{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Lecture 28: Sum of a random number of random variables; Inequalities (Cauchy-Schwarz, Jensen, Markov, Chebyshev)\n", "\n", "\n", "## Stat 110, Prof. Joe Blitzstein, Harvard University\n", "\n", "----" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Sum of Random Number of Random Variables\n", "\n", "Let's say there's a store. At this store, there is a random number of customers. Each customer chooses to spend a random amount of money at the store. What can we say about the expected total amount spent at the store? What about its variance?\n", "\n", "* $N$ is the number of customers (maybe Poisson? it doesn't matter at this point)\n", "* $X_j$ be the amount customer $j$ spends\n", "* $X_j$ has mean $\\mu$ and variance $\\sigma^2$\n", "* assume that $N, \\, X_1, \\, N_2, \\dots$ are independent\n", "\n", "Find the mean and variance of $X = \\sum_{j=1}^{N} X_j$.\n", "\n", "Be careful, because $N$ is itself a random variable, and so we are summing up a _random number of random variables_. At this point, $N$ could be any distribution, perhaps Poisson, but we can put that question aside for now. Let's see if we can find some statistics about $X$ in terms of random variable $N$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $\\mathbb{E}(X)$\n", "\n", "Now, if we blindly tried to apply linearity, we might have:\n", "\n", "\\begin{align}\n", " \\mathbb{E}(X) &=? \\, N \\, \\mu &\\quad \\text{... no, this cannot be right}\n", "\\end{align}\n", "\n", "But this is a category error, since the right-hand side of that equation is a _value_, while the left-hand side is a _random variable_. However, it does give us a clue: all we need to do is find a way to treat $N$ with a _value_, which we can do by thinking conditionally.\n", "\n", "There are two ways we can use to find $\\mathbb{E}(X)$:\n", "\n", "\\begin{align}\n", " &\\text{...conditioning on } N \\text{, the long way} \\\\\n", " \\mathbb{E}(X) &= \\sum_{n=0}^{\\infty} \\mathbb{E}(X|N=n) \\, P(N=n) &\\quad \\text{conditioning on }N \\\\\n", " &= \\sum_{n=0}^{\\infty} \\mu \\, n \\, P(N=n) &\\quad \\text{by linearity, independence} \\\\\n", " &= \\mu \\, \\sum_{n=0}^{\\infty} n \\, P(N=n) &\\quad \\text{but by definition of }\\mathbb{E} \\\\\n", " &= \\boxed{\\mu \\, \\mathbb{E}(N)} \\\\\\\\\n", " \\\\\n", " & \\text{... or using Adam's Law} \\\\\n", " \\mathbb{E}(X) &= \\mathbb{E}\\left( \\mathbb{E}(X|N) \\right) &\\quad \\text{Adam's Law} \\\\\n", " &= \\mathbb{E}(\\mu \\, N) &\\quad \\text{treating } N \\text{ as constant, and by linearity} \\\\\n", " &= \\boxed{\\mu \\, \\mathbb{E}(N)} \n", "\\end{align}\n", "\n", "Note that unlike the 2-envelope paradox covered in lectures 25 and 26, we _can_ forget about the condition $N=n$ _because of the independence of $X_j$_. That fact allows us to use linearity in the second step of finding $\\mathbb{E}(X)$ the long way by conditioning on $N$. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $\\operatorname{Var}(X)$\n", "\n", "Here, we will just apply EVvE's Law:\n", "\n", "\\begin{align}\n", " \\operatorname{Var}(X) &= \\mathbb{E}\\left( \\operatorname{Var}(X|N) \\right) + \\operatorname{Var}\\left( \\mathbb{E}(X|N) \\right) \\quad \\text{and working on each part} \\\\\\\\\n", " \\mathbb{E}\\left( \\operatorname{Var}(X|N) \\right) &= \\mathbb{E}(N \\sigma^2) \\quad \\text{treating } N \\text{ as fixed, independence} \\\\\n", " &= \\sigma^2 \\, \\mathbb{E}(N) \\\\\n", " \\\\\n", " \\operatorname{Var}\\left( \\mathbb{E}(X|N) \\right) &= \\operatorname{Var}(\\mu \\, N) \\quad \\text{} \\\\\n", " &= \\mu^2 \\, \\operatorname{Var}(N) \\\\\\\\\n", " \\Rightarrow \\operatorname{Var}(X) &= \\boxed{ \\sigma^2 \\, \\mathbb{E}(N) + \\mu^2 \\, \\operatorname{Var}(N) }\n", "\\end{align}\n", "\n", "Idiot-checking that result, we confirm that\n", "\n", "* $\\mathbb{E}(X) = \\mu \\, \\mathbb{E}(N)$ means that the average total sales is the average per-customer spending amount times the average number of customers\n", "* $\\sigma^2$ is in terms of money-squared; $\\mu$ is in terms of money, so $\\operatorname{Var}(X)$ is in terms of money-squared\n", "* both $\\mathbb{E}(N)$ and $\\operatorname{Var}(N)$ are dimensionless values (\"people\" are not units in this case)\n", "* if we knew the MGF of each $X_j$, then we could condition on $N$ to treat it as fixed in order to get the MGF of $X$ (multiply by $n$)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Statistical Inequality\n", "\n", "In the field of law, it is a lot easier to stand in court to use statistical inequalities rather than approximations.\n", "\n", "Basically, approximations could be painted to be subjective. If you use a Poisson approximation and had to explain yourself in court, you might be hard-pressed to justify why you think your approximation is good. \"Good\" is subject to interpretation; we would claim that an approximation is good if it is close to the truth. But a lawyer might then grill on that, asking you \"Is there some accepted standard to closeness? Do you know how close your approximation is?\"\n", "\n", "Now if you knew that, then you would already *know* the actual distribution and you wouldn't be using an approximation in the first place!\n", "\n", "...\n", "\n", "That sort of argument probably would not fly very well in front of the judge.\n", "\n", "But inequalities are pretty much unassailable, as they provide a _definite fact about something that is random_." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### (1) Cauchy-Schwarz Inequality: $\\lvert \\mathbb{E}(XY) \\rvert \\le \\sqrt{\\mathbb{E}(X^2)(Y^2)}$\n", "\n", "Recall that $\\mathbb{E}(XY)$ is an analog of the dot-product for random variables $X,Y$.\n", "\n", "Recall the definition of a dot-product in linear algebra.\n", "\n", "\\begin{align}\n", " \\textbf{a} \\cdot \\textbf{b} &= \\lVert \\mathbf{a} \\rVert \\lVert \\mathbf{b} \\rVert \\mathbf{cos}\\theta \\\\\n", " \\\\\n", " \\mathbb{E}(XY) &\\le \\sqrt{\\mathbb{E}(X^2)(Y^2)}\n", "\\end{align}\n", "\n", "Note that $X,Y$ are uncorrelated, then by definition $\\mathbb{E}(XY) = \\mathbb{E}(X)\\mathbb{E}(Y)$, and so using the Cauchy-Schwarz Inequality is really unnecessary.\n", "\n", "The Cauchy-Schwarz Inequality really comes in handy when the random variables $X,Y$ are correlated. In such a case where $X,Y$ are correlated and you are interested in $\\mathbb{E}(XY)$, then without using a 2D LOTUS or calculating the distribution of $XY$, we can still find an upper bound to $\\mathbb{E}(XY)$.\n", "\n", "#### Statistical Interpretation\n", "\n", "The interpretation is easiest to see if we let $X,Y$ have mean 0.\n", "\n", "Then:\n", "\\begin{align}\n", " \\lvert \\operatorname{Corr}(X,Y) \\rvert &= \\lvert \\frac{\\operatorname{Cov}(X,Y)}{\\sqrt{\\operatorname{Var}(X)} \\sqrt{\\operatorname{Var}(Y)}} \\rvert \\\\\n", " &= \\lvert \\frac{\\mathbb{E}(XY) - \\mathbb{E}(X)\\mathbb{E}(Y)}{\\sqrt{\\mathbb{E}(X^2)-(\\mathbb{E}(X))^2}\\sqrt{\\mathbb{E}(Y^2)-(\\mathbb{E}(Y))^2}} \\rvert \\\\\n", " &= \\lvert \\frac{\\mathbb{E}(XY)}{\\left(\\mathbb{E}(X^2) \\mathbb{E}(Y^2)\\right)^{\\frac{1}{2}}} \\rvert \\le 1\n", "\\end{align}\n", "\n", "... and the left-hand side of the above equation is *exactly* the Cauchy-Schwarz Inequality. Note that this would hold for any value of mean for $X,Y$.\n", "\n", "So for statistics, the Cauchy-Schwarz Inequality really just means that the correlation of two random variables will be between -1 and 1. For providing upper bounds to $\\mathbb{E}(XY)$, the Cauchy-Schwarz Inequality's strengths are simplicity and generality." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### (2) Jensen's Inequality: if $g$ is a convex function, then $\\mathbb{E} g(X) \\ge g(\\mathbb{E}X)$\n", "\n", "\\begin{align}\n", " \\mathbb{E}g(x) &\\le g(\\mathbb{E}X) &\\quad \\text{if }g(x) \\text{ is concave} \\\\\\\\\n", " \\mathbb{E}h(x) &\\ge h(\\mathbb{E}X) &\\quad \\text{if }h(x) \\text{ is convex} \n", "\\end{align}\n", "\n", "This tells you when $g$ is a convex function which way the inequality goes.\n", "\n", "Now, a convex function is such that $g^{\\prime\\prime}(x) \\ge 0$ (if $g^{\\prime\\prime}(x)$ exists). Another way of describing it is that if you pick any two points on $g(x)$, the line segment they make will lay above $g(x)$.\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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03nnn0dixY+mss86iBQsWKJXgPXv2hF34RNS4Y7zrrrtU445+9913ES+aWbdu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2grEdhWncRJbKOdsF76+tKP8VdDW30bRNv1EB5f8q6FAMTXkSwp4coW+Uh+HHbfmhN2S70Cf7zJ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\nwsICcrkcMpkMlpaWkM1mRdxJt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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "%matplotlib inline \n", "\n", "plt.xkcd()\n", "\n", "x = np.linspace(-10, 10, 1000)\n", "y = x**2\n", "x2 = [-4,7]\n", "y2 = [16,49]\n", "\n", "_, ax = plt.subplots(figsize=(12,8))\n", "ax.axhline(y=0, color='k')\n", "ax.axvline(x=0, color='k')\n", "\n", "ax.set_xticks([])\n", "ax.set_yticks([])\n", "\n", "plt.text(2.0, 80.0, '$g(x)=x^2$,\\n$g^{\\prime\\prime}(x)=2$', color='xkcd:azure')\n", "plt.text(-5.5, 10.0, 'A')\n", "plt.text(7.5, 42.5, 'B')\n", "\n", "ax.plot(x, y, 'xkcd:azure', lw=4, alpha=0.6)\n", "ax.plot(x2,y2,'xkcd:orange', linestyle='--', lw=4, alpha=0.6)\n", "\n", "plt.suptitle(r'Convexity: $\\overline{AB}$ lies above $g(x)$ for all points $A,B$')\n", "\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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F0Rp22bjb7XZYrVaYzWbo9Xro9XooigJFUSCEgKqqSCQSiMfj2oFCNnQyNHu9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\nAbW1tdaP7Ha74fP54PV6UVNTA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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = np.linspace(0.001, 5, 1000)\n", "y = 1/x\n", "x2 = np.linspace(0.001,5,1000)\n", "y2 = np.log(x2)\n", "\n", "_, ax = plt.subplots(figsize=(12,8))\n", "ax.axhline(y=0, color='k')\n", "ax.axvline(x=0, color='k')\n", "\n", "ax.set_xticks([])\n", "ax.set_yticks([])\n", "ax.set_xlim([-2,5])\n", "ax.set_ylim([-2,5])\n", "ax.text(0.85, 2.2, (r'$\\mathbb{E}(\\frac{1}{X}) \\geq \\mathbb{E}(X)$' '\\n$X$ is positive, $g(X)$ is convex'), color='xkcd:azure')\n", "ax.text(0.85, -1.2, '$\\mathbb{E}(\\ln X) \\leq \\ln \\mathbb{E}(X)$\\n$h(X)$ is concave, inequality flips', color='xkcd:orange')\n", "ax.plot(x, y, 'xkcd:azure', lw=4, alpha=0.6)\n", "ax.plot(x2, y2, 'xkcd:orange', lw=4, alpha=0.6)\n", "\n", "plt.suptitle('Examples: Jensen\\'s Inequality')\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Proof\n", "\n", "We will try a graphical approach to prove Jensen's Inequality." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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o1+r4a3j89UK/MdPpT7b0rgL6j3690B9d6121ioqKjOyofg3XM83peuo3Xnpj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waplyClrEeRiIiIxlUtcqetLg1jHfyeiGojnU4DAMLhcGVZZTa1QhLKGZ+E8LNvPRERTR21yJ22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\nOp0O2u02j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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = np.linspace(-10, 10, 1000)\n", "y = x**2\n", "\n", "f = 1 + 2*(x - 1)\n", "\n", "_, ax = plt.subplots(figsize=(12,8))\n", "ax.axhline(y=0, color='k')\n", "ax.axvline(x=0, color='k')\n", "\n", "ax.text(1.9, 2.2, ('$y = a + bx$'), color='xkcd:orange')\n", "ax.text(1.22, 0.95, ('($x, g(x))$'))\n", "\n", "ax.set_xticks([])\n", "ax.set_yticks([])\n", "ax.set_xticks([])\n", "ax.set_yticks([])\n", "ax.set_xlim([-2,4])\n", "ax.set_ylim([-2,4])\n", "ax.plot(x, y, 'xkcd:azure', lw=4, alpha=0.6)\n", "ax.plot(x, f, 'xkcd:orange', linestyle='--', lw=4, alpha=0.6)\n", "\n", "plt.suptitle(r'Line $y$ tangent to $g(x)$')\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Referring to the figure above...\n", "\n", "\\begin{align}\n", " g(x) &\\geq a + bx &\\quad g(x) \\text{ lies above the tangent } y = a + bx \\\\ \n", " \\Rightarrow g(X) &\\geq a + bX &\\quad \\text{inequality applies to r.v. as well} \\\\\n", " \\mathbb{E}\\left( g(X) \\right) &\\geq \\mathbb{E}\\left( a + bX \\right) &\\quad \\mathbb{E} \\text{ as operator} \\\\\\\\\n", " \\mathbb{E}\\left( a + bX \\right) &= a + b \\, \\mathbb{E}(X) \\\\\n", " &= a + b \\, \\mu &\\quad \\text{let } \\mu = \\mathbb{E}(X) \\\\ \n", " &= g(\\mu) \\\\\n", " &= g\\left(\\mathbb{E}(X)\\right) \\\\\\\\\n", " \\therefore \\mathbb{E}\\left( g(X) \\right) &\\geq g\\left(\\mathbb{E}(X)\\right) &\\quad \\blacksquare\n", "\\end{align}\n", "\n", "It is also possible to use Taylor expansions to prove Jensen's Inequality, but that is something for another time." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### (3) Markov's Inequality: $P(|X| \\geq a) \\leq \\frac{\\mathbb{E}|X|}{a}$ for any $a \\gt 0$ \n", "\n", "The strength of Markov's Inequality is not in its accuracy of _approximation_, but rather in it simplicity and generality.\n", "\n", "#### Proof\n", "\n", "We will use the fundamental bridge to convert the probability of an event to the expected value of the indicator of said event.\n", "\n", "\\begin{align}\n", " I_{|X| \\geq a} &\\geq |X| &\\quad \\text{always true for } I_{|X| \\geq a} = 0 \\text{ or } I_{|X| \\geq a} = 1 \\\\\n", " a \\, \\mathbb{E} \\, I_{|X| \\geq a} \\geq \\mathbb{E} \\, |X| \\\\\n", " \\mathbb{E} \\, I_{|X| \\geq a} \\geq \\frac{\\mathbb{E} \\, |X|}{a}\n", "\\end{align}\n", "\n", "#### Intuition\n", "\n", "Suppose that we have 100 people.\n", "\n", "Q1: Is it possible that at least 95% of the people are younger than the mean age of the group?\n", "\n", "A1: Yes, if one of those people is really, really old, then that person's age can raise the mean age above a large portion of group.\n", "\n", "Q2: Is it possible that at least 50% of the people are older than *twice* the mean age?\n", "\n", "A2: No, since that 50% will raise the mean age beyond that!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### (4) Chebyshev's Inequality: $P(|X-\\mu| \\leq \\frac{Var(X)}{a^2}$ for $\\mu = \\mathbb{E}X, a \\gt 0$\n", "\n", "An alternate way of putting Chebyshev's Inequality is\n", "\n", "\\begin{align}\n", " P\\left(|X-\\mu| > c \\, SD(X) \\right) \\leq \\frac{1}{c^2} &\\quad \\text{for } c > 0 \\\\\n", "\\end{align}\n", "\n", "Consider this: the probability that $X$ is more than 2 standard deviations from its mean is at most $\\frac{1}{4}$. Recall the Normal distribution and the 68,95,98.7 percent rule? Compared to that rule, Chebyshev's Inequality provides a cruder but still correct relation.\n", "\n", "#### Proof\n", "\n", "\\begin{align}\n", " P(|X-\\mu| \\ge a) &= P\\left( (X-\\mu)^2 \\ge a^2 \\right) \\\\\n", " &\\leq \\frac{\\mathbb{E}(X-\\mu)^2}{a^2} &\\quad \\text{by Markov's Inequality} \\\\\n", " &\\leq \\frac{Var(X)}{a^2} &\\quad \\blacksquare\n", "\\end{align}\n", "\n", "----" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "View [Lecture 28: Inequalities | Statistics 110](http://bit.ly/2CzCOoM) 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": 2 }