Random Variable Modulo Addition.ipynb

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  "name": "",
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import random\n",
      "import matplotlib.pyplot as plt\n",
      "import matplotlib\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def random_sum(sampler, count):\n",
      "    return sum(sampler() for i in range(count))\n",
      "\n",
      "def random_xor(sampler, count):\n",
      "    return reduce(lambda a,b: a^b, (sampler() for i in range(count)))\n",
      "\n",
      "def sample_sum(samples):\n",
      "    return sum(samples)\n",
      "\n",
      "def sample_xor(samples):\n",
      "    return reduce(lambda x, y: x^y, samples)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 181
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "count = 100000\n",
      "n = 10\n",
      "max_value = 2**8-1\n",
      "sampler = lambda: random.randint(0, max_value)\n",
      "samples =  [[sampler() for i in range(n)] for i in range(count)]\n",
      "standard_samples = [sample_sum(sample) for sample in samples]\n",
      "mod_samples = [sample_sum(sample) % max_value for sample in samples]\n",
      "xor_samples = [sample_xor(sample) % max_value for sample in samples]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 182
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(standard_samples);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x112432710>"
       ]
      }
     ],
     "prompt_number": 183
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The sum of independent random variables approaches a normal distribution due to the [central limit theorem](http://en.wikipedia.org/wiki/Central_limit_theorem)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(mod_samples);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10e85a4d0>"
       ]
      }
     ],
     "prompt_number": 184
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      " The distribution of the modulo sum of random variables can become uniform again due to boundary conditions of a sort that are removed due to the modulo operation. See <http://emmettmcquinn.com/blog/> for more."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(xor_samples);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x112a731d0>"
       ]
      }
     ],
     "prompt_number": 185
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "XOR is a common simple technique to combine entropy [(see here for more)](https://tools.ietf.org/html/rfc4086#section-5.1)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}