{
 "metadata": {
  "name": "4.Throwing darts.ipynb"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Geometric Distribution\n",
      "## *If I throw 100 darts at the number line between 1 and 1000, how far apart do we expect them to be?*"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import random\n",
      "import math\n",
      "\n",
      "line_len = 1000\n",
      "num_darts = 100\n",
      "\n",
      "line = [0 for x in xrange(line_len)]\n",
      "\n",
      "print \"throwing darts...\"\n",
      "for i in xrange(num_darts):\n",
      "    pos = random.randint(0,line_len-1)\n",
      "    line[pos] += 1\n",
      "    \n",
      "    if (i < 10):\n",
      "        print \"%d: %d\" % (i, pos)\n",
      "    \n",
      "\n",
      "print \"The maximum number of darts at one position was %d\" % (max(line))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "throwing darts...\n",
        "0: 287\n",
        "1: 2521\n",
        "2: 5384\n",
        "3: 6135\n",
        "4: 576\n",
        "5: 7624\n",
        "6: 1546\n",
        "7: 2526\n",
        "8: 4493\n",
        "9: 9576\n",
        "The maximum number of darts at one position was 2\n"
       ]
      }
     ],
     "prompt_number": 111
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure(figsize=(15,4), dpi=100)\n",
      "plt.bar(range(line_len), line)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10b7cbed0>"
       ]
      }
     ],
     "prompt_number": 112
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## compute the distances between darts\n",
      "\n",
      "distances = []\n",
      "\n",
      "last_dart = -1\n",
      "left_distance = line_len\n",
      "\n",
      "for i in xrange(line_len):\n",
      "    if (line[i] > 0):\n",
      "        right_dist = i - last_dart\n",
      "        min_dist = right_dist\n",
      "        if (left_distance < min_dist):\n",
      "            min_distance = left_distance\n",
      "        \n",
      "        if (last_dart != -1):\n",
      "            distances.append(min_dist)\n",
      "        \n",
      "        if (line[i] > 1):\n",
      "            for dd in xrange(line[i]-1):\n",
      "                distances.append(0)\n",
      "            left_distance = 0\n",
      "        else:\n",
      "            left_distance = right_dist\n",
      "            \n",
      "        last_dart = i\n",
      "\n",
      "distances.append(last_distance)\n",
      "\n",
      "mean = (sum(distances) + 0.) / len(distances)\n",
      "expected_mean = float(line_len) / num_darts\n",
      "print \"The observed mean distance was %.02f and the expected mean was %.02f\" % (mean, expected_mean)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The observed mean distance was 9.97 and the expected mean was 10.00\n"
       ]
      }
     ],
     "prompt_number": 113
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "The expected mean distance is intuitive, but what do we expect for the standard deviation and the shape of the distribution?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sumdiff = 0.0\n",
      "for d in distances:\n",
      "    diff = (d - mean) ** 2\n",
      "    sumdiff += diff\n",
      "\n",
      "sumdiff /= len(distances)\n",
      "stdev = math.sqrt(sumdiff)\n",
      "\n",
      "print \"The range in distances was %d to %d\" % (min(distances), max(distances))\n",
      "print \"The average distance between darts was: %.02f +/- %.02f\" % (mean, stdev)\n",
      "print \"The expected distances was %.02f\" % (float(line_len) / num_darts)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The range in distances was 0 to 70\n",
        "The average distance between darts was: 9.97 +/- 9.69\n",
        "The expected distances was 10.00\n"
       ]
      }
     ],
     "prompt_number": 114
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure()\n",
      "plt.hist(distances)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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DgKUIPABYiis64Ssk+5cNlLh0IOxB4PEVkv3LBkpcOhD24BANAFiKwAOApQg8AFiKwAOA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\nAGApAg8AliLwAGApAg8AliLwAGCp/wNJ3A1w3U/VygAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10d37c090>"
       ]
      }
     ],
     "prompt_number": 115
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure()\n",
      "plt.hist(distances, bins=range(max(distances)+1), normed=True)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10d36f090>"
       ]
      }
     ],
     "prompt_number": 116
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "What happens if you increase the number of darts or the length of the number line?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Notice that the probability of a given dart landing at a given spot is \n",
      "\n",
      "$$\n",
      "p = \\text{line_len} / \\text{num_darts} \\text{ (10%)}\n",
      "$$\n",
      "\n",
      "So the probability of a given dart not landing at the next position is\n",
      "\n",
      "$$\n",
      "1-p \\text{ (90%)}\n",
      "$$\n",
      "\n",
      "And the probability of not having 2 darts in a row is\n",
      "\n",
      "$$\n",
      "(1-p)^2 \\text{ (90% * 90% = .81%) }\n",
      "$$\n",
      "\n",
      "And not having 3 in a row\n",
      "\n",
      "$$\n",
      "(1-p)^3 \\text{ (90% * 90% * 90% = 72.9%) }\n",
      "$$\n",
      "\n",
      "So the probability of not having several darts in a row followed by 1 dart is\n",
      "\n",
      "$$\n",
      "pdf = (1-p)^{k} p\n",
      "$$\n",
      "\n",
      "## This is called the geometric distribution"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p = .1\n",
      "geom_dist = []\n",
      "for i in xrange(max(distances)+1):\n",
      "    geom_prob = (1-p)**i * p\n",
      "    geom_dist.append(geom_prob)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 117
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure()\n",
      "plt.bar(range(len(geom_dist)), geom_dist, color=\"green\")\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10c41cb90>"
       ]
      }
     ],
     "prompt_number": 118
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure()\n",
      "plt.hist(distances, bins=range(max(distances)+1), normed=True, label=\"Observed\")\n",
      "plt.plot(range(len(geom_dist)), geom_dist, color=\"green\", linewidth=4, label=\"Geometric\")\n",
      "plt.legend()\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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J0He7oe+OzneQ/ud0nOycEEIIU2nWu146OjoC5VfkZGVlsXv3boKDg42OCQ4O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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10b7fd9d0>"
       ]
      }
     ],
     "prompt_number": 119
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### The mean of the geometric distribution is 1/p (also p = 1/expected_mean)\n",
      "### The stdev is sqrt((1-p)/p^2)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expected_stdev = math.sqrt((1-p)/(p**2))\n",
      "print \"The expected stdev was %.02f and we observed %.02f\" % (expected_stdev, stdev)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The expected stdev was 9.49 and we observed 9.69\n"
       ]
      }
     ],
     "prompt_number": 120
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The geometric distribution can be well approximated by an exponential distribution\n",
      "\n",
      "$$\n",
      "pdf(x) = p * e^{-p * x}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p = .1\n",
      "exp_dist = []\n",
      "for i in xrange(max(distances)+1):\n",
      "    exp_prob = p * math.exp(-p * i)\n",
      "    exp_dist.append(exp_prob)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 121
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure()\n",
      "plt.bar(range(len(exp_dist)), exp_dist, color=\"orange\")\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10c460450>"
       ]
      }
     ],
     "prompt_number": 122
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure()\n",
      "plt.hist(distances, bins=range(max(distances)+1), normed=True, label=\"Observed\")\n",
      "plt.plot(range(len(geom_dist)), geom_dist, color=\"green\", linewidth=4, label=\"Geometric\")\n",
      "plt.plot(range(len(exp_dist)),  exp_dist, color=\"orange\", linewidth=4, linestyle=\"dashed\", label=\"Exponential\")\n",
      "plt.legend()\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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m5rbPVynVZepy2dnZBAcHk5SUBMDUqVMZNWoUY8eO1dWJiIhAo9HwwgsvAODl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\nranK6Qj/jYX9LwKQVVqe6AH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Roby8PDQaDVZWViYV/7lz5/j+++956qmndMOVTSl+oMow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\nAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10c41cb90>"
       ]
      }
     ],
     "prompt_number": 123
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Try to set the number of darts so that the average spacing is 42\n",
      "## *What does this imply when we see \"clusters\" of events?*"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Further reading\n",
      "* http://www.matplotlib.org - The project web page for matplotlib.\n",
      "* http://matplotlib.org/gallery.html - A large gallery showcaseing various types of plots matplotlib can create. Highly recommended!\n",
      "* http://www.loria.fr/~rougier/teaching/matplotlib - A good matplotlib tutorial.\n",
      "* http://scipy-lectures.github.io/matplotlib/matplotlib.html - Another good matplotlib reference.\n",
      "* https://docs.python.org/2/tutorial/ Python Tutorial\n",
      "* http://en.wikipedia.org/wiki/Probability_distribution Probability distributions\n",
      "* https://schatzlab.cshl.edu/teaching All my teaching materials"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}