{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## The normal random variable is special\n", "\n", "This is the only r.v. that we are going to actively study. If you take a course specifically on probability you will study tens of r.v. how they relate and thier properties. In addition you will learn how to mathematically derive their distributions and their summary statistics instead of approximating them as we do in this course. \n", "\n", "Why is the normal random variable so special?? Well let's first discuss some properies of a normal random variable first.\n", "\n", "To start off, let's look at a normal r.v. distribution:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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cslM1zn/ufLCW4UzqFnZcyOVywWw2D97WarUIBALQ6XTQ6/VITk6GJEn4+c9/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\njh1YuXLlmBZNytPRM4CX3j0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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "\n", "from numpy.random import normal\n", "import seaborn as sns\n", "\n", "sns.distplot(normal(size=1000))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Question: is this what a normal distribution looks like?\n", "\n", "Almost! It is important to remember that this is approximately what a normal distribution looks like. We can only see what a normal distribution looks like by taking infinite samples. \n", "\n", "Most people say that the normal distribution looks like a bell, and thus call it a bell curve. For your information, a normal distribution has no skewness, it is a symetric distribution.\n", "\n", "Normal distributions can have any mean or standard deviation. The below normal random variable has a mean of 3.5 and a standard deviation of .25:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "3.4914305021\n", "0.244714943362\n" ] }, { "data": { "image/png": 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C6HX6UJRjQX5O8rTzo+2YCfPLdHj/VA+au93IzLJArYrPqIFSPl9KaQfAtsTLtC7xjEaj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\nDVKXQzLDEKZpGfYEcexML4x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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "\n", "sns.distplot(normal(loc=3.5, scale=0.25, size=1000))\n", "\n", "print np.mean(normal(loc=3.5, scale=0.25, size=1000))\n", "print np.std(normal(loc=3.5, scale=0.25, size=1000))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Central limit theorem\n", "\n", "So all that I have said above is rather normal (pun) for a r.v. They all have distributions and they all have means. So what gives?\n", "\n", "Well it all has to do with the central limit theorem (arguably the most important theorem in all of statistics). This theorem says:\n", "\n", "
\n", "The sum of independent and identically distributed (iid) r.v. with finite mean and variance is approximatly distributed normally (that means if we take the samples from the sum of iid r.v., those samples are approximately distributed normally).\n", "
\n", "\n", "This is incredibly important! \n", "\n", "Why?\n", "\n", "Well, because even if we don't know what the random variables are (what the distributions look like), we know that their sum is distributed normally! So we know something about them. Let's prove it. \n", "\n", "We start off with the simplest of distributions, a uniform one:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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UFcSV10RUjRzW6XnlKBd7zRdDWUEcviaiasS9ygvHUFZQMJKEcXpRBBFRtcid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\nDCtdSskwlAt468gwkmkZG69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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sns.distplot(np.random.uniform(size=10000))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "this is not a normal distribution, but if we make a r.v. that is the sum of uniform random variables, what will it be distributed?" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def almost_normal():\n", " return sum(np.random.uniform(size=50))" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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BzrXY0T3owp3zsmBK5oSsmzGx49S16LQqAOOjDBWl3FmJEk/QIuxwOGAwGCZvq1Qq+P1+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\nWISJiIgEYREmIiIS5P8HhmBojlP81f8AAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "dist = np.array([almost_normal() for _ in range(10000)])\n", "\n", "sns.distplot(dist)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pretty normal right ;)\n", "\n", "Now if we reduce the number of normals that we sum:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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9NtReGESESYcirhNMAUKjVmFxXiwUBSir7REdh3wUS5iEGnN6cKyyEyoJWF6Q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\nae3atSgtLcW2bdvGZxB78cUXx2cQ27F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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def almost_normal():\n", " return sum(np.random.uniform(size=2))\n", "\n", "dist = np.array([almost_normal() for _ in range(10000)])\n", "\n", "sns.distplot(dist)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This becomes less normal.\n", "\n", "## Why does this matter\n", "\n", "Well, by making a very simple assumption (that each x_i is an iid sample from a r.v.) we were able to know something very specific about our samples. But there are probably two things that are bothering you now:\n", "\n", "1. What does iid mean?\n", "2. Okay so we know it is normal, now what?\n", "\n", "And both of these are really good questions, and for that reason we will be doing a video on each of them." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "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": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 2 }