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   "cells": [
    {
     "source": [
      "> This is one of the 100 recipes of the [IPython Cookbook](http://ipython-books.github.io/), the definitive guide to high-performance scientific computing and data science in Python.\n"
     ], 
     "cell_type": "markdown", 
     "metadata": []
    }, 
    {
     "source": [
      "# 15.4. Computing exact probabilities and manipulating random variables"
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "from sympy import *\n", 
      "from sympy.stats import *\n", 
      "init_printing()"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "## Rolling dice"
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "source": [
      "Let's roll two dices X and Y."
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "X, Y = Die('X', 6), Die('Y', 6)"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "We can compute probabilities defined by equalities (with the Eq operator) or inequalities..."
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "P(Eq(X, 3))"
     ], 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "P(X>3)"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "Conditions can also involve multiple random variables..."
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "P(X>Y)"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "Conditional probabilities..."
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "P(X+Y>6, X<5)"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "## Continuous random variables"
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "source": [
      "We can also work with arbitrary discrete or continuous random variables."
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "Z = Normal('Z', 0, 1)  # Gaussian variable"
     ], 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "P(Z>pi)"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "We can compute expectancies and variances..."
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "E(Z**2), variance(Z**2)"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "as well as densities."
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "f = density(Z)"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "This is a lambda function, it can be evaluated on a SymPy symbol:"
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "var('x')\n", 
      "f(x)"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "We can plot this density."
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "%matplotlib inline\n", 
      "plot(f(x), (x, -6, 6));"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "SymPy.stats works by using integrals and summations for computing probabilistic quantities. For example, P(Z>pi) is:"
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }, 
    {
     "cell_type": "code", 
     "language": "python", 
     "outputs": [], 
     "collapsed": false, 
     "input": [
      "Eq(Integral(f(x), (x, pi, oo)), \n", 
      "   simplify(integrate(f(x), (x, pi, oo))))"
     ], 
     "metadata": {}
    }, 
    {
     "source": [
      "> You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).\n\n> [IPython Cookbook](http://ipython-books.github.io/), by [Cyrille Rossant](http://cyrille.rossant.net), Packt Publishing, 2014 (500 pages)."
     ], 
     "cell_type": "markdown", 
     "metadata": {}
    }
   ], 
   "metadata": {}
  }
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