{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Lecture 10: Expectation Continued\n", "\n", "\n", "## Stat 110, Prof. Joe Blitzstein, Harvard University\n", "\n", "----" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## A Proof of Linearity (discrete case)\n", "\n", "Let $T = X + Y$, and show that $\\mathbb{E}(T) = \\mathbb{E}(X) + \\mathbb{E}(Y)$.\n", "\n", "We will also show that $\\mathbb{E}(cX) = c \\mathbb{E}(X)$.\n", "\n", "In general, we'd like to be in a position where\n", "\n", "\\begin{align}\n", " \\sum_{t} t P(T=t) \\stackrel{?}{=} \\sum_{x} x P(X=x) + \\sum_{y} y P(Y=y)\n", "\\end{align}\n", "\n", "so, let's try attacking this from the l.h.s." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Considering the image above of a discrete r.v. in Pebble World, note that\n", "\n", "\n", "\\begin{align}\n", " \\mathbb{E}(X) &= \\sum_{x} x P(X=x) & &\\text{grouping the pebbles per X value; weighted average} \\\\\n", " &= \\sum_{s}X(s)P(\\{s\\}) & &\\text{ungrouped; sum each pebble separately} \\\\\n", " \\\\\n", " \\\\\n", " \\Rightarrow \\mathbb{E}(T) &= \\sum_{s} (X+Y)(s)P(\\{s\\}) \\\\\n", " &= \\sum_{s}X(s)P(\\{s\\}) + \\sum_{s}Y(s)P(\\{s\\}) \\\\\n", " &= \\sum_{x} x P(X=x) + \\sum_{y} y P(Y=y) \\\\\n", " &= \\mathbb{E}(X) + \\mathbb{E}(Y) ~~~~ \\blacksquare \\\\\n", " \\\\\n", " \\\\\n", " \\Rightarrow \\mathbb{E}(cX) &= \\sum_{x} cx P(X=x) \\\\\n", " &= c \\sum_{x} x P(X=x) \\\\\n", " &= c \\mathbb{E}(X) ~~~~ \\blacksquare\n", "\\end{align}\n", "\n", "----" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Negative Binomial Distribution\n", "\n", "### Description\n", "\n", "A misnomer: this distribution is actually non-negative, and neither is it binomial.\n", "\n", "The Negative Binomial is a generalization of the Geometric distribution, where we have a series of independent $Bern(p)$ trials and we want to know # failures before the r<sup>th</sup> success.\n", "\n", "We can codify this using a bit string:\n", "\n", "\\begin{align}\n", " & \\text{1000100100001001} & \\text{0 denotes failure, 1 denotes success} & \\\\\n", " & r = 5 \\\\\n", " & n = 11 & \\text{failures} \n", "\\end{align}\n", "\n", "Note that the very last bit position is, of course, a success.\n", "\n", "Note also that we can permutate the preceding $r-1$ successes amongst the $n+r-1$ slots that come before that final r<sup>th</sup> success.\n", "\n", "### Notation\n", "\n", "$X \\sim \\operatorname{NB}(r,p)$\n", "\n", "### Parameters\n", "\n", "* $r$ - the total number of successes before we stop counting\n", "* $p$ - probability of success" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Probability mass function\n", "\n", "\\begin{align}\n", " P(X=n) &= \\binom{n+r-1}{r-1} p^r (1-p)^n & &\\text{for } n = 0,1,2,\\dots\\\\\n", " &= \\binom{n+r-1}{n} p^r (1-p)^n & &\\text{or conversely}\\\\\n", "\\end{align}\n", "\n", "### Expected value\n", "\n", "Let $X_j$ be the # failures before the $(j-1)^{\\text{st}}$ and $j^{\\text{th}}$ success. Then we could write\n", "\n", "\\begin{align}\n", " \\mathbb{E}(X) &= \\mathbb{E}(X_1 + X_2 + \\dots + X_r) \\\\\n", " &= \\mathbb{E}(X_1) + \\mathbb{E}(X_2) + \\dots + \\mathbb{E}(X_r) & &\\text{by Linearity} \\\\\n", " &= r \\mathbb{E}(X_1) & &\\text{by symmetry} \\\\\n", " &= r \\frac{q}{p} ~~~~ \\blacksquare\n", "\\end{align}\n", "\n", "----" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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AGjBggEN9kydPJgB08uRJIspNzyg2SuLfDazr1q2T5cXgwD7P+vjx4wkAXb16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YKU5n6nauQhAlCscn7iHiyinh/HseI+TrIOhuwpaeF9MQqgtHViHzv7eUj35/G+GABy0cRpRl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\nZAhVq1aN1TulUmmgK4fDKQpfTsPhfESYm5s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"text/plain": [ "<matplotlib.figure.Figure at 0x18422719828>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "from matplotlib.ticker import (MultipleLocator, FormatStrFormatter,\n", " AutoMinorLocator)\n", "from scipy.stats import nbinom\n", "\n", "%matplotlib inline\n", "\n", "plt.xkcd()\n", "_, ax = plt.subplots(figsize=(12,8))\n", "\n", "# seme Negative Binomial parameters\n", "r_values = [1, 2, 4, 8]\n", "p_values = [0.25]*len(r_values)\n", "#p_values = [0.25, 0.24, 0.23, 0.22]\n", "params = list(zip(r_values, p_values))\n", "\n", "# colorblind-safe, divergent color scheme\n", "colors = ['#018571', '#80cdc1', '#dfc27d', '#a6611a']\n", "\n", "for i,(r,p) in enumerate(params):\n", " x = np.arange(nbinom.ppf(0.01, r, p), nbinom.ppf(0.99, r, p))\n", " pmf = nbinom.pmf(x, r, p)\n", " ax.plot(x, pmf, 'o', color=colors[i], ms=8, label='r={}, p={}'.format(r,p))\n", " ax.vlines(x, 0, pmf, lw=2, color=colors[i], alpha=0.3)\n", "\n", "# legend styling\n", "legend = ax.legend()\n", "for label in legend.get_texts():\n", " label.set_fontsize('large')\n", "for label in legend.get_lines():\n", " label.set_linewidth(1.5)\n", "\n", "# y-axis\n", "ax.set_ylim([0.0, 0.251])\n", "ax.set_ylabel(r'$P(X=n)$')\n", "\n", "# x-axis\n", "ax.set_xlim([0, 55])\n", "ax.set_xlabel(r'total # of failures $n$ before seeing $r^{th}$ success')\n", "\n", "# x-axis tick formatting\n", "majorLocator = MultipleLocator(5)\n", "majorFormatter = FormatStrFormatter('%d')\n", "minorLocator = MultipleLocator(1)\n", "ax.xaxis.set_major_locator(majorLocator)\n", "ax.xaxis.set_major_formatter(majorFormatter)\n", "ax.xaxis.set_minor_locator(minorLocator)\n", "\n", "ax.grid(color='grey', linestyle='-', linewidth=0.3)\n", "\n", "plt.suptitle(r'Negative Binomial PMF: $P(X=n) = \\binom{n+r-1}{r-1} p^r (1-p)^n$')\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## Revisting the Geometric: the First Success Distribution\n", "\n", "$X \\sim \\operatorname{FS}(p)$ is the geometric distribution that counts the trials until first success, *including that first success*.\n", "\n", "Let $Y = X - 1$. \n", "\n", "Then $Y \\sim \\operatorname{Geom}(p)$\n", "\n", "Expected value of $\\operatorname{FS}(p)$ is\n", "\n", "\\begin{align}\n", " \\mathbb{E}(X) &= E(Y) + 1 \\\\\n", " &= \\frac{q}{p} + 1 \\\\\n", " &= \\boxed{\\frac{1}{p}}\n", "\\end{align}\n", "\n", "----" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## Putnam Problem\n", "\n", "Consider a random permutation of $1, 2, 3, \\dots , n$, where $n \\ge 2$.\n", "\n", "Find expected # local maxima. For example, given the permuation $\\boxed{3} ~~ 2 ~~ 1 ~~ 4 ~~ \\boxed{7} ~~ 5 ~~ \\boxed{6}$ we have 3 local maxima:\n", "\n", "- $\\boxed{3} \\gt 2$\n", "- $4 \\lt \\boxed{7} \\gt 5$\n", "- $ 5 \\lt \\boxed{6}$\n", "\n", "Now, there are 2 kinds of cases we need to consider:\n", "\n", "- non-edge case: $4 ~~ \\boxed{7} ~~ 5$ has probability of $\\frac{1}{3}$ that the largest number is in the middle position\n", "- edge case: in both left-edge $\\boxed{3} ~~ 2$ and right-edge $5 ~~ \\boxed{6}$, the probability that the larger number is in the right position is $\\frac{1}{2}$\n", "\n", "Let $I_j$ be the indicator r.v. of position $j$ having a local maximum, $1 \\le j \\le n$.\n", "\n", "Using Linearity, we can say that the expected number of local maxima is given by\n", "\n", "\\begin{align}\n", " \\mathbb{E}(I_j) &= \\mathbb{E}(I_1 + I_2 + \\dots + I_n) \\\\\n", " &= \\mathbb{E}(I_1) + \\mathbb{E}(I_2) + \\dots + \\mathbb{E}(I_n) & &\\text{by Linearity} \\\\\n", " &= (n-2) \\frac{1}{3} + 2 \\frac{1}{2} \\\\\n", " &= \\boxed{\\frac{n+1}{3}}\n", "\\end{align}\n", "\n", "Idiot-checking this, we have:\n", "\n", "\\begin{align}\n", " \\mathbb{E}(I_{n=2}) &= \\frac{2+1}{3} & &\\text{... case where } n=2 \\\\\n", " &= 1 \\\\\n", " \\\\\n", " \\\\\n", " \\mathbb{E}(I_{n=\\infty}) &= \\frac{\\infty+1}{3} & &\\text{... case where } n= \\infty \\\\\n", " &= \\infty \\\\\n", "\\end{align}\n", "\n", "----" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## St. Petersburg Paradox\n", "\n", "Consider a game of chance involving a fair coin. We will flip the coin until the very first heads shows (hypergeometric distribution). \n", "\n", "- If heads shows on the very first flip, you get $\\$2$.\n", "- If the first heads shows on the second flip, you get $\\$4$.\n", "- If the first heads shows on the third flip, you get $\\$8$.\n", "\n", "So you will get $\\$2^n$ if the first heads shows up on the n<sup>th</sup> trial, including the heads flip.\n", "\n", "_How much would you be willing to pay to play this game?_\n", "\n", "Let's tackle this by thinking about the expected amount of $\\$\\$\\$$ we stand to make. \n", "\n", "Given $Y = 2^n$, find $\\mathbb{E}(Y)$:\n", "\n", "\\begin{align}\n", " \\mathbb{E}(Y) &= \\sum_{k=1}^\\infty 2^k \\frac{1}{2^{k-1}} ~ \\frac{1}{2}\\\\\n", " &= \\sum_{k=1}^\\infty 2^k \\frac{1}{2^k}\\\\\n", " &= \\sum_{k=1}^\\infty 1\\\\\n", " \\\\\n", " \\\\\n", " \\mathbb{E}(Y_{k=40}) &= \\sum_{k=1}^{40} 1 \\\\\n", " &= 40\n", "\\end{align}\n", "\n", "So, the \"paradox\" here is that even if we capped the payout to $2^{40} \\approx \\$1000000000$, Linearity shows us we would only pay $40. It is very hard to grasp this, but the truth is that if you were offered this game at any price, you should take it.\n", "\n", "----" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "View [Lecture 10: Expectation Continued | Statistics 110](http://bit.ly/2vXxPsj) on YouTube." ] } ], "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": "3.6.3" } }, "nbformat": 4, "nbformat_minor": 1 }