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@TheDataLeek
Created October 27, 2016 23:54
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{
"cells": [
{
"cell_type": "code",
"execution_count": 57,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"import scipy.stats as st\n",
"import sympy as sp\n",
"from sympy import symbols\n",
"import matplotlib.pyplot as plt\n",
"import seaborn\n",
"\n",
"sp.init_printing()\n",
"\n",
"%matplotlib inline"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Solve the following problems in Section 1.12 of your textbook:\n",
"\n",
"## 1.58(b)\n",
"\n",
"**Six children (Dick, Helen, Joni, Mark, Sam, and Tony) play catch. If Dick has the ball he is equally likely to throw it to Helen, Mark, Sam, and Tony. If Helen has the ball she is equally likely to throw it to Dick, Joni, Sam, and Tony. If Sam has the ball he is equally likely to throw it to Dick, Helen, Mark, and Tony. If either Joni or Tony gets the ball, they keep throwing it to each other. If Mark gets the ball he runs away with it. (b) Suppose Dick has the ball at the beginning of the game. What is the probability Mark will end up with it?**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's first write out our transition probabilities."
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# D H J M S T\n",
"p = np.array([[0, 0.25, 0, 0.25, 0.25, 0.25], # D\n",
" [0.25, 0, 0.25, 0, 0.25, 0.25], # H\n",
" [0, 0, 0, 0, 0, 1], # J\n",
" [0, 0, 0, 1, 0, 0], # M\n",
" [0.25, 0.25, 0, 0.25, 0, 0.25], # S\n",
" [0, 0, 1, 0, 0, 0]]) # T"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can verify this numerically."
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"array([ 0. , 0. , 0.41111111, 0.4 , 0. ,\n",
" 0.18888889])"
]
},
"execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dist = np.array([1, 0, 0, 0, 0, 0])\n",
"for _ in range(100000):\n",
" dist = dist @ p\n",
"dist"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1.77\n",
"\n",
"**Consider a branching process as defined in Example 1.8, in which each family has a number of children that follows a shifted geometric distribution: $p_k = p {(1 - p)}^k$ for $k \\ge 0$, which counts the number of failures before the first success when success has probability $p$. Compute the probability that starting from one individual the chain will be absorbed at 0.**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1.75\n",
"\n",
"**The opposite of the aging chain is the renewal chain with state space\n",
"$\\{0, 1, 2, \\ldots\\}$ in which $p(i, i - 1) = 1$ when $i > 0$. The only nontrivial part of the transition probability is $p(0, i) = p_i$. Show that this chain is always recurrent but is positive recurrent if and only if $\\sum_n n p_n < \\infty$**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1.73\n",
"\n",
"**Consider the Markov chain with state space $\\{1, 2, \\ldots\\}$ and transition probability**\n",
"\n",
"$$\n",
"\\begin{cases}\n",
"p(m, m + 1) = \\frac{m}{2m + 2} \\quad & m \\ge 1\\\\\n",
"p(m, m - 1) = \\frac{1}{2} \\quad & m \\ge 2\\\\\n",
"p(m, m) = \\frac{1}{2m + 2} \\quad & m \\ge 2\n",
"\\end{cases}\n",
"$$\n",
"\n",
"**and $p(1, 1) = 1 - p(1, 2) = 3/4$. Show that there is no stationary distribution.**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# In addition, solve the following problems in Section 2.6 of your textbook:\n",
"\n",
"## 2.1\n",
"\n",
"**Suppose that the time to repair a machine is exponentially distributed random\n",
"variable with mean 2.**\n",
"\n",
"### A\n",
"\n",
"**What is the probability the repair takes more than 2h.**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is just an exponential distribution with parameter $\\lambda = \\frac{1}{2}$, with pdf\n",
"\n",
"$$\n",
"f(x) = \\frac{1}{2} e^{-\\frac{x}{2}}\n",
"$$\n",
"\n",
"And corresponding cdf\n",
"\n",
"$$\n",
"F(x) = 1 - e^{-\\frac{x}{2}}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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r2Nd2+eXT+fjjj92OuSkM48ySGyF8VVFlMcv3/8g/1j3LI6ue4tM9X9YkDlaL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\nrkJggg==\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x7f589c470be0>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x = np.linspace(0, 4, 1000)\n",
"pdf = lambda l, x: l * np.exp(-l * x)\n",
"cdf = lambda l, x: 1 - np.exp(-l * x)\n",
"plt.figure()\n",
"plt.plot(x, pdf(1/2, x), label=r'PDF with $\\lambda = \\frac{1}{2}$')\n",
"plt.plot(x, cdf(1/2, x), label=r'CDF with $\\lambda = \\frac{1}{2}$')\n",
"plt.title('Exponential Distribution')\n",
"plt.legend(loc=0)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So the question is basically asking for $P(x > 2) \\equiv 1 - P(x \\le 2) = 1 - 1 + e^{-1}$"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAIsAAAAPBAMAAAAmFiECAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\naf9h/QUsUMpODc7VUwAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$0.367879441171$$"
],
"text/plain": [
"0.367879441171"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.exp(-1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Therefore it has probability of 0.368."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### B\n",
"\n",
"**What is the probability that the repair takes more than 5h given that it takes more than 3h.**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So this is asking $P(x > 5 \\,\\vert\\, x > 3)$. Unfortunately these two events are not independent, so we can't do a fancy trick here. By the definition of conditional probability however, we can rewrite this as the following\n",
"\n",
"$$\n",
"P(x > 5 \\,\\vert\\, x > 3) = \\frac{P(x > 5 \\cap x > 3)}{P(x > 3)}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To understand what is happening with the numerator, let's graph it."
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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a28QlM+oIZM/geGZNB5/97nKee2m/loOKSNFQiBAZAY7jMKmpktde2kZbfTkA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\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x7f588d72c908>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x = np.linspace(2, 8, 1000)\n",
"p1 = np.linspace(3, 8, 1000)\n",
"p2 = np.linspace(5, 8, 1000)\n",
"pdf_vals = pdf(1/2, x)\n",
"plt.figure()\n",
"plt.plot(x, pdf_vals)\n",
"plt.fill_between(x, 0, pdf_vals, where=pdf_vals < pdf(1/2, 3), alpha=0.5)\n",
"plt.fill_between(x, 0, pdf_vals, where=pdf_vals < pdf(1/2, 5), alpha=0.5)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The spot where they overlap is simply $P(x > 5)$.\n",
"\n",
"Therefore our answer is\n",
"\n",
"$$\n",
"P(x > 5 \\,\\vert\\, x > 3) = \\frac{P(x > 5 \\cap x > 3)}{P(x > 3)} = \\frac{P(x > 5)}{P(x > 3)}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAIsAAAAPBAMAAAAmFiECAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\naf9h/QUsUMpODc7VUwAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$0.367879441171$$"
],
"text/plain": [
"0.367879441171"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.exp(-5/2) / np.exp(-3/2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 2.2\n",
"\n",
"**The lifetime of a radio is exponentially distributed with mean 5 years. If Ted buys a 7 year-old radio, what is the probability it will be working 3 years later?**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Again, this is simply the exponential distribution with $\\lambda = \\frac{1}{5}$. All we're interested in is $P(x > 10 \\,\\lvert\\, x > 7)$. So same question as before.\n",
"\n",
"Using the same approach, let's rewrite this\n",
"\n",
"$$\n",
"P(x > 10 \\,\\vert\\, x > 7) = \\frac{P(x > 10 \\cap x > 7)}{P(x > 7)} = \\frac{P(x > 10)}{P(x > 7)}\n",
"$$\n",
"\n",
"This is literally the exact same, just with different parameters."
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAIwAAAAPBAMAAADEyjp7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9S/lTHOT/JWdXxOSU73katT4/88fvvHgOySKRUYAAAAASUVORK5CYII=\n",
"text/latex": [
"$$0.548811636094$$"
],
"text/plain": [
"0.548811636094"
]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.exp(-10/5) / np.exp(-7/5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So there is a 55% chance that it will still work."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 2.5\n",
"\n",
"**Three people are fishing and each catches fish at rate 2 per hour. How long do we have to wait until everyone has caught at least one fish?**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can model this as three independent random variables, $Y_1, Y_2,$ and $Y_3$ where each is Poisson-distributed with $\\lambda = 2$."
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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FTYqrFmWZVgd5oZsv3GLe22/nvYPneWNvLe0XRwZhzk4wVsK8cVV+yO8SG24xjwQSc/NJ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\nl/8P9UZ4Yqw9W0cAAAAASUVORK5CYII=\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x7f588d2b3c50>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"rv = st.poisson(2)\n",
"x = np.arange(6)\n",
"fig, axarr = plt.subplots(2, 1)\n",
"axarr[0].plot(x, rv.pmf(x))\n",
"axarr[0].set_title('PMF')\n",
"axarr[1].plot(x, rv.cdf(x))\n",
"axarr[1].set_title('CDF')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The probability of someone catching at least 1 fish"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAIwAAAAPBAMAAADEyjp7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAACb0lEQVQ4Ea2Tv2tTURTHPy/vxTRNXhMtiHRo\nYguCUuFhWhCF8lwEpwZdHISmoDhUoUsXKSQg6NAOAZ1cTKcOVQwuoiKGWhcJNrjopMGCg4O1Whpt\nq/Gce9P+BT5435x7vud+7o93AgeGR9Cnf/W4aG41L3peXndU8kaIZonODeZyktVHkxeXcrnA2ndy\nTyR5lUNlNbMkpzjYiBQkngXnOkesQFcVr91u78CNdMcZk3HN1MSzFEO8Cq7OjIXECzwmUYWeRUhk\neW4FvlaJyFI1Z2lFMMY5Ckkb9bRIlUjUiK0Lxq3ilPzfEsHdSfgk86wQP1vFlYo8rAnGOAGM2Mh/\nSKZAqobfkhn+dtoNExWJIBDMMw2M4O6r6uCjvIqxSfzmbqSHymTx/2rVysYwqSujcs9eWTCbFz43\nrHDOYpSlGOPI2fYi5ybcD4j+VIzbniBzme4p+pnE2QxZMILTNBg3tBibhGFbA/GThwUjH9NgZj5s\nNTLrROYJFNNOc2y/StrFYDK6luzGOPLzfTeCU429Q3k11uZTJbw/sbxg+AVjDSPTFvOlg7EOPYVO\njaS75/WKY3rF3WWira4C3kYvirklmFDlTGAxs7sY42gv2Zp4mURL+8TTD657Xk5WZDcz9frWW97o\nblTe1+srD5o40nz2io1DpoatSa0rRtovUpAC2Q1Po3I3FRkswrhcixXxZOXoD8mbL2UcxrOdGjG7\npN1u05d3NvEn8Jq8om9KqrelRwNnwQqkBBPbwxiHomBM5IUUm9D77TXMwaVBaRl3QAacbi/DiaF8\nR9yXO/JPWRBj6NG1sJO8p8uZmncDLyT8L88/i3bYn18AzzUAAAAASUVORK5CYII=\n",
"text/latex": [
"$$0.864664716763$$"
],
"text/plain": [
"0.864664716763"
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rv = st.poisson(2)\n",
"1 - rv.cdf(0)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 2.7\n",
"\n",
"**Let $S$ and $T$ be exponentially distributed with rates $\\lambda$ and $\\mu$. Let $U=\\min\\{S, T\\}$ and $V = \\max\\{S, T\\}$.**\n",
"\n",
"### A\n",
"\n",
"**Find $EU$.**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### B\n",
"\n",
"**Find $E(V - U)$.**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### C\n",
"\n",
"**Find $EV$.**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### D\n",
"\n",
"**Use the identity $V = S + T - U$ to get a different looking formula for $EV$ and verify the two are equal.**"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.5.2"
}
},
"nbformat": 4,
"nbformat_minor": 1
}
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