{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import matplotlib.pyplot as plt\n",
      "import numpy as np"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.random.seed(42)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Part (a)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Estimate and plot the number of U(0,1) numbers you can multiply before underflowing."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "num_prods = [np.nonzero(np.cumproduct(np.random.random(1000)) == 0)[0][0] + 1 for _ in xrange(200000)]\n",
      "mean = np.mean(num_prods)\n",
      "sd = np.std(num_prods)\n",
      "print \"{0} +/- {1} products before underflowing\".format(mean, sd)\n",
      "\n",
      "rounded_mean = np.round(mean)\n",
      "plt.hist(num_prods, bins=np.arange(rounded_mean - 3*sd, rounded_mean + 3*sd, 4))\n",
      "plt.title(\"Histogram of the Number of Products Before Underflow\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "746.410685 +/- 27.3135432127 products before underflowing\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "<matplotlib.text.Text at 0xaf64d4c>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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zZswwsbGxxt/f31x66aXmk08+MUOHDvU4xqWtOyMjo9i5ZIwx//73v43L5TJL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\n",
       "text": [
        "<matplotlib.figure.Figure at 0xac2b18c>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Part (b)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "I first estimate the mean and variance of log(U). It seems that the mean is -1 and the variance is -1."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "log_uniform = np.log(np.random.random(1000000))\n",
      "mu = np.mean(log_uniform)\n",
      "var = np.var(log_uniform)\n",
      "mu, var"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "(-1.0005811922459742, 1.0003386611570766)"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "By the central limit theorem, the sum of N random variables $log(U)$ should be distributed as $\\mathcal{N}(-N, N)$. Experimentally, this seems accurate."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "num_sums = 10000\n",
      "sum_logs = [np.sum(np.log(np.random.random(num_sums))) for _ in xrange(20000)]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mean = np.mean(sum_logs)\n",
      "sd = np.std(sum_logs)\n",
      "print \"{0} +/- {1} (variance {2})\".format(mean, sd, sd**2)\n",
      "plt.hist(sum_logs, bins=25)\n",
      "plt.title('Histogram of sum of {0} log(U)'.format(num_sums))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "-9998.68065929 +/- 100.524298534 (variance 10105.1345958)\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "<matplotlib.text.Text at 0xb0ba34c>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Mubm5+uijjxQSEqLU1FQVFBS4fRzHUZcuXZSXl+c+3n//fb/9jBo1SgsXLtQb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\nSUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xac2634c>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Part (c)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In part (a), I estimated that you can multiply an average of about 746 $U(0,1)$ values before underflowing.\n",
      "\n",
      "As we know, adding the log of numbers and then exp'ing them is the same as multiplying the numbers.\n",
      "In part (b), I showed that the mean of the sum of $N$ $log(U)$ values is $-N$.\n",
      "Thus we would expect that we could sum about 746 $log(U)$ values before the exp'ing would underflow"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "for s in xrange(-740, -750, -1):\n",
      "    print s, np.exp(s)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "-740 4.19955798965e-322\n",
        "-741 1.53160350211e-322\n",
        "-742 5.43472210425e-323\n",
        "-743 1.97626258336e-323\n",
        "-744 9.88131291682e-324\n",
        "-745 4.94065645841e-324\n",
        "-746 0.0\n",
        "-747 0.0\n",
        "-748 0.0\n",
        "-749 0.0\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "WHAMMY!"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}