{ "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", "