{
 "metadata": {
  "name": "2.Flipping a coin.ipynb"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Binomial Distribution\n",
      "## *If I flip a coin 100 times, what is the probability of having exactly 42 heads?*\n",
      "---"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## simulate 100 trials of flipping a coin 100 times\n",
      "\n",
      "num_trials = 100\n",
      "num_flips = 100\n",
      "\n",
      "head_counts = []\n",
      "\n",
      "target_number = 42\n",
      "target_count = 0.\n",
      "\n",
      "## for each trial\n",
      "for t in xrange(num_trials):\n",
      "    \n",
      "    ## simulate a single trial of 100 flips\n",
      "    num_heads = 0\n",
      "    for f in xrange(num_flips):\n",
      "        r = random.randint(0,1)\n",
      "        \n",
      "        ## if the number of 0, record it as a head\n",
      "        if (r == 0):\n",
      "            num_heads += 1\n",
      "\n",
      "    head_counts.append(num_heads)\n",
      "    \n",
      "    if (num_heads == target_number):\n",
      "        target_count += 1\n",
      "\n",
      "target_prob = target_count / num_trials\n",
      "print \"I saw %d %d of %d times for a probability of %0.5f\" % (target_number, target_count, num_trials, target_prob)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "I saw 42 2 of 100 times for a probability of 0.02000\n"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## print the number of times we saw heads in the first 50 trials\n",
      "print head_counts[0:50]\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[62, 51, 51, 51, 42, 46, 45, 48, 46, 48, 46, 58, 44, 45, 49, 54, 61, 45, 49, 58, 46, 48, 57, 45, 57, 54, 52, 56, 52, 50, 43, 44, 61, 58, 44, 57, 49, 53, 45, 53, 55, 44, 50, 51, 55, 45, 45, 44, 51, 49]\n"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## Plot the histogram\n",
      "#import matplotlib.pyplot as plt\n",
      "plt.figure()\n",
      "plt.hist(head_counts)\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10b5b8610>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## Plot the histogram using integer values by creating more bins\n",
      "plt.figure()\n",
      "plt.hist(head_counts, bins=range(num_flips))\n",
      "plt.title(\"Using integer values\")\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10b7b5790>"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## Plot the density function, notice bars sum to exactly 1 \n",
      "plt.figure()\n",
      "plt.hist(head_counts, bins=range(num_flips), normed=True)\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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1rV27VitXrpTH41FLS0vsKgcARM6KQFFRkXXhwgWrv7/fWrFihTU4OBi2PhAIWGvWrLGG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\nAAxA2AOAAQh7ADDA/wHDj+HvZ0mnqQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10b707550>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## Compute mean & stdev of distribution\n",
      "mean = (sum(head_counts) + 0.) / num_trials\n",
      "sumdiff = 0.0\n",
      "for count in head_counts:\n",
      "    diff = (count - mean) ** 2\n",
      "    sumdiff += diff\n",
      "\n",
      "sumdiff /= num_trials\n",
      "stdev = math.sqrt(sumdiff)\n",
      "\n",
      "print \"The average value was %0.2f +/- %0.2f\" % (mean, stdev)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The average value was 49.81 +/- 5.17\n"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## *What happens to the mean or standard deviation if we change the number of trials or flips?*\n",
      "---"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Can we figure out what the distribution should look like analytically?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "Ah, the good old binomial distribution!\n",
      "\n",
      "$$\n",
      "p(\\text{k heads in n flips}) = {n \\choose k} p^{k} (1-p)^{(n-k)}\n",
      "$$\n",
      "\n",
      "$$\n",
      "p \\leftarrow \\text { probability of heads in a single flip }\n",
      "$$\n",
      "\n",
      "$$\n",
      "p^k \\leftarrow \\text{ total probability of k heads}\n",
      "$$\n",
      "\n",
      "$$\n",
      "1-p \\leftarrow \\text{ probabilty of one tails}\n",
      "$$\n",
      "\n",
      "$$\n",
      "n-k \\leftarrow \\text{ number of tails}\n",
      "$$\n",
      "\n",
      "$$\n",
      "(1-p)^{(n-k)} \\leftarrow \\text{ probability of all the tails}\n",
      "$$\n",
      "\n",
      "$$\n",
      "{n \\choose k} \\leftarrow \\text{ all the different orderings of k heads in n flips}\n",
      "$$\n",
      "\n",
      "**Reminder: **\n",
      "\n",
      "$$\n",
      "{n \\choose k} = \\frac{n!}{k!(n-k)!}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## Compute the probability of seeing 42 heads\n",
      "prob_heads = .5\n",
      "prob_42 = math.factorial(num_flips) / (math.factorial(42) * math.factorial(num_flips-42)) * (prob_heads**42) * ((1-prob_heads)**(num_flips-42))\n",
      "print \"The probability of seeing 42 heads in %d trials is %.05f\" % (num_trials, prob_42)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The probability of seeing 42 heads in 100 trials is 0.02229\n"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## evaluate the probability density function for the range [0, num_flips]\n",
      "prob_dist = []\n",
      "for k in xrange(num_flips+1):\n",
      "    prob_k = math.factorial(num_flips) / (math.factorial(k) * math.factorial(num_flips-k)) * (prob_heads**k) * ((1-prob_heads)**(num_flips-k))\n",
      "    prob_dist.append(prob_k)\n",
      "\n",
      "plt.figure()\n",
      "plt.plot(prob_dist, color='red', linewidth=4)\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10049c0d0>"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## overlay the analytic distribution over the data\n",
      "plt.figure()\n",
      "plt.hist(head_counts, bins=range(num_flips), normed=True)\n",
      "plt.plot(prob_dist, color='red', linewidth=4)\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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VA3zJFfQ28OnIwd1ACxfCihWWqjnAeg3uOpIGdyX+/P3fW0If4CE4G/rSq2XL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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10b7f7f90>"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## FACT: The mean of the binomial distribution is \n",
      "mean = n * p\n",
      "\n",
      "## FACT: The standard deviation of the binomial distribution is \n",
      "$$\n",
      "stdev = \\sqrt(np(1 \u2212 p))\n",
      "$$\n",
      "\n",
      "(For a fixed probability, grows with the sqrt of the number of trials)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expected_mean  = num_flips * prob_heads\n",
      "expected_stdev = math.sqrt(num_flips * prob_heads * (1 - prob_heads)) \n",
      "\n",
      "print \"The expected frequency is %.02f +/- %.02f\" % (expected_mean, expected_stdev)\n",
      "print \"The observed frequency was %0.2f +/- %0.2f\" % (mean, stdev)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The expected frequency is 50.00 +/- 5.00\n",
        "The observed frequency was 49.81 +/- 5.17\n"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## It is imprecise to compute the factorial of very large numbers, use the gaussian (normal) approximation instead\n",
      "\n",
      "$$\n",
      "p(x)=\\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\exp \\left( -\\frac{(x-\\mu)^2}{2 \\sigma^2} \\right)\n",
      "$$\n",
      "where $\\sigma$ is the standard deviation and $\\mu$ is the mean\n",
      "\n",
      "This is nice because you can estimate the distribution in your head if you know the mean and standard deviation\n",
      "\n",
      "![Normal Distributions](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/500px-Standard_deviation_diagram.svg.png)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gaus_prob_dist = []\n",
      "pi = 3.1415926\n",
      "\n",
      "for k in xrange(num_flips+1):\n",
      "    var = expected_stdev**2\n",
      "    num = math.exp(-(float(k)-expected_mean)**2/(2*var))\n",
      "    denom = math.sqrt(2*pi*var)\n",
      "    gaus_prob = num/denom\n",
      "    gaus_prob_dist.append(gaus_prob)\n",
      "\n",
      "plt.figure()\n",
      "plt.plot(gaus_prob_dist, color='green', linewidth=4, linestyle='dashed')\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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GwmRhuH3C7XCRsD1GA+NwTnJ4Jy6fMCV9AAjyDrL7pA8AP535U7FDICfEpgXZ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\nEk7jISB4nQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10b76fe10>"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## overlay the analytic distributions over the data\n",
      "plt.figure()\n",
      "plt.hist(head_counts, bins=range(num_flips), normed=True)\n",
      "plt.plot(prob_dist, color='red', linewidth=4)\n",
      "plt.plot(gaus_prob_dist, color='green', linewidth=4, linestyle='dashed')\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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69OlahaMJk8nEwoULWbJkCQsWLADMxyU/3zy3Qn5+PpGRkVqGaHOf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\nQuiMJH4hhNCZ/w+KjDSvTeYzaQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10bb0f750>"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## *How can you shift the distribution to be centered at higher or lower values, or have a broader/narrower peak?*"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 27
    }
   ],
   "metadata": {}
  }
 ]
}