Another way to compute `e`

find_e.ipynb

{
 "metadata": {
  "name": "Finding e"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Estimate `e` by sampling from a uniform distribution\n",
      "#### by pmhobson"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Function to sample from a uniform distribution between 0 and 1 until the sum of all samples is at least 1\n",
      "This happens `num_iter` times and an array of results are returned"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def sumToOne(num_iter):\n",
      "    '''\n",
      "    sum up random samples until you reach 1,\n",
      "    return the array of the number of samples required  \n",
      "    '''\n",
      "    num_samples = np.empty((num_iter,))\n",
      "    for k in range(num_iter):\n",
      "        total = 0\n",
      "        ns = 0\n",
      "        while total <= 1:\n",
      "            total += np.random.uniform(0,1)\n",
      "            ns += 1\n",
      "        num_samples[k] = ns\n",
      "    \n",
      "    return num_samples"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Function to make a plot"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def plot_e_guesses(num_trials, num_samples):\n",
      "    '''plot the results from estimated `e`'''\n",
      "    fig, ax1 = plt.subplots(figsize=(6,4), dpi=300)\n",
      "    ax1.plot(num_trials, num_samples, '.', markersize=4, markeredgecolor='k', \n",
      "             markerfacecolor='k', alpha=0.87, zorder=5, label='Average values')\n",
      "    ax1.axhline(np.e, color='b', linewidth=1.25, alpha=0.75, \n",
      "                zorder=0, label=r'e = %0.10f' % (np.e,))\n",
      "    \n",
      "    ax1.set_ylim([2.5,3.0])\n",
      "    ax1.set_xlabel('Number of iterations')\n",
      "    ax1.set_ylabel('Avg. number of samples to reach 1')\n",
      "    \n",
      "    ax1.legend(loc='lower right')\n",
      "    \n",
      "    ax1.set_xscale('log')\n",
      "    ax1.xaxis.grid(True, which='major', linestyle='-', alpha=0.35)\n",
      "    ax1.xaxis.grid(True, which='minor', linestyle='-', alpha=0.20)\n",
      "    ax1.yaxis.grid(True, which='major', linestyle='-', alpha=0.35)\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Execute everything"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "num_samples = np.array([])\n",
      "num_trials = np.logspace(2,6,100)\n",
      "for nt in num_trials:\n",
      "    ns = sumToOne(int(nt))\n",
      "    num_samples = np.hstack([num_samples, np.mean(ns)])\n",
      "\n",
      "plot_e_guesses(num_trials, num_samples)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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IbSa4BZGWlobw8HDY29vDwcEBkZGRuHv3rj5iI4QQYkCCCWLAgAGIiYnBo0eP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\nVasK8WzYsAFisRitW7fG9OnTAQDffPM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      }
     ],
     "prompt_number": 4
    }
   ],
   "metadata": {}
  }
 ]
}