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probability_basis/probability_distributions.ipynb
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{
"cells": [
{
"metadata": {},
"cell_type": "markdown",
"source": "# 1.0 正規分布(normal distribution)について\n\n2つのパラメータ \n平均$\\mu$(mean)と分散$\\sigma^2$(variance) \nこのとき$\\sigma$を標準偏差(standard deviation)という \n$N(\\mu,\\sigma^2)$と表記 \n$f(x)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}$ \n\n特に$N(0,1)$を標準正規分布(the standard normal distribution)という。\n\n$0-1\\sigma > 34.1\\%$ \n$1-2\\sigma > 13.6\\%$ \n$2-3\\sigma > 2.1\\%$ \n$3\\sigma- > 0.2\\%$ \n\nそのグラフを描いてみよう!"
},
{
"metadata": {
"trusted": false,
"collapsed": false
},
"cell_type": "code",
"source": "%matplotlib inline\n%config inlineBackend.figure_format = 'retina'\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# use np.exp for array\ndef normal_distribution(mu, sig2, x):\n return np.exp(-(x-mu)**2 / (2*sig2)) / np.sqrt(2*np.math.pi*sig2)\n\nx = np.arange(-5, 5, 0.1)\ny1 = normal_distribution(0, 1, x)\ny2 = normal_distribution(2, 2, x)\ny3 = normal_distribution(0, 4, x)\n\nplt.rcParams[\"figure.figsize\"] = (8,3)\nplt.title('normal distribution')\nplt.grid(which='major',color='black',linestyle='-') \nplt.plot(x, y1)\nplt.plot(x, y2)\nplt.plot(x, y3)\nplt.show()",
"execution_count": 3,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"text/plain": "<matplotlib.figure.Figure at 0x11fb32208>",
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iUkZEsgFtgenuK4hIebe/bwWyqepRT7a1LMsIxEZkl7v9dsid2xSBW5blmZDUVlDVRBHp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\niIioWAp1RERExVKoIyIiKvYbQVXmwOsNxlwAAAAASUVORK5CYII=\n"
}
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "# 1.1 正規分布のパラメータをデータから推定する \n$\\hat{\\mu}= \\frac{1}{n}\\sum{x_i}$ \n$\\hat{\\sigma^2}=\\frac{1}{n-1}\\sum{(x_i-\\hat{\\mu})^2}$\n\n"
},
{
"metadata": {
"trusted": false,
"collapsed": false
},
"cell_type": "code",
"source": "#サイコロを12回振ってその合計が何になるか?\n#それを100000回試してみてグラフにする。\nM = 4\nN = 100000\ndata = np.zeros(M*6+1, dtype=int)\nfor i in range(N):\n a = 0;\n for j in range(M):\n a = a + np.random.randint(1, 7)\n data[a] = data[a] + 1\n\n#平均と分散を推定してみる\nmu = 0.0\nfor i in range(data.size):\n if data[i] != 0:\n mu = mu + i * data[i]\nmu = mu / N\nsig2 = 0.0\nfor i in range(data.size):\n if data[i] != 0:\n sig2 = sig2 + (i - mu)**2 * data[i]\nsig2 = sig2 / (N - 1)\nprint('mu=', mu, ' sig2=', sig2, ' sig=', np.sqrt(sig2)) \n\nx = np.arange(M, M*6+1, 1)\ny = normal_distribution(mu, sig2, x)\n\n# 表示する\nplt.rcParams[\"figure.figsize\"] = (8,3)\nplt.grid(which='major',color='black',linestyle='-') \nplt.xlim(M, M*6)\nplt.plot(x, y)\nfor i in range(data.size):\n if data[i] != 0:\n plt.plot(i, data[i] / N, 'o')\nplt.show()\n",
"execution_count": 4,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "mu= 14.0051 sig2= 11.7212512025 sig= 3.42363128893\n"
},
{
"metadata": {},
"output_type": "display_data",
"data": {
"text/plain": "<matplotlib.figure.Figure at 0x11fc88cf8>",
"image/png": 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}
}
]
},
{
"metadata": {
"collapsed": true
},
"cell_type": "markdown",
"source": "# 2.0 ポアソン分布(poisson distribution)について\n\nある一定の間隔(時間や距離)に平均で$\\lambda$回起きる事象がある。 \nそのときに丁度$k$回起こる離散確率分布をあらわす。 \n例えば平均で2年に一回起こる事象がある。この時に8年で何回起こるかを推定する場合、8年では平均で4回起こるはずなので$\\lambda=4$のポアソン分布で推定すれば良い。 \n同様に1年で何回起こるかを推定する場合は、1年では0.5回起こるはずなので、$\\lambda=0.5$のポアソン分布で推定すれば良い。 \n距離の場合の例を示すと、例えばガソリンスタンドが平均2キロ毎に1軒あるものとする。この場合に4キロ行く時に何軒のガソリンスタンドがあるだろうか? \nそれを推定する場合は$\\lambda=2$のポアソン分布で推定すれば良いのだ。 \n後のグラフではっきりするが、0軒の場合は約15%ほどになる。給油するかどうかはあなたの判断にまかせる。\n\n1つのパラメータ \n生起間隔$\\lambda$ \n$P(X=k)=\\frac{\\lambda^k e^{-\\lambda}}{k!}$ \n\n平均$E(X)$と分散$V(X)$は \n$E(X)=\\lambda$ \n$V(X)=\\lambda$ \n\nそのグラフを描いてみよう!"
},
{
"metadata": {
"trusted": false,
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},
"cell_type": "code",
"source": "%matplotlib inline\n%config inlineBackend.figure_format = 'retina'\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef poisson_distribution(lamda, ks):\n return [lamda**k * np.exp(-lamda) / np.math.factorial(k) for k in ks]\n\nks = np.arange(0, 9+1, 1)\ny1 = poisson_distribution(0.5, ks)\ny2 = poisson_distribution(1, ks)\ny3 = poisson_distribution(2, ks)\ny4 = poisson_distribution(4, ks)\n# P(2)をN(2,2)で近似してみる\nks2 = np.arange(0, 9+0.1, 0.1)\ny32 = normal_distribution(2, 2, ks2)\n\nplt.rcParams[\"figure.figsize\"] = (8,3)\nplt.title('poisson distribution')\nplt.grid(which='major',color='black',linestyle='-') \nplt.plot(ks, y1)\nplt.plot(ks, y1, 'o')\nplt.plot(ks, y2)\nplt.plot(ks, y2, 'o')\nplt.plot(ks, y3)\nplt.plot(ks, y3, 'o')\nplt.plot(ks, y4)\nplt.plot(ks, y4, 'o')\nplt.plot(ks2, y32)\nplt.show()",
"execution_count": 8,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"text/plain": "<matplotlib.figure.Figure at 0x12003be10>",
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/YTgoBx6b8xgvrXiJqItRVopeCCHynoeuGCfdSOLo+KPs676PapOqUffnuhSp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\ntm3FVFVtp/MUrYULb+WOOy5j1KhRJCUlkZSUFMj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jY2NjYxNkbDG2sbGxsbEJMrYY29jY\n2NjYBBlbjG1sbGxsbIKMLcY2NjY2NjZB5v8B790bCbzyUj4AAAAASUVORK5CYII=\n"
}
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "# 3.0 二項分布(binomial distribution)について\n\n生起する確率が独立で$p$の事象がある。 \nこの場合、$n$回の試行をした時に丁度$k$回生起する離散確率分布をあらわす。 \n例えばコインを投げた場合の表の出る確率を0.5とする。 \nそれを9回繰り返した時に表が何回出るかを推定する場合は$p=0.5,n=9$の二項分布で推定すれば良い。 \n同様に用意したインチキコインは表の出る確率が0.2だった。 \nそれを9回繰り返した時に表が何回出るかを推定する場合は$p=0.2,n=9$の二項分布で推定すれば良い。 \n\n2つのパラメータ \n生起確率$p$ \n試行回数$n$ \nこの二項分布を$B(n,p)$と記述する。 \n$P(X=k)=\\begin{pmatrix}n\\\\k\\end{pmatrix}p^k (1-p)^{n-k}$ \n$\\begin{pmatrix}n\\\\k\\end{pmatrix} = n個からk個選ぶ組み合わせ = {}_n C_k = \\frac{n!}{k!(n-k)!}$\n\n平均$E(X)$と分散$V(X)$は \n$E(X)=np$ \n$V(X)=np(1-p)$ \n\nそのグラフを描いてみよう!"
},
{
"metadata": {
"trusted": false,
"collapsed": false
},
"cell_type": "code",
"source": "%matplotlib inline\n%config inlineBackend.figure_format = 'retina'\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef binomial_distribution(n, p, ks):\n res = []\n for k in ks:\n c = np.math.factorial(n) / \\\n (np.math.factorial(k) * np.math.factorial(n - k))\n res.append(c * p**k * (1 - p)**(n - k))\n return res\n\nks = np.arange(0, 9+1, 1)\ny1 = binomial_distribution(9, 0.5, ks)\ny2 = binomial_distribution(9, 0.2, ks)\n# B(9,0.1)をN(9*0.1,9*0.1*(10-0.1))で近似してみる\nks2 = np.arange(0, 9+0.1, 0.1)\ny22 = normal_distribution(9*0.2, 9*0.2*(1.0-0.2), ks2)\n\nplt.rcParams[\"figure.figsize\"] = (8,3)\nplt.title('binomial distribution')\nplt.grid(which='major',color='black',linestyle='-') \nplt.plot(ks, y1)\nplt.plot(ks, y1, 'o')\nplt.plot(ks, y2)\nplt.plot(ks, y2, 'o')\nplt.plot(ks2, y22)\n\nplt.show()",
"execution_count": 16,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"text/plain": "<matplotlib.figure.Figure at 0x120417f98>",
"image/png": 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pVhUEWIHCwmDhQpg50/49DRH3RVDn/DrE3xFf2HFQKaWUmzRRn45p06BxYxg0qMyiBbkF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\nkZycTLLLDIPLli3DGFP2umbGmFIvwPn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}
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "# 3.1 二項分布(binomial distribution)の例\n\nある製品を生産するが、1個あたりの不良品率が0.05であったとする。 \n大量に生産しているこの製品から無作為に100個を選んでチェックをする。 \nこの時に不良品が2個以上である確率を求めてみよう。 \nまた、何個の不良品が出るか、その期待値を求めてみよう。 \nこれを推定するには$p=0.05, n=100$の二項分布で推定すれば良い。 \n2個以上の不良品率はkを2以上の確率を全て加算すれば求められる。 \n期待値は個数と確率を掛けたものを全て合算すれば良い。\n\nそのグラフを描いてみよう! \nそして答えを求めてみよう!"
},
{
"metadata": {
"trusted": false,
"collapsed": false
},
"cell_type": "code",
"source": "ks = np.arange(0, 100+1, 1)\ny = binomial_distribution(100, 0.05, ks)\n#yn = normal_distribution(100*0.05, 100*0.05*(1-0.05), ks)\n\nplt.rcParams[\"figure.figsize\"] = (8,3)\nplt.title('binomial distribution example')\nplt.grid(which='major',color='black',linestyle='-') \nplt.plot(ks, y)\nplt.plot(ks, y, 'o')\n#plt.plot(ks, yn)\nplt.show()\n\npp = 0.0\nfor k in ks[2:]:\n pp = pp + y[k]\nprint('2個以上の不良品率=', pp)\n\npe = 0.0\nfor k in ks:\n pe = pe + k * y[k]\nprint('不良品個数の期待値=', pe)\n\nprint('推定平均np=', 100*0.05)\nprint('推定分散np(1-p)=', 100*0.05*(1.0-0.05))\nprint('推定標準偏差sqrt(np(1-p))=', np.math.sqrt(100*0.05*(1.0-0.05)))",
"execution_count": 15,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"text/plain": "<matplotlib.figure.Figure at 0x1200ce080>",
"image/png": 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3AX9CMgIft27FNsZviJmZ2QxT7xH4ADC/4v0J6bKRZU4cpUxLFXWB6X0IMzOz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\nAAAAAElFTkSuQmCC\n"
}
},
{
"name": "stdout",
"output_type": "stream",
"text": "2個以上の不良品率= 0.962918790673\n不良品個数の期待値= 5.0\n推定平均np= 5.0\n推定分散np(1-p)= 4.75\n推定標準偏差sqrt(np(1-p))= 2.179449471770337\n"
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "# 3.2 二項分布(binomial distribution)の例\n\nある珈琲のブレンドは2種の豆を1対1で混ぜて大きな袋に保存している。 \nその袋から無作為に100粒を選んで1杯の珈琲を淹れる。 \nこの時に2種の豆の差が10未満であれば良いブレンド珈琲を飲むことができる。 \nその確率を求めてみよう。 \nこれを推定するには$p=0.5, n=100$の二項分布で推定すれば良い。 \n46以上54以下となる確率を全て加算すれば求められる。 \n\nそのグラフを描いてみよう! \nそして答えを求めてみよう!"
},
{
"metadata": {
"trusted": false,
"collapsed": false
},
"cell_type": "code",
"source": "ks = np.arange(0, 100+1, 1)\ny = binomial_distribution(100, 0.5, ks)\n#yn = normal_distribution(100*0.5, 100*0.5*(1-0.5), ks)\n\nplt.rcParams[\"figure.figsize\"] = (8,3)\nplt.title('binomial distribution example')\nplt.grid(which='major',color='black',linestyle='-') \nplt.plot(ks, y)\nplt.plot(ks, y, 'o')\n#plt.plot(ks, yn)\nplt.show()\n\npp = 0.0\nfor k in ks[46:55]:\n pp = pp + y[k]\nprint('良いブレンドを飲める確率=', pp)\n\n",
"execution_count": 14,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"text/plain": "<matplotlib.figure.Figure at 0x120011198>",
"image/png": 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}
},
{
"name": "stdout",
"output_type": "stream",
"text": "良いブレンドを飲める確率= 0.631798382673\n"
}
]
},
{
"metadata": {
"collapsed": true
},
"cell_type": "markdown",
"source": "# 3.3 二項分布(binomial distribution)を正規分布で考える例\n\n二項分布の$n$が大きい時は二項分布の代わりに正規分布を使うこともできる。 \n先の問題を正規分布に当てはめて考えてみよう。 \n\n二項分布の平均$E(X)$と分散$V(X)$は \n$E(X)=np$ \n$V(X)=np(1-p)$ \nだったよね? \n\nそのグラフを描いてみよう! \nそして答えを求めてみよう!"
},
{
"metadata": {
"trusted": false,
"collapsed": false
},
"cell_type": "code",
"source": "ks = np.arange(0, 100+1, 1)\ny = normal_distribution(100*0.5, 100*0.5*(1-0.5), ks)\n\nplt.rcParams[\"figure.figsize\"] = (8,3)\nplt.title('binomial distribution example')\nplt.grid(which='major',color='black',linestyle='-') \nplt.plot(ks, y)\nplt.plot(ks, y, 'o')\n#plt.plot(ks, yn)\nplt.show()\n\npp = 0.0\nfor k in ks[46:55]:\n pp = pp + y[k]\nprint('良いブレンドを飲める確率=', pp)",
"execution_count": 13,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"text/plain": "<matplotlib.figure.Figure at 0x12013f240>",
"image/png": 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BXdRrej5soYd8GHUClxwys2HAvcDVYUs8vceiejD2kZl9HlgfXuno\n7jKX6rh/dgdOAH7h7icQPOVyLfosZ42Z7U/QKhwNlBG0xKegOh4ofa7XqBN4JoPESB+El8LuBe50\n9wfC1evNbET4+iHAX6OKrwicDJxjZm8CdwNnmNmdwNuq46xaCzS7+wvh8v8SJHR9lrPnc8Cb7v6u\nu+8A7iMYDV51nBtd1WtXA6J1KeoEroFecufXwOvuflPKugXAJeHPFwMPpG8kmXH3H7n7oe5+GMHn\n9jF3/xrwIKrjrAkvNTab2ZHhqs8Cr6HPcjatIZg9cq+w09RnCTpmqo6zw+h4la6rel0AXBA+ARAH\nDgee73bHUT8HHs4XfhPtA71cH2lARcDMTgaeBF4huDzjwI8IPgz3EHzLWw2c5+4bo4qzWJjZ6cD3\n3P0cMzsQ1XFWmdkYgo6CexDM4n4pQacr1XOWmNl0gi+iHwIvAd8A9kV13C9mdhfBzJzDgfXAdOB+\nYD6d1KuZ/RD4OsH7cLW7L+p2/1EncBEREem9qC+hi4iISB8ogYuIiBQgJXAREZECpAQuIiJSgJTA\nRURECpASuIiISAFSAhcRESlA/x+FsdccYrWhrgAAAABJRU5ErkJggg==\n"
}
},
{
"name": "stdout",
"output_type": "stream",
"text": "良いブレンドを飲める確率= 0.632680052053\n"
}
]
},
{
"metadata": {
"collapsed": true
},
"cell_type": "markdown",
"source": "# 4.0 指数分布(exponential distribution)について\n\n指数分布は生存期間(時間・距離)の確率密度関数である。 \nパラメータの$\\lambda$はポアソン分布の単位期間あたりの生起回数期待値と同じ意味である。 \n例えば平均で2年に1回起こる事象がある。これは年あたり0.5回なので、これが起きた時からの生存時間は$\\lambda$=0.5の指数分布に従う。\nまた平均で1年に4回起こる事象がある。これが起きた時からの生存時間は$\\lambda$=4の指数分布に従う。\n\n1つのパラメータ \n生起間隔$\\lambda$ \n$P(x|\\lambda)=\\lambda e^{-\\lambda x}$ \n\n$E(X)=\\frac{1}{\\lambda}$ \n$V(X)=\\frac{1}{\\lambda ^2}$ \n\nそのグラフを描いてみよう!"
},
{
"metadata": {
"trusted": false,
"collapsed": false
},
"cell_type": "code",
"source": "%matplotlib inline\n%config inlineBackend.figure_format = 'retina'\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef exponential_distribution(xs, lamda):\n return [lamda * np.exp(-lamda * x) for x in xs]\n\nxs = np.arange(0, 5+0.1, 0.1)\ny1 = exponential_distribution(xs, 0.5)\ny2 = exponential_distribution(xs, 1)\ny3 = exponential_distribution(xs, 4)\n\nplt.rcParams[\"figure.figsize\"] = (8,3)\nplt.title('exponential distribution')\nplt.grid(which='major',color='black',linestyle='-') \nplt.plot(xs, y1)\nplt.plot(xs, y2)\nplt.plot(xs, y3)\nplt.show()",
"execution_count": 8,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"text/plain": "<matplotlib.figure.Figure at 0x11edd81d0>",
"image/png": 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OAc4FXjOzlwEHrgemE2T724DPmdnlQAvQAJy1V1EVFMDhh8Pzz8MnP7lXmxIRERns+kzG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\n"
}
}
]
},
{
"metadata": {
"trusted": false,
"collapsed": true
},
"cell_type": "code",
"source": "",
"execution_count": null,
"outputs": []
}
],
"metadata": {
"_draft": {
"nbviewer_url": "https://gist.github.com/4e45e938dd0c3303b34b272082af91a4"
},
"language_info": {
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