0\" width=\"2000\" height=\"2000\"/>\n </clipPath>\n</defs>\n<defs>\n <clipPath id=\"clip2201\">\n <rect x=\"0\" y=\"0\" width=\"3200\" height=\"2400\"/>\n </clipPath>\n</defs>\n<polygon clip-path=\"url(#clip2201)\" points=\"\n0,2400 3200,2400 3200,0 0,0 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip2202\">\n <rect x=\"640\" y=\"0\" width=\"2241\" height=\"2241\"/>\n </clipPath>\n</defs>\n<polygon clip-path=\"url(#clip2201)\" points=\"\n188.156,1040.48 1507.88,1040.48 1507.88,47.2441 188.156,47.2441 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip2203\">\n <rect x=\"188\" y=\"47\" width=\"1321\" height=\"994\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 225.507,1040.48 225.507,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 433.011,1040.48 433.011,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 640.514,1040.48 640.514,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 848.018,1040.48 848.018,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1055.52,1040.48 1055.52,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1263.02,1040.48 1263.02,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1470.53,1040.48 1470.53,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,1022.9 1507.88,1022.9 \n \"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,737.882 1507.88,737.882 \n \"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,452.865 1507.88,452.865 \n \"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,167.848 1507.88,167.848 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,1040.48 1507.88,1040.48 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,1040.48 188.156,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 225.507,1040.48 225.507,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 433.011,1040.48 433.011,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 640.514,1040.48 640.514,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 848.018,1040.48 848.018,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1055.52,1040.48 1055.52,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1263.02,1040.48 1263.02,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1470.53,1040.48 1470.53,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,1022.9 207.952,1022.9 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,737.882 207.952,737.882 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,452.865 207.952,452.865 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,167.848 207.952,167.848 \n \"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 225.507, 1098.48)\" x=\"225.507\" y=\"1098.48\">0</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 433.011, 1098.48)\" x=\"433.011\" y=\"1098.48\">1</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 640.514, 1098.48)\" x=\"640.514\" y=\"1098.48\">2</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 848.018, 1098.48)\" x=\"848.018\" y=\"1098.48\">3</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1055.52, 1098.48)\" x=\"1055.52\" y=\"1098.48\">4</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1263.02, 1098.48)\" x=\"1263.02\" y=\"1098.48\">5</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1470.53, 1098.48)\" x=\"1470.53\" y=\"1098.48\">6</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.156, 1040.4)\" x=\"156.156\" y=\"1040.4\">0.0</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.156, 755.382)\" x=\"156.156\" y=\"755.382\">0.1</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.156, 470.365)\" x=\"156.156\" y=\"470.365\">0.2</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.156, 185.348)\" x=\"156.156\" y=\"185.348\">0.3</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 848.018, 1190.4)\" x=\"848.018\" y=\"1190.4\">k</text>\n</g>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"225.507\" cy=\"889.921\" r=\"12\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"225.507\" cy=\"889.921\" r=\"9\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"433.011\" cy=\"490.989\" r=\"12\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"433.011\" cy=\"490.989\" r=\"9\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"640.514\" cy=\"136.382\" r=\"12\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"640.514\" cy=\"136.382\" r=\"9\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"848.018\" cy=\"234.884\" r=\"12\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"848.018\" cy=\"234.884\" r=\"9\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1055.52\" cy=\"628.891\" r=\"12\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1055.52\" cy=\"628.891\" r=\"9\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1263.02\" cy=\"917.83\" r=\"12\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1263.02\" cy=\"917.83\" r=\"9\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1470.53\" cy=\"1011.22\" r=\"12\"/>\n<circle clip-path=\"url(#clip2203)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1470.53\" cy=\"1011.22\" r=\"9\"/>\n<polyline clip-path=\"url(#clip2203)\" style=\"stroke:#e26f46; stroke-width:6; stroke-opacity:1; fill:none\" stroke-dasharray=\"24, 15\" points=\"\n 225.507,894.662 228.62,891.426 231.732,888.13 234.845,884.772 237.957,881.353 241.07,877.871 244.182,874.328 247.295,870.721 250.407,867.052 253.52,863.318 \n 256.633,859.521 259.745,855.66 262.858,851.734 265.97,847.744 269.083,843.688 272.195,839.567 275.308,835.381 278.42,831.129 281.533,826.811 284.646,822.427 \n 287.758,817.978 290.871,813.462 293.983,808.88 297.096,804.232 300.208,799.519 303.321,794.739 306.433,789.893 309.546,784.982 312.658,780.005 315.771,774.963 \n 318.884,769.855 321.996,764.683 325.109,759.446 328.221,754.145 331.334,748.78 334.446,743.351 337.559,737.86 340.671,732.306 343.784,726.691 346.897,721.014 \n 350.009,715.276 353.122,709.478 356.234,703.621 359.347,697.705 362.459,691.731 365.572,685.7 368.684,679.613 371.797,673.471 374.91,667.275 378.022,661.025 \n 381.135,654.723 384.247,648.369 387.36,641.966 390.472,635.513 393.585,629.013 396.697,622.466 399.81,615.874 402.923,609.238 406.035,602.56 409.148,595.84 \n 412.26,589.081 415.373,582.284 418.485,575.45 421.598,568.581 424.71,561.679 427.823,554.745 430.935,547.782 434.048,540.789 437.161,533.771 440.273,526.728 \n 443.386,519.662 446.498,512.575 449.611,505.469 452.723,498.346 455.836,491.208 458.948,484.058 462.061,476.896 465.174,469.726 468.286,462.549 471.399,455.368 \n 474.511,448.184 477.624,441 480.736,433.819 483.849,426.642 486.961,419.473 490.074,412.312 493.187,405.163 496.299,398.028 499.412,390.909 502.524,383.809 \n 505.637,376.73 508.749,369.674 511.862,362.645 514.974,355.645 518.087,348.676 521.2,341.74 524.312,334.84 527.425,327.98 530.537,321.16 533.65,314.384 \n 536.762,307.655 539.875,300.974 542.987,294.345 546.1,287.77 549.212,281.252 552.325,274.792 555.438,268.394 558.55,262.061 561.663,255.794 564.775,249.596 \n 567.888,243.47 571,237.418 574.113,231.443 577.225,225.547 580.338,219.733 583.451,214.003 586.563,208.359 589.676,202.804 592.788,197.339 595.901,191.969 \n 599.013,186.694 602.126,181.517 605.238,176.44 608.351,171.466 611.464,166.596 614.576,161.833 617.689,157.179 620.801,152.636 623.914,148.205 627.026,143.889 \n 630.139,139.69 633.251,135.61 636.364,131.65 639.477,127.812 642.589,124.098 645.702,120.51 648.814,117.049 651.927,113.717 655.039,110.515 658.152,107.444 \n 661.264,104.507 664.377,101.705 667.489,99.0383 670.602,96.5087 673.715,94.1172 676.827,91.8651 679.94,89.7534 683.052,87.7829 686.165,85.9547 689.277,84.2697 \n 692.39,82.7285 695.502,81.3319 698.615,80.0805 701.728,78.975 704.84,78.0158 707.953,77.2035 711.065,76.5383 714.178,76.0206 717.29,75.6507 720.403,75.4286 \n 723.515,75.3546 726.628,75.4286 729.741,75.6507 732.853,76.0206 735.966,76.5383 739.078,77.2035 742.191,78.0158 745.303,78.975 748.416,80.0805 751.528,81.3319 \n 754.641,82.7285 757.754,84.2697 760.866,85.9547 763.979,87.7829 767.091,89.7534 770.204,91.8651 773.316,94.1172 776.429,96.5087 779.541,99.0383 782.654,101.705 \n 785.766,104.507 788.879,107.444 791.992,110.515 795.104,113.717 798.217,117.049 801.329,120.51 804.442,124.098 807.554,127.812 810.667,131.65 813.779,135.61 \n 816.892,139.69 820.005,143.889 823.117,148.205 826.23,152.636 829.342,157.179 832.455,161.833 835.567,166.596 838.68,171.466 841.792,176.44 844.905,181.517 \n 848.018,186.694 851.13,191.969 854.243,197.339 857.355,202.804 860.468,208.359 863.58,214.003 866.693,219.733 869.805,225.547 872.918,231.443 876.03,237.418 \n 879.143,243.47 882.256,249.596 885.368,255.794 888.481,262.061 891.593,268.394 894.706,274.792 897.818,281.252 900.931,287.77 904.043,294.345 907.156,300.974 \n 910.269,307.655 913.381,314.384 916.494,321.16 919.606,327.98 922.719,334.84 925.831,341.74 928.944,348.676 932.056,355.645 935.169,362.645 938.282,369.674 \n 941.394,376.73 944.507,383.809 947.619,390.909 950.732,398.028 953.844,405.163 956.957,412.312 960.069,419.473 963.182,426.642 966.295,433.819 969.407,441 \n 972.52,448.184 975.632,455.368 978.745,462.549 981.857,469.726 984.97,476.896 988.082,484.058 991.195,491.208 994.307,498.346 997.42,505.469 1000.53,512.575 \n 1003.65,519.662 1006.76,526.728 1009.87,533.771 1012.98,540.789 1016.1,547.782 1019.21,554.745 1022.32,561.679 1025.43,568.581 1028.55,575.45 1031.66,582.284 \n 1034.77,589.081 1037.88,595.84 1041,602.56 1044.11,609.238 1047.22,615.874 1050.33,622.466 1053.45,629.013 1056.56,635.513 1059.67,641.966 1062.78,648.369 \n 1065.9,654.723 1069.01,661.025 1072.12,667.275 1075.23,673.471 1078.35,679.613 1081.46,685.7 1084.57,691.731 1087.68,697.705 1090.8,703.621 1093.91,709.478 \n 1097.02,715.276 1100.13,721.014 1103.25,726.691 1106.36,732.306 1109.47,737.86 1112.58,743.351 1115.7,748.78 1118.81,754.145 1121.92,759.446 1125.03,764.683 \n 1128.15,769.855 1131.26,774.963 1134.37,780.005 1137.48,784.982 1140.6,789.893 1143.71,794.739 1146.82,799.519 1149.94,804.232 1153.05,808.88 1156.16,813.462 \n 1159.27,817.978 1162.39,822.427 1165.5,826.811 1168.61,831.129 1171.72,835.381 1174.84,839.567 1177.95,843.688 1181.06,847.744 1184.17,851.734 1187.29,855.66 \n 1190.4,859.521 1193.51,863.318 1196.62,867.052 1199.74,870.721 1202.85,874.328 1205.96,877.871 1209.07,881.353 1212.19,884.772 1215.3,888.13 1218.41,891.426 \n 1221.52,894.662 1224.64,897.838 1227.75,900.955 1230.86,904.012 1233.97,907.01 1237.09,909.951 1240.2,912.834 1243.31,915.661 1246.42,918.431 1249.54,921.146 \n 1252.65,923.805 1255.76,926.41 1258.87,928.961 1261.99,931.46 1265.1,933.905 1268.21,936.299 1271.32,938.641 1274.44,940.933 1277.55,943.175 1280.66,945.368 \n 1283.77,947.512 1286.89,949.609 1290,951.658 1293.11,953.661 1296.23,955.618 1299.34,957.53 1302.45,959.397 1305.56,961.221 1308.68,963.002 1311.79,964.74 \n 1314.9,966.437 1318.01,968.093 1321.13,969.708 1324.24,971.284 1327.35,972.821 1330.46,974.32 1333.58,975.782 1336.69,977.206 1339.8,978.595 1342.91,979.948 \n 1346.03,981.266 1349.14,982.55 1352.25,983.801 1355.36,985.018 1358.48,986.204 1361.59,987.358 1364.7,988.481 1367.81,989.574 1370.93,990.637 1374.04,991.671 \n 1377.15,992.677 1380.26,993.654 1383.38,994.605 1386.49,995.529 1389.6,996.427 1392.71,997.3 1395.83,998.147 1398.94,998.97 1402.05,999.77 1405.16,1000.55 \n 1408.28,1001.3 1411.39,1002.03 1414.5,1002.74 1417.61,1003.43 1420.73,1004.1 1423.84,1004.75 1426.95,1005.37 1430.06,1005.98 1433.18,1006.57 1436.29,1007.15 \n 1439.4,1007.7 1442.52,1008.24 1445.63,1008.76 1448.74,1009.26 1451.85,1009.75 1454.97,1010.23 1458.08,1010.68 1461.19,1011.13 1464.3,1011.56 1467.42,1011.97 \n 1470.53,1012.37 \n \"/>\n<polygon clip-path=\"url(#clip2201)\" points=\"\n425.389,848.483 958.333,848.483 958.333,689.723 425.389,689.723 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 425.389,848.483 958.333,848.483 958.333,689.723 425.389,689.723 425.389,848.483 \n \"/>\n<circle clip-path=\"url(#clip2201)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"569.389\" cy=\"742.643\" r=\"25\"/>\n<circle clip-path=\"url(#clip2201)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"569.389\" cy=\"742.643\" r=\"21\"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:41px; text-anchor:start;\" transform=\"rotate(0, 681.389, 757.643)\" x=\"681.389\" y=\"757.643\">Bin(6, 0.4)</text>\n</g>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#e26f46; stroke-width:6; stroke-opacity:1; fill:none\" stroke-dasharray=\"24, 15\" points=\"\n 457.389,795.563 649.389,795.563 \n \"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:41px; text-anchor:start;\" transform=\"rotate(0, 681.389, 810.563)\" x=\"681.389\" y=\"810.563\">Normal dist</text>\n</g>\n<polygon clip-path=\"url(#clip2201)\" points=\"\n1774.78,1040.48 3121.26,1040.48 3121.26,47.2441 1774.78,47.2441 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip2204\">\n <rect x=\"1774\" y=\"47\" width=\"1347\" height=\"994\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1812.88,1040.48 1812.88,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2077.52,1040.48 2077.52,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2342.16,1040.48 2342.16,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2606.8,1040.48 2606.8,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2871.44,1040.48 2871.44,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1774.78,1012.44 3121.26,1012.44 \n \"/>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1774.78,813.126 3121.26,813.126 \n \"/>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1774.78,613.814 3121.26,613.814 \n \"/>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1774.78,414.501 3121.26,414.501 \n \"/>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1774.78,215.188 3121.26,215.188 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1774.78,1040.48 3121.26,1040.48 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1774.78,1040.48 1774.78,47.2441 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1812.88,1040.48 1812.88,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2077.52,1040.48 2077.52,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2342.16,1040.48 2342.16,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2606.8,1040.48 2606.8,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2871.44,1040.48 2871.44,1025.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1774.78,1012.44 1794.97,1012.44 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1774.78,813.126 1794.97,813.126 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1774.78,613.814 1794.97,613.814 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1774.78,414.501 1794.97,414.501 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1774.78,215.188 1794.97,215.188 \n \"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1812.88, 1098.48)\" x=\"1812.88\" y=\"1098.48\">0.0</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2077.52, 1098.48)\" x=\"2077.52\" y=\"1098.48\">2.5</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2342.16, 1098.48)\" x=\"2342.16\" y=\"1098.48\">5.0</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2606.8, 1098.48)\" x=\"2606.8\" y=\"1098.48\">7.5</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2871.44, 1098.48)\" x=\"2871.44\" y=\"1098.48\">10.0</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1742.78, 1029.94)\" x=\"1742.78\" y=\"1029.94\">0.00</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1742.78, 830.626)\" x=\"1742.78\" y=\"830.626\">0.05</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1742.78, 631.314)\" x=\"1742.78\" y=\"631.314\">0.10</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1742.78, 432.001)\" x=\"1742.78\" y=\"432.001\">0.15</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1742.78, 232.688)\" x=\"1742.78\" y=\"232.688\">0.20</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 2448.02, 1190.4)\" x=\"2448.02\" y=\"1190.4\">k</text>\n</g>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1812.88\" cy=\"1003.76\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1812.88\" cy=\"1003.76\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1918.74\" cy=\"943.021\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1918.74\" cy=\"943.021\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2024.59\" cy=\"757.908\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2024.59\" cy=\"757.908\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2130.45\" cy=\"446.814\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2130.45\" cy=\"446.814\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2236.31\" cy=\"164.001\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2236.31\" cy=\"164.001\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2342.16\" cy=\"107.439\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2342.16\" cy=\"107.439\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2448.02\" cy=\"308.55\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2448.02\" cy=\"308.55\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2553.87\" cy=\"610.217\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2553.87\" cy=\"610.217\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2659.73\" cy=\"844.846\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2659.73\" cy=\"844.846\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2765.58\" cy=\"962.782\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2765.58\" cy=\"962.782\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2871.44\" cy=\"1002.51\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2871.44\" cy=\"1002.51\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2977.3\" cy=\"1011.24\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2977.3\" cy=\"1011.24\" r=\"9\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"3083.15\" cy=\"1012.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2204)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"3083.15\" cy=\"1012.37\" r=\"9\"/>\n<polyline clip-path=\"url(#clip2204)\" style=\"stroke:#e26f46; stroke-width:6; stroke-opacity:1; fill:none\" stroke-dasharray=\"24, 15\" points=\"\n 1812.88,995.276 1816.06,994.398 1819.23,993.482 1822.41,992.526 1825.59,991.528 1828.76,990.487 1831.94,989.401 1835.11,988.269 1838.29,987.089 1841.46,985.86 \n 1844.64,984.58 1847.82,983.248 1850.99,981.861 1854.17,980.419 1857.34,978.919 1860.52,977.359 1863.69,975.739 1866.87,974.056 1870.05,972.308 1873.22,970.494 \n 1876.4,968.611 1879.57,966.658 1882.75,964.633 1885.92,962.534 1889.1,960.359 1892.28,958.107 1895.45,955.774 1898.63,953.36 1901.8,950.863 1904.98,948.28 \n 1908.15,945.609 1911.33,942.849 1914.5,939.998 1917.68,937.053 1920.86,934.014 1924.03,930.877 1927.21,927.641 1930.38,924.304 1933.56,920.865 1936.73,917.321 \n 1939.91,913.671 1943.09,909.913 1946.26,906.045 1949.44,902.066 1952.61,897.973 1955.79,893.767 1958.96,889.443 1962.14,885.003 1965.32,880.443 1968.49,875.762 \n 1971.67,870.96 1974.84,866.035 1978.02,860.986 1981.19,855.812 1984.37,850.512 1987.55,845.084 1990.72,839.529 1993.9,833.845 1997.07,828.031 2000.25,822.088 \n 2003.42,816.015 2006.6,809.812 2009.77,803.478 2012.95,797.013 2016.13,790.418 2019.3,783.692 2022.48,776.836 2025.65,769.85 2028.83,762.736 2032,755.493 \n 2035.18,748.122 2038.36,740.625 2041.53,733.003 2044.71,725.257 2047.88,717.388 2051.06,709.398 2054.23,701.29 2057.41,693.064 2060.59,684.723 2063.76,676.269 \n 2066.94,667.705 2070.11,659.033 2073.29,650.256 2076.46,641.378 2079.64,632.4 2082.82,623.327 2085.99,614.162 2089.17,604.908 2092.34,595.569 2095.52,586.15 \n 2098.69,576.654 2101.87,567.086 2105.05,557.45 2108.22,547.751 2111.4,537.993 2114.57,528.181 2117.75,518.321 2120.92,508.418 2124.1,498.477 2127.27,488.504 \n 2130.45,478.504 2133.63,468.484 2136.8,458.448 2139.98,448.404 2143.15,438.356 2146.33,428.313 2149.5,418.279 2152.68,408.262 2155.86,398.268 2159.03,388.304 \n 2162.21,378.376 2165.38,368.492 2168.56,358.657 2171.73,348.88 2174.91,339.168 2178.09,329.526 2181.26,319.963 2184.44,310.486 2187.61,301.101 2190.79,291.815 \n 2193.96,282.637 2197.14,273.573 2200.32,264.629 2203.49,255.814 2206.67,247.135 2209.84,238.597 2213.02,230.209 2216.19,221.977 2219.37,213.908 2222.54,206.009 \n 2225.72,198.286 2228.9,190.746 2232.07,183.395 2235.25,176.24 2238.42,169.286 2241.6,162.54 2244.77,156.008 2247.95,149.696 2251.13,143.608 2254.3,137.751 \n 2257.48,132.13 2260.65,126.749 2263.83,121.614 2267,116.729 2270.18,112.098 2273.36,107.727 2276.53,103.618 2279.71,99.7756 2282.88,96.2036 2286.06,92.9049 \n 2289.23,89.8828 2292.41,87.1399 2295.59,84.6788 2298.76,82.5018 2301.94,80.6109 2305.11,79.008 2308.29,77.6944 2311.46,76.6715 2314.64,75.9401 2317.82,75.501 \n 2320.99,75.3546 2324.17,75.501 2327.34,75.9401 2330.52,76.6715 2333.69,77.6944 2336.87,79.008 2340.04,80.6109 2343.22,82.5018 2346.4,84.6788 2349.57,87.1399 \n 2352.75,89.8828 2355.92,92.9049 2359.1,96.2036 2362.27,99.7756 2365.45,103.618 2368.63,107.727 2371.8,112.098 2374.98,116.729 2378.15,121.614 2381.33,126.749 \n 2384.5,132.13 2387.68,137.751 2390.86,143.608 2394.03,149.696 2397.21,156.008 2400.38,162.54 2403.56,169.286 2406.73,176.24 2409.91,183.395 2413.09,190.746 \n 2416.26,198.286 2419.44,206.009 2422.61,213.908 2425.79,221.977 2428.96,230.209 2432.14,238.597 2435.31,247.135 2438.49,255.814 2441.67,264.629 2444.84,273.573 \n 2448.02,282.637 2451.19,291.815 2454.37,301.101 2457.54,310.486 2460.72,319.963 2463.9,329.526 2467.07,339.168 2470.25,348.88 2473.42,358.657 2476.6,368.492 \n 2479.77,378.376 2482.95,388.304 2486.13,398.268 2489.3,408.262 2492.48,418.279 2495.65,428.313 2498.83,438.356 2502,448.404 2505.18,458.448 2508.36,468.484 \n 2511.53,478.504 2514.71,488.504 2517.88,498.477 2521.06,508.418 2524.23,518.321 2527.41,528.181 2530.58,537.993 2533.76,547.751 2536.94,557.45 2540.11,567.086 \n 2543.29,576.654 2546.46,586.15 2549.64,595.569 2552.81,604.908 2555.99,614.162 2559.17,623.327 2562.34,632.4 2565.52,641.378 2568.69,650.256 2571.87,659.033 \n 2575.04,667.705 2578.22,676.269 2581.4,684.723 2584.57,693.064 2587.75,701.29 2590.92,709.398 2594.1,717.388 2597.27,725.257 2600.45,733.003 2603.63,740.625 \n 2606.8,748.122 2609.98,755.493 2613.15,762.736 2616.33,769.85 2619.5,776.836 2622.68,783.692 2625.86,790.418 2629.03,797.013 2632.21,803.478 2635.38,809.812 \n 2638.56,816.015 2641.73,822.088 2644.91,828.031 2648.08,833.845 2651.26,839.529 2654.44,845.084 2657.61,850.512 2660.79,855.812 2663.96,860.986 2667.14,866.035 \n 2670.31,870.96 2673.49,875.762 2676.67,880.443 2679.84,885.003 2683.02,889.443 2686.19,893.767 2689.37,897.973 2692.54,902.066 2695.72,906.045 2698.9,909.913 \n 2702.07,913.671 2705.25,917.321 2708.42,920.865 2711.6,924.304 2714.77,927.641 2717.95,930.877 2721.13,934.014 2724.3,937.053 2727.48,939.998 2730.65,942.849 \n 2733.83,945.609 2737,948.28 2740.18,950.863 2743.35,953.36 2746.53,955.774 2749.71,958.107 2752.88,960.359 2756.06,962.534 2759.23,964.633 2762.41,966.658 \n 2765.58,968.611 2768.76,970.494 2771.94,972.308 2775.11,974.056 2778.29,975.739 2781.46,977.359 2784.64,978.919 2787.81,980.419 2790.99,981.861 2794.17,983.248 \n 2797.34,984.58 2800.52,985.86 2803.69,987.089 2806.87,988.269 2810.04,989.401 2813.22,990.487 2816.4,991.528 2819.57,992.526 2822.75,993.482 2825.92,994.398 \n 2829.1,995.276 2832.27,996.115 2835.45,996.919 2838.63,997.687 2841.8,998.422 2844.98,999.124 2848.15,999.795 2851.33,1000.44 2854.5,1001.05 2857.68,1001.63 \n 2860.85,1002.19 2864.03,1002.72 2867.21,1003.23 2870.38,1003.71 2873.56,1004.17 2876.73,1004.61 2879.91,1005.03 2883.08,1005.43 2886.26,1005.81 2889.44,1006.17 \n 2892.61,1006.51 2895.79,1006.83 2898.96,1007.14 2902.14,1007.44 2905.31,1007.71 2908.49,1007.98 2911.67,1008.23 2914.84,1008.47 2918.02,1008.69 2921.19,1008.91 \n 2924.37,1009.11 2927.54,1009.3 2930.72,1009.49 2933.9,1009.66 2937.07,1009.82 2940.25,1009.98 2943.42,1010.12 2946.6,1010.26 2949.77,1010.39 2952.95,1010.51 \n 2956.12,1010.63 2959.3,1010.74 2962.48,1010.84 2965.65,1010.94 2968.83,1011.03 2972,1011.12 2975.18,1011.2 2978.35,1011.28 2981.53,1011.35 2984.71,1011.42 \n 2987.88,1011.49 2991.06,1011.55 2994.23,1011.6 2997.41,1011.66 3000.58,1011.71 3003.76,1011.76 3006.94,1011.8 3010.11,1011.84 3013.29,1011.88 3016.46,1011.92 \n 3019.64,1011.95 3022.81,1011.98 3025.99,1012.01 3029.17,1012.04 3032.34,1012.07 3035.52,1012.1 3038.69,1012.12 3041.87,1012.14 3045.04,1012.16 3048.22,1012.18 \n 3051.4,1012.2 3054.57,1012.21 3057.75,1012.23 3060.92,1012.24 3064.1,1012.26 3067.27,1012.27 3070.45,1012.28 3073.62,1012.29 3076.8,1012.3 3079.98,1012.31 \n 3083.15,1012.32 \n \"/>\n<polygon clip-path=\"url(#clip2201)\" points=\"\n2487.56,345.084 3025.26,345.084 3025.26,186.324 2487.56,186.324 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2487.56,345.084 3025.26,345.084 3025.26,186.324 2487.56,186.324 2487.56,345.084 \n \"/>\n<circle clip-path=\"url(#clip2201)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2631.56\" cy=\"239.244\" r=\"25\"/>\n<circle clip-path=\"url(#clip2201)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2631.56\" cy=\"239.244\" r=\"21\"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:41px; text-anchor:start;\" transform=\"rotate(0, 2743.56, 254.244)\" x=\"2743.56\" y=\"254.244\">Bin(12, 0.4)</text>\n</g>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#e26f46; stroke-width:6; stroke-opacity:1; fill:none\" stroke-dasharray=\"24, 15\" points=\"\n 2519.56,292.164 2711.56,292.164 \n \"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:41px; text-anchor:start;\" transform=\"rotate(0, 2743.56, 307.164)\" x=\"2743.56\" y=\"307.164\">Normal dist</text>\n</g>\n<polygon clip-path=\"url(#clip2201)\" points=\"\n188.156,2240.48 1507.88,2240.48 1507.88,1247.24 188.156,1247.24 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip2205\">\n <rect x=\"188\" y=\"1247\" width=\"1321\" height=\"994\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 225.507,2240.48 225.507,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 474.511,2240.48 474.511,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 723.515,2240.48 723.515,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 972.52,2240.48 972.52,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1221.52,2240.48 1221.52,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1470.53,2240.48 1470.53,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,2212.37 1507.88,2212.37 \n \"/>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,1924.71 1507.88,1924.71 \n \"/>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,1637.05 1507.88,1637.05 \n \"/>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 188.156,1349.38 1507.88,1349.38 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,2240.48 1507.88,2240.48 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,2240.48 188.156,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 225.507,2240.48 225.507,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 474.511,2240.48 474.511,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 723.515,2240.48 723.515,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 972.52,2240.48 972.52,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1221.52,2240.48 1221.52,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1470.53,2240.48 1470.53,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,2212.37 207.952,2212.37 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,1924.71 207.952,1924.71 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,1637.05 207.952,1637.05 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 188.156,1349.38 207.952,1349.38 \n \"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 225.507, 2298.48)\" x=\"225.507\" y=\"2298.48\">0</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 474.511, 2298.48)\" x=\"474.511\" y=\"2298.48\">5</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 723.515, 2298.48)\" x=\"723.515\" y=\"2298.48\">10</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 972.52, 2298.48)\" x=\"972.52\" y=\"2298.48\">15</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1221.52, 2298.48)\" x=\"1221.52\" y=\"2298.48\">20</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1470.53, 2298.48)\" x=\"1470.53\" y=\"2298.48\">25</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.156, 2229.87)\" x=\"156.156\" y=\"2229.87\">0.00</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.156, 1942.21)\" x=\"156.156\" y=\"1942.21\">0.05</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.156, 1654.55)\" x=\"156.156\" y=\"1654.55\">0.10</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 156.156, 1366.88)\" x=\"156.156\" y=\"1366.88\">0.15</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 848.018, 2390.4)\" x=\"848.018\" y=\"2390.4\">k</text>\n</g>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"225.507\" cy=\"2212.36\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"225.507\" cy=\"2212.36\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"275.308\" cy=\"2212.1\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"275.308\" cy=\"2212.1\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"325.109\" cy=\"2210.19\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"325.109\" cy=\"2210.19\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"374.91\" cy=\"2201.23\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"374.91\" cy=\"2201.23\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"424.71\" cy=\"2171.5\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"424.71\" cy=\"2171.5\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"474.511\" cy=\"2097.93\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"474.511\" cy=\"2097.93\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"524.312\" cy=\"1958.06\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"524.312\" cy=\"1958.06\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"574.113\" cy=\"1752.19\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"574.113\" cy=\"1752.19\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"623.914\" cy=\"1522.1\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"623.914\" cy=\"1522.1\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"673.715\" cy=\"1343.14\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"673.715\" cy=\"1343.14\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"723.515\" cy=\"1285.19\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"723.515\" cy=\"1285.19\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"773.316\" cy=\"1369.48\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"773.316\" cy=\"1369.48\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"823.117\" cy=\"1556.79\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"823.117\" cy=\"1556.79\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"872.918\" cy=\"1775.32\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"872.918\" cy=\"1775.32\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"922.719\" cy=\"1962.63\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"922.719\" cy=\"1962.63\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"972.52\" cy=\"2090.27\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"972.52\" cy=\"2090.27\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1022.32\" cy=\"2161.5\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1022.32\" cy=\"2161.5\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1072.12\" cy=\"2194.42\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1072.12\" cy=\"2194.42\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1121.92\" cy=\"2207.05\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1121.92\" cy=\"2207.05\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1171.72\" cy=\"2211.07\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1171.72\" cy=\"2211.07\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1221.52\" cy=\"2212.11\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1221.52\" cy=\"2212.11\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1271.32\" cy=\"2212.33\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1271.32\" cy=\"2212.33\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1321.13\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1321.13\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1370.93\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1370.93\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1420.73\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1420.73\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1470.53\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2205)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1470.53\" cy=\"2212.37\" r=\"9\"/>\n<polyline clip-path=\"url(#clip2205)\" style=\"stroke:#e26f46; stroke-width:6; stroke-opacity:1; fill:none\" stroke-dasharray=\"24, 15\" points=\"\n 225.507,2212.15 228.62,2212.12 231.732,2212.1 234.845,2212.07 237.957,2212.03 241.07,2212 244.182,2211.96 247.295,2211.91 250.407,2211.86 253.52,2211.81 \n 256.633,2211.75 259.745,2211.69 262.858,2211.62 265.97,2211.55 269.083,2211.46 272.195,2211.37 275.308,2211.27 278.42,2211.17 281.533,2211.05 284.646,2210.92 \n 287.758,2210.78 290.871,2210.63 293.983,2210.47 297.096,2210.29 300.208,2210.1 303.321,2209.89 306.433,2209.66 309.546,2209.41 312.658,2209.15 315.771,2208.86 \n 318.884,2208.55 321.996,2208.21 325.109,2207.85 328.221,2207.46 331.334,2207.03 334.446,2206.58 337.559,2206.09 340.671,2205.57 343.784,2205 346.897,2204.39 \n 350.009,2203.74 353.122,2203.04 356.234,2202.3 359.347,2201.5 362.459,2200.64 365.572,2199.72 368.684,2198.74 371.797,2197.7 374.91,2196.58 378.022,2195.39 \n 381.135,2194.13 384.247,2192.78 387.36,2191.34 390.472,2189.82 393.585,2188.2 396.697,2186.48 399.81,2184.66 402.923,2182.73 406.035,2180.68 409.148,2178.52 \n 412.26,2176.23 415.373,2173.81 418.485,2171.26 421.598,2168.56 424.71,2165.72 427.823,2162.73 430.935,2159.58 434.048,2156.26 437.161,2152.78 440.273,2149.12 \n 443.386,2145.29 446.498,2141.26 449.611,2137.04 452.723,2132.63 455.836,2128.01 458.948,2123.18 462.061,2118.13 465.174,2112.87 468.286,2107.38 471.399,2101.66 \n 474.511,2095.7 477.624,2089.5 480.736,2083.06 483.849,2076.37 486.961,2069.42 490.074,2062.22 493.187,2054.76 496.299,2047.03 499.412,2039.04 502.524,2030.78 \n 505.637,2022.25 508.749,2013.45 511.862,2004.38 514.974,1995.04 518.087,1985.42 521.2,1975.53 524.312,1965.38 527.425,1954.95 530.537,1944.26 533.65,1933.31 \n 536.762,1922.1 539.875,1910.63 542.987,1898.92 546.1,1886.96 549.212,1874.77 552.325,1862.35 555.438,1849.71 558.55,1836.85 561.663,1823.79 564.775,1810.54 \n 567.888,1797.11 571,1783.51 574.113,1769.76 577.225,1755.85 580.338,1741.82 583.451,1727.68 586.563,1713.43 589.676,1699.1 592.788,1684.7 595.901,1670.24 \n 599.013,1655.76 602.126,1641.26 605.238,1626.76 608.351,1612.29 611.464,1597.86 614.576,1583.49 617.689,1569.21 620.801,1555.03 623.914,1540.97 627.026,1527.06 \n 630.139,1513.31 633.251,1499.76 636.364,1486.41 639.477,1473.3 642.589,1460.44 645.702,1447.85 648.814,1435.56 651.927,1423.58 655.039,1411.94 658.152,1400.66 \n 661.264,1389.75 664.377,1379.24 667.489,1369.15 670.602,1359.49 673.715,1350.27 676.827,1341.53 679.94,1333.27 683.052,1325.51 686.165,1318.26 689.277,1311.54 \n 692.39,1305.37 695.502,1299.74 698.615,1294.67 701.728,1290.18 704.84,1286.27 707.953,1282.95 711.065,1280.22 714.178,1278.1 717.29,1276.57 720.403,1275.66 \n 723.515,1275.35 726.628,1275.66 729.741,1276.57 732.853,1278.1 735.966,1280.22 739.078,1282.95 742.191,1286.27 745.303,1290.18 748.416,1294.67 751.528,1299.74 \n 754.641,1305.37 757.754,1311.54 760.866,1318.26 763.979,1325.51 767.091,1333.27 770.204,1341.53 773.316,1350.27 776.429,1359.49 779.541,1369.15 782.654,1379.24 \n 785.766,1389.75 788.879,1400.66 791.992,1411.94 795.104,1423.58 798.217,1435.56 801.329,1447.85 804.442,1460.44 807.554,1473.3 810.667,1486.41 813.779,1499.76 \n 816.892,1513.31 820.005,1527.06 823.117,1540.97 826.23,1555.03 829.342,1569.21 832.455,1583.49 835.567,1597.86 838.68,1612.29 841.792,1626.76 844.905,1641.26 \n 848.018,1655.76 851.13,1670.24 854.243,1684.7 857.355,1699.1 860.468,1713.43 863.58,1727.68 866.693,1741.82 869.805,1755.85 872.918,1769.76 876.03,1783.51 \n 879.143,1797.11 882.256,1810.54 885.368,1823.79 888.481,1836.85 891.593,1849.71 894.706,1862.35 897.818,1874.77 900.931,1886.96 904.043,1898.92 907.156,1910.63 \n 910.269,1922.1 913.381,1933.31 916.494,1944.26 919.606,1954.95 922.719,1965.38 925.831,1975.53 928.944,1985.42 932.056,1995.04 935.169,2004.38 938.282,2013.45 \n 941.394,2022.25 944.507,2030.78 947.619,2039.04 950.732,2047.03 953.844,2054.76 956.957,2062.22 960.069,2069.42 963.182,2076.37 966.295,2083.06 969.407,2089.5 \n 972.52,2095.7 975.632,2101.66 978.745,2107.38 981.857,2112.87 984.97,2118.13 988.082,2123.18 991.195,2128.01 994.307,2132.63 997.42,2137.04 1000.53,2141.26 \n 1003.65,2145.29 1006.76,2149.12 1009.87,2152.78 1012.98,2156.26 1016.1,2159.58 1019.21,2162.73 1022.32,2165.72 1025.43,2168.56 1028.55,2171.26 1031.66,2173.81 \n 1034.77,2176.23 1037.88,2178.52 1041,2180.68 1044.11,2182.73 1047.22,2184.66 1050.33,2186.48 1053.45,2188.2 1056.56,2189.82 1059.67,2191.34 1062.78,2192.78 \n 1065.9,2194.13 1069.01,2195.39 1072.12,2196.58 1075.23,2197.7 1078.35,2198.74 1081.46,2199.72 1084.57,2200.64 1087.68,2201.5 1090.8,2202.3 1093.91,2203.04 \n 1097.02,2203.74 1100.13,2204.39 1103.25,2205 1106.36,2205.57 1109.47,2206.09 1112.58,2206.58 1115.7,2207.03 1118.81,2207.46 1121.92,2207.85 1125.03,2208.21 \n 1128.15,2208.55 1131.26,2208.86 1134.37,2209.15 1137.48,2209.41 1140.6,2209.66 1143.71,2209.89 1146.82,2210.1 1149.94,2210.29 1153.05,2210.47 1156.16,2210.63 \n 1159.27,2210.78 1162.39,2210.92 1165.5,2211.05 1168.61,2211.17 1171.72,2211.27 1174.84,2211.37 1177.95,2211.46 1181.06,2211.55 1184.17,2211.62 1187.29,2211.69 \n 1190.4,2211.75 1193.51,2211.81 1196.62,2211.86 1199.74,2211.91 1202.85,2211.96 1205.96,2212 1209.07,2212.03 1212.19,2212.07 1215.3,2212.1 1218.41,2212.12 \n 1221.52,2212.15 1224.64,2212.17 1227.75,2212.19 1230.86,2212.21 1233.97,2212.22 1237.09,2212.24 1240.2,2212.25 1243.31,2212.27 1246.42,2212.28 1249.54,2212.29 \n 1252.65,2212.3 1255.76,2212.3 1258.87,2212.31 1261.99,2212.32 1265.1,2212.32 1268.21,2212.33 1271.32,2212.33 1274.44,2212.34 1277.55,2212.34 1280.66,2212.34 \n 1283.77,2212.35 1286.89,2212.35 1290,2212.35 1293.11,2212.35 1296.23,2212.36 1299.34,2212.36 1302.45,2212.36 1305.56,2212.36 1308.68,2212.36 1311.79,2212.36 \n 1314.9,2212.36 1318.01,2212.37 1321.13,2212.37 1324.24,2212.37 1327.35,2212.37 1330.46,2212.37 1333.58,2212.37 1336.69,2212.37 1339.8,2212.37 1342.91,2212.37 \n 1346.03,2212.37 1349.14,2212.37 1352.25,2212.37 1355.36,2212.37 1358.48,2212.37 1361.59,2212.37 1364.7,2212.37 1367.81,2212.37 1370.93,2212.37 1374.04,2212.37 \n 1377.15,2212.37 1380.26,2212.37 1383.38,2212.37 1386.49,2212.37 1389.6,2212.37 1392.71,2212.37 1395.83,2212.37 1398.94,2212.37 1402.05,2212.37 1405.16,2212.37 \n 1408.28,2212.37 1411.39,2212.37 1414.5,2212.37 1417.61,2212.37 1420.73,2212.37 1423.84,2212.37 1426.95,2212.37 1430.06,2212.37 1433.18,2212.37 1436.29,2212.37 \n 1439.4,2212.37 1442.52,2212.37 1445.63,2212.37 1448.74,2212.37 1451.85,2212.37 1454.97,2212.37 1458.08,2212.37 1461.19,2212.37 1464.3,2212.37 1467.42,2212.37 \n 1470.53,2212.37 \n \"/>\n<polygon clip-path=\"url(#clip2201)\" points=\"\n874.176,1545.08 1411.88,1545.08 1411.88,1386.32 874.176,1386.32 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 874.176,1545.08 1411.88,1545.08 1411.88,1386.32 874.176,1386.32 874.176,1545.08 \n \"/>\n<circle clip-path=\"url(#clip2201)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1018.18\" cy=\"1439.24\" r=\"25\"/>\n<circle clip-path=\"url(#clip2201)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1018.18\" cy=\"1439.24\" r=\"21\"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:41px; text-anchor:start;\" transform=\"rotate(0, 1130.18, 1454.24)\" x=\"1130.18\" y=\"1454.24\">Bin(25, 0.4)</text>\n</g>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#e26f46; stroke-width:6; stroke-opacity:1; fill:none\" stroke-dasharray=\"24, 15\" points=\"\n 906.176,1492.16 1098.18,1492.16 \n \"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:41px; text-anchor:start;\" transform=\"rotate(0, 1130.18, 1507.16)\" x=\"1130.18\" y=\"1507.16\">Normal dist</text>\n</g>\n<polygon clip-path=\"url(#clip2201)\" points=\"\n1801.54,2240.48 3121.26,2240.48 3121.26,1247.24 1801.54,1247.24 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<defs>\n <clipPath id=\"clip2206\">\n <rect x=\"1801\" y=\"1247\" width=\"1321\" height=\"994\"/>\n </clipPath>\n</defs>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1838.89,2240.48 1838.89,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2087.89,2240.48 2087.89,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2336.9,2240.48 2336.9,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2585.9,2240.48 2585.9,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 2834.91,2240.48 2834.91,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 3083.91,2240.48 3083.91,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1801.54,2212.37 3121.26,2212.37 \n \"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1801.54,2008.96 3121.26,2008.96 \n \"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1801.54,1805.56 3121.26,1805.56 \n \"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1801.54,1602.15 3121.26,1602.15 \n \"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"\n 1801.54,1398.74 3121.26,1398.74 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1801.54,2240.48 3121.26,2240.48 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1801.54,2240.48 1801.54,1247.24 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1838.89,2240.48 1838.89,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2087.89,2240.48 2087.89,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2336.9,2240.48 2336.9,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2585.9,2240.48 2585.9,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2834.91,2240.48 2834.91,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 3083.91,2240.48 3083.91,2225.58 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1801.54,2212.37 1821.33,2212.37 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1801.54,2008.96 1821.33,2008.96 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1801.54,1805.56 1821.33,1805.56 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1801.54,1602.15 1821.33,1602.15 \n \"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 1801.54,1398.74 1821.33,1398.74 \n \"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1838.89, 2298.48)\" x=\"1838.89\" y=\"2298.48\">0</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2087.89, 2298.48)\" x=\"2087.89\" y=\"2298.48\">10</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2336.9, 2298.48)\" x=\"2336.9\" y=\"2298.48\">20</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2585.9, 2298.48)\" x=\"2585.9\" y=\"2298.48\">30</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2834.91, 2298.48)\" x=\"2834.91\" y=\"2298.48\">40</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 3083.91, 2298.48)\" x=\"3083.91\" y=\"2298.48\">50</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1769.54, 2229.87)\" x=\"1769.54\" y=\"2229.87\">0.000</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1769.54, 2026.46)\" x=\"1769.54\" y=\"2026.46\">0.025</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1769.54, 1823.06)\" x=\"1769.54\" y=\"1823.06\">0.050</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1769.54, 1619.65)\" x=\"1769.54\" y=\"1619.65\">0.075</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 1769.54, 1416.24)\" x=\"1769.54\" y=\"1416.24\">0.100</text>\n</g>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 2461.4, 2390.4)\" x=\"2461.4\" y=\"2390.4\">k</text>\n</g>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1838.89\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1838.89\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1863.79\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1863.79\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1888.69\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1888.69\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1913.59\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1913.59\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1938.49\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1938.49\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1963.39\" cy=\"2212.35\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1963.39\" cy=\"2212.35\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"1988.29\" cy=\"2212.28\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"1988.29\" cy=\"2212.28\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2013.19\" cy=\"2211.99\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2013.19\" cy=\"2211.99\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2038.09\" cy=\"2210.99\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2038.09\" cy=\"2210.99\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2062.99\" cy=\"2208.09\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2062.99\" cy=\"2208.09\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2087.89\" cy=\"2200.66\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2087.89\" cy=\"2200.66\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2112.79\" cy=\"2183.97\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2112.79\" cy=\"2183.97\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2137.69\" cy=\"2150.84\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2137.69\" cy=\"2150.84\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2162.59\" cy=\"2092.46\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2162.59\" cy=\"2092.46\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2187.49\" cy=\"2001.1\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2187.49\" cy=\"2001.1\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2212.39\" cy=\"1874.33\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2212.39\" cy=\"1874.33\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2237.29\" cy=\"1719.4\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2237.29\" cy=\"1719.4\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2262.2\" cy=\"1555.08\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2262.2\" cy=\"1555.08\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2287.1\" cy=\"1409.01\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2287.1\" cy=\"1409.01\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2312\" cy=\"1310.35\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2312\" cy=\"1310.35\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2336.9\" cy=\"1280.29\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2336.9\" cy=\"1280.29\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2361.8\" cy=\"1324.67\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2361.8\" cy=\"1324.67\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2386.7\" cy=\"1432.27\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2386.7\" cy=\"1432.27\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2411.6\" cy=\"1579.25\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2411.6\" cy=\"1579.25\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2436.5\" cy=\"1737.53\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2436.5\" cy=\"1737.53\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2461.4\" cy=\"1883.15\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2461.4\" cy=\"1883.15\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2486.3\" cy=\"2001.33\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2486.3\" cy=\"2001.33\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2511.2\" cy=\"2087.31\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2511.2\" cy=\"2087.31\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2536.1\" cy=\"2143.89\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2536.1\" cy=\"2143.89\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2561\" cy=\"2177.74\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2561\" cy=\"2177.74\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2585.9\" cy=\"2196.21\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2585.9\" cy=\"2196.21\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2610.8\" cy=\"2205.42\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2610.8\" cy=\"2205.42\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2635.7\" cy=\"2209.62\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2635.7\" cy=\"2209.62\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2660.6\" cy=\"2211.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2660.6\" cy=\"2211.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2685.5\" cy=\"2212.04\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2685.5\" cy=\"2212.04\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2710.4\" cy=\"2212.27\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2710.4\" cy=\"2212.27\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2735.3\" cy=\"2212.34\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2735.3\" cy=\"2212.34\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2760.2\" cy=\"2212.36\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2760.2\" cy=\"2212.36\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2785.1\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2785.1\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2810\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2810\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2834.91\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2834.91\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2859.81\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2859.81\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2884.71\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2884.71\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2909.61\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2909.61\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2934.51\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2934.51\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2959.41\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2959.41\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2984.31\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2984.31\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"3009.21\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"3009.21\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"3034.11\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"3034.11\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"3059.01\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"3059.01\" cy=\"2212.37\" r=\"9\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"3083.91\" cy=\"2212.37\" r=\"12\"/>\n<circle clip-path=\"url(#clip2206)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"3083.91\" cy=\"2212.37\" r=\"9\"/>\n<polyline clip-path=\"url(#clip2206)\" style=\"stroke:#e26f46; stroke-width:6; stroke-opacity:1; fill:none\" stroke-dasharray=\"24, 15\" points=\"\n 1838.89,2212.37 1842,2212.37 1845.11,2212.37 1848.23,2212.37 1851.34,2212.37 1854.45,2212.37 1857.56,2212.37 1860.68,2212.37 1863.79,2212.37 1866.9,2212.37 \n 1870.01,2212.37 1873.13,2212.37 1876.24,2212.37 1879.35,2212.37 1882.46,2212.37 1885.58,2212.37 1888.69,2212.37 1891.8,2212.37 1894.91,2212.37 1898.03,2212.37 \n 1901.14,2212.37 1904.25,2212.37 1907.36,2212.37 1910.48,2212.37 1913.59,2212.37 1916.7,2212.37 1919.81,2212.36 1922.93,2212.36 1926.04,2212.36 1929.15,2212.36 \n 1932.26,2212.36 1935.38,2212.35 1938.49,2212.35 1941.6,2212.35 1944.71,2212.34 1947.83,2212.34 1950.94,2212.33 1954.05,2212.32 1957.17,2212.31 1960.28,2212.3 \n 1963.39,2212.29 1966.5,2212.28 1969.62,2212.26 1972.73,2212.25 1975.84,2212.23 1978.95,2212.2 1982.07,2212.17 1985.18,2212.14 1988.29,2212.11 1991.4,2212.06 \n 1994.52,2212.02 1997.63,2211.96 2000.74,2211.9 2003.85,2211.83 2006.97,2211.75 2010.08,2211.66 2013.19,2211.55 2016.3,2211.43 2019.42,2211.3 2022.53,2211.15 \n 2025.64,2210.98 2028.75,2210.79 2031.87,2210.57 2034.98,2210.32 2038.09,2210.05 2041.2,2209.74 2044.32,2209.4 2047.43,2209.01 2050.54,2208.58 2053.65,2208.1 \n 2056.77,2207.57 2059.88,2206.98 2062.99,2206.32 2066.1,2205.59 2069.22,2204.78 2072.33,2203.88 2075.44,2202.89 2078.55,2201.81 2081.67,2200.61 2084.78,2199.29 \n 2087.89,2197.84 2091,2196.26 2094.12,2194.53 2097.23,2192.63 2100.34,2190.56 2103.46,2188.31 2106.57,2185.86 2109.68,2183.2 2112.79,2180.31 2115.91,2177.18 \n 2119.02,2173.8 2122.13,2170.14 2125.24,2166.2 2128.36,2161.96 2131.47,2157.4 2134.58,2152.51 2137.69,2147.26 2140.81,2141.65 2143.92,2135.66 2147.03,2129.26 \n 2150.14,2122.45 2153.26,2115.21 2156.37,2107.52 2159.48,2099.36 2162.59,2090.74 2165.71,2081.62 2168.82,2072 2171.93,2061.88 2175.04,2051.23 2178.16,2040.05 \n 2181.27,2028.34 2184.38,2016.09 2187.49,2003.3 2190.61,1989.96 2193.72,1976.07 2196.83,1961.65 2199.94,1946.69 2203.06,1931.21 2206.17,1915.21 2209.28,1898.71 \n 2212.39,1881.73 2215.51,1864.28 2218.62,1846.39 2221.73,1828.07 2224.84,1809.37 2227.96,1790.3 2231.07,1770.91 2234.18,1751.22 2237.29,1731.29 2240.41,1711.15 \n 2243.52,1690.84 2246.63,1670.42 2249.75,1649.93 2252.86,1629.43 2255.97,1608.96 2259.08,1588.59 2262.2,1568.37 2265.31,1548.36 2268.42,1528.62 2271.53,1509.21 \n 2274.65,1490.18 2277.76,1471.61 2280.87,1453.55 2283.98,1436.06 2287.1,1419.2 2290.21,1403.03 2293.32,1387.61 2296.43,1372.98 2299.55,1359.21 2302.66,1346.34 \n 2305.77,1334.41 2308.88,1323.49 2312,1313.59 2315.11,1304.77 2318.22,1297.06 2321.33,1290.48 2324.45,1285.06 2327.56,1280.83 2330.67,1277.79 2333.78,1275.96 \n 2336.9,1275.35 2340.01,1275.96 2343.12,1277.79 2346.23,1280.83 2349.35,1285.06 2352.46,1290.48 2355.57,1297.06 2358.68,1304.77 2361.8,1313.59 2364.91,1323.49 \n 2368.02,1334.41 2371.13,1346.34 2374.25,1359.21 2377.36,1372.98 2380.47,1387.61 2383.58,1403.03 2386.7,1419.2 2389.81,1436.06 2392.92,1453.55 2396.04,1471.61 \n 2399.15,1490.18 2402.26,1509.21 2405.37,1528.62 2408.49,1548.36 2411.6,1568.37 2414.71,1588.59 2417.82,1608.96 2420.94,1629.43 2424.05,1649.93 2427.16,1670.42 \n 2430.27,1690.84 2433.39,1711.15 2436.5,1731.29 2439.61,1751.22 2442.72,1770.91 2445.84,1790.3 2448.95,1809.37 2452.06,1828.07 2455.17,1846.39 2458.29,1864.28 \n 2461.4,1881.73 2464.51,1898.71 2467.62,1915.21 2470.74,1931.21 2473.85,1946.69 2476.96,1961.65 2480.07,1976.07 2483.19,1989.96 2486.3,2003.3 2489.41,2016.09 \n 2492.52,2028.34 2495.64,2040.05 2498.75,2051.23 2501.86,2061.88 2504.97,2072 2508.09,2081.62 2511.2,2090.74 2514.31,2099.36 2517.42,2107.52 2520.54,2115.21 \n 2523.65,2122.45 2526.76,2129.26 2529.87,2135.66 2532.99,2141.65 2536.1,2147.26 2539.21,2152.51 2542.33,2157.4 2545.44,2161.96 2548.55,2166.2 2551.66,2170.14 \n 2554.78,2173.8 2557.89,2177.18 2561,2180.31 2564.11,2183.2 2567.23,2185.86 2570.34,2188.31 2573.45,2190.56 2576.56,2192.63 2579.68,2194.53 2582.79,2196.26 \n 2585.9,2197.84 2589.01,2199.29 2592.13,2200.61 2595.24,2201.81 2598.35,2202.89 2601.46,2203.88 2604.58,2204.78 2607.69,2205.59 2610.8,2206.32 2613.91,2206.98 \n 2617.03,2207.57 2620.14,2208.1 2623.25,2208.58 2626.36,2209.01 2629.48,2209.4 2632.59,2209.74 2635.7,2210.05 2638.81,2210.32 2641.93,2210.57 2645.04,2210.79 \n 2648.15,2210.98 2651.26,2211.15 2654.38,2211.3 2657.49,2211.43 2660.6,2211.55 2663.71,2211.66 2666.83,2211.75 2669.94,2211.83 2673.05,2211.9 2676.16,2211.96 \n 2679.28,2212.02 2682.39,2212.06 2685.5,2212.11 2688.62,2212.14 2691.73,2212.17 2694.84,2212.2 2697.95,2212.23 2701.07,2212.25 2704.18,2212.26 2707.29,2212.28 \n 2710.4,2212.29 2713.52,2212.3 2716.63,2212.31 2719.74,2212.32 2722.85,2212.33 2725.97,2212.34 2729.08,2212.34 2732.19,2212.35 2735.3,2212.35 2738.42,2212.35 \n 2741.53,2212.36 2744.64,2212.36 2747.75,2212.36 2750.87,2212.36 2753.98,2212.36 2757.09,2212.37 2760.2,2212.37 2763.32,2212.37 2766.43,2212.37 2769.54,2212.37 \n 2772.65,2212.37 2775.77,2212.37 2778.88,2212.37 2781.99,2212.37 2785.1,2212.37 2788.22,2212.37 2791.33,2212.37 2794.44,2212.37 2797.55,2212.37 2800.67,2212.37 \n 2803.78,2212.37 2806.89,2212.37 2810,2212.37 2813.12,2212.37 2816.23,2212.37 2819.34,2212.37 2822.45,2212.37 2825.57,2212.37 2828.68,2212.37 2831.79,2212.37 \n 2834.91,2212.37 2838.02,2212.37 2841.13,2212.37 2844.24,2212.37 2847.36,2212.37 2850.47,2212.37 2853.58,2212.37 2856.69,2212.37 2859.81,2212.37 2862.92,2212.37 \n 2866.03,2212.37 2869.14,2212.37 2872.26,2212.37 2875.37,2212.37 2878.48,2212.37 2881.59,2212.37 2884.71,2212.37 2887.82,2212.37 2890.93,2212.37 2894.04,2212.37 \n 2897.16,2212.37 2900.27,2212.37 2903.38,2212.37 2906.49,2212.37 2909.61,2212.37 2912.72,2212.37 2915.83,2212.37 2918.94,2212.37 2922.06,2212.37 2925.17,2212.37 \n 2928.28,2212.37 2931.39,2212.37 2934.51,2212.37 2937.62,2212.37 2940.73,2212.37 2943.84,2212.37 2946.96,2212.37 2950.07,2212.37 2953.18,2212.37 2956.29,2212.37 \n 2959.41,2212.37 2962.52,2212.37 2965.63,2212.37 2968.74,2212.37 2971.86,2212.37 2974.97,2212.37 2978.08,2212.37 2981.19,2212.37 2984.31,2212.37 2987.42,2212.37 \n 2990.53,2212.37 2993.65,2212.37 2996.76,2212.37 2999.87,2212.37 3002.98,2212.37 3006.1,2212.37 3009.21,2212.37 3012.32,2212.37 3015.43,2212.37 3018.55,2212.37 \n 3021.66,2212.37 3024.77,2212.37 3027.88,2212.37 3031,2212.37 3034.11,2212.37 3037.22,2212.37 3040.33,2212.37 3043.45,2212.37 3046.56,2212.37 3049.67,2212.37 \n 3052.78,2212.37 3055.9,2212.37 3059.01,2212.37 3062.12,2212.37 3065.23,2212.37 3068.35,2212.37 3071.46,2212.37 3074.57,2212.37 3077.68,2212.37 3080.8,2212.37 \n 3083.91,2212.37 \n \"/>\n<polygon clip-path=\"url(#clip2201)\" points=\"\n2487.56,1545.08 3025.26,1545.08 3025.26,1386.32 2487.56,1386.32 \n \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 2487.56,1545.08 3025.26,1545.08 3025.26,1386.32 2487.56,1386.32 2487.56,1545.08 \n \"/>\n<circle clip-path=\"url(#clip2201)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2631.56\" cy=\"1439.24\" r=\"25\"/>\n<circle clip-path=\"url(#clip2201)\" style=\"fill:#009af9; stroke:none; fill-opacity:1\" cx=\"2631.56\" cy=\"1439.24\" r=\"21\"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:41px; text-anchor:start;\" transform=\"rotate(0, 2743.56, 1454.24)\" x=\"2743.56\" y=\"1454.24\">Bin(50, 0.4)</text>\n</g>\n<polyline clip-path=\"url(#clip2201)\" style=\"stroke:#e26f46; stroke-width:6; stroke-opacity:1; fill:none\" stroke-dasharray=\"24, 15\" points=\"\n 2519.56,1492.16 2711.56,1492.16 \n \"/>\n<g clip-path=\"url(#clip2201)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:41px; text-anchor:start;\" transform=\"rotate(0, 2743.56, 1507.16)\" x=\"2743.56\" y=\"1507.16\">Normal dist</text>\n</g>\n</svg>\n"
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "",
"execution_count": null,
"outputs": []
}
],
"metadata": {
"_draft": {
"nbviewer_url": "https://gist.github.com/f6746b8a903fabe24d89d771b6fe23f2"
},
"gist": {
"id": "f6746b8a903fabe24d89d771b6fe23f2",
"data": {
"description": "11 Kullback-Leibler information",
"public": true
}
},
"kernelspec": {
"name": "julia-1.1",
"display_name": "Julia 1.1.0",
"language": "julia"
},
"language_info": {
"file_extension": ".jl",
"name": "julia",
"mimetype": "application/julia",
"version": "1.1.0"
},
"toc": {
"nav_menu": {},
"number_sections": true,
"sideBar": true,
"skip_h1_title": true,
"title_cell": "目次",
"title_sidebar": "目次",
"toc_cell": true,
"toc_position": {},
"toc_section_display": true,
"toc_window_display": false
}
},
"nbformat": 4,
"nbformat_minor": 2
}
Sign up for free to join this conversation on GitHub. Already have an account? Sign in to comment