{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Introduction to Central Limit Theorem" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is a notebook to learn about the central limit theorem. The basic idea is to draw N random numbers $\\{x_i\\}$ (for $i=1\\ldots N$) from some probability distribution $p(x)$ and calculate the sum $y=\\sum_{i=1}^N x_i$.Note that in general that $y$ is a random variable. These means that if I draw a different set of $M$ numbers, I will get a slightly different value for $y$. \n", "\n", "In statstical physics, we are often interested in the behavior of such extensive variables (variables that scale with $N$). We would like to understand its average value, its fluctuations , and how these scale with $N$.\n", "\n", "In this notebook, we will try to get an intuition for this by repeatedly calculating $y$ for different draws of $N$ random. Let $y_\\alpha$ (with $\\alpha=1\\ldots M$) be the sum on $\\alpha$'th time I draw $N$ numbers. Then, we can make a histogram of these $y_\\alpha$. This historgram tells us about the probability of observing a $y_\\alpha$. \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Binary Variables\n", "\n", "We now perform this when the $x_i$ are binary variables with $x_i=\\pm 1$ with\n", "$$\n", "p(x_i=1)=q\\\\\n", "p(x_i=0)=1-q\n", "$$\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "('The empirical mean is', 90.090000000000003)\n", "('The empirical std is', 3.1814933600433615)\n" ] } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "import seaborn as sns\n", "%matplotlib inline \n", "#Draw M sets of N random numbers\n", "\n", "N=100\n", "M=100\n", "\n", "q=0.9\n", "\n", "Data=np.random.binomial(1,q,(M,N))\n", "\n", "#Draw from random distribution\n", "#mean=10;\n", "#sigma=5;\n", "#Data=np.random.normal(mean,sigma,(M,N))\n", "\n", "\n", "#Draw from Gamma distribution\n", "#shape=2\n", "#scale=2\n", "#Data=np.random.gamma(shape,scale,(M,N))\n", "\n", "\n", "y_vector=np.sum(Data, axis=1)\n", "\n", "plt.clf()\n", "sns.distplot(y_vector, kde='False');\n", "plt.show()\n", "\n", "#Calculate mean value\n", "\n", "mean_y=np.mean(y_vector)\n", "print(\"The empirical mean is\", mean_y)\n", "std_y=np.std(y_vector)\n", "print(\"The empirical std is\", std_y)\n", "#Print Theoretical std: print(\"The theoretical std for bernoulli is:\", np.sqrt(N*q*(1-q)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Continuous Variables\n", "\n", "We now perform a similar simulation when the $x_i$ are continuous variables drawn from some other distributions: Normal Distribution or even Gamma Distribution (look up on Wikipedia). Here fix $M=5000$.\n", "\n", "\n", "" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.12" } }, "nbformat": 4, "nbformat_minor": 1 }