{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Chapter 10: Inequalities and limit theorems\n", " \n", "This Jupyter notebook is the Python equivalent of the R code in section 10.6 R, pp. 447 - 450, [Introduction to Probability, 1st Edition](https://www.crcpress.com/Introduction-to-Probability/Blitzstein-Hwang/p/book/9781466575578), Blitzstein & Hwang.\n", "\n", "----" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Jensen's inequality\n", "\n", "Python/NumPy/SciPy make it easy to compare the expectations of $X$ and $g(X)$ for a given choice of $g$, and this allows us to verify some special cases of Jensen's inequality. For example, suppose we simulate 104 times from the $Expo(1)$ distribution:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "np.random.seed(24157817)\n", "\n", "from scipy.stats import expon\n", "\n", "x = expon.rvs(size=10**4)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "According to Jensen's inequality, $\\mathbb{E}(log \\, X) \\leq log \\, \\mathbb{E} \\, X$. The former can be approximated by numpy.mean(numpy.log(x)) and the latter can be approximated by numpy.log(numpy.mean(x)), so compute both" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "numpy.mean(numpy.log(x)) = -0.5600958563379892\n", "numpy.log(numpy.mean(x)) = 0.00800014338803644\n" ] } ], "source": [ "meanlog = np.mean(np.log(x))\n", "print('numpy.mean(numpy.log(x)) = {}'.format(meanlog))\n", "\n", "logmean = np.log(np.mean(x))\n", "print('numpy.log(numpy.mean(x)) = {}'.format(logmean))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the $Expo(1)$ distribution, we find that numpy.mean(numpy.log(x)) is approximately −0.56 (the true value is around −0.577), while numpy.log(numpy.mean(x)) is approximately 0 (the true value is 0). This indeed suggests $\\mathbb{E}(log \\, X) \\leq log \\, \\mathbb{E} \\, X$. We could also compare numpy.mean(x**3) to numpy.mean(x)**3, or numpy.mean(numpy.sqrt(x)) to numpy.sqrt(numpy.mean(x)) - the possibilities are endless." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Visualization of the law of large numbers\n", "\n", "To plot the running proportion of Heads in a sequence of independent fair coin tosses, we first generate the coin tosses themselves:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "np.random.seed(39088169)\n", "\n", "from scipy.stats import binom\n", "\n", "nsim = 300\n", "p = 1/2\n", "x = binom.rvs(1, p, size=nsim)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Then we compute $\\bar{X}_n$ for each value of $n$ and store the results in xbar:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "# divide by sequence from 1 to nsim, inclusive\n", "xbar = np.cumsum(x) / np.arange(1, nsim+1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The above line of code performs element-wise division of the two arrays numpy.cumsum(x) and np.arange(1, nsim+1). Finally, we plot xbar against the number of coin tosses:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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5c6azjXz22WcAbNy4ke+++w6AH374gR9//BGAb7/9lo0bNwLw6aefOtvIBx98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\n", 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