{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Week 2-3: Discrete random variables\n",
    "\n",
    "\n",
    " #### [Back to main page](https://petrosyan.page/fall2020math3215)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "### Example 1\n",
    "\n",
    "Consider the  double coin flip experiment $S=\\{HH, HT, TH, TT\\}$, and define a random variable which counts the number of heads in each outcome. Then \n",
    "$$X(HH)=2,\\;X(HT)=1,\\;X(TH)=1,\\; X(TT)=0.$$\n",
    " \n",
    "Observe that Range$(X)=\\{0,1,3\\}$ and the pmf of the random variable is \n",
    "        \n",
    "$$f(0)=0.25,\\; f(1)=0.5,\\; f(2)=0.15.$$\n",
    "\n",
    "Below we draw the line graph and the histogram of the pmf of this random variable. Red circles represent the range of $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkgEAoCOSAQCgI5IBAKAjkleaJ5547p8cW/Z78uw502TnzuS220Z/Mhn2fLJmeL9F8kpy3XXJBReM3r/ggtFtjh37PXn2nGly3XXJ2WcnF144+tPr+diz55M14/tdrbWh13CQdevWtVe/+tVDL2Oy9u1Lbr01X9tf+XpOzLo8leOyPznppKRq6NVNn9aS3buzP8flqayz35PwnD0/Lefliaw97pnk/POTE04YenXL5qabbrq9tbZ16HVM0nLM7D17kj/7s+TlL0/Wrl2mhR1L+55N7rgjafu/dV8dl7zylckJa4Zb1yLNzSX33JOcc06yfv3Qq1kkez5Zq3y/9+xJ7rwzOe+8Q8+UQ83slf9Vzoq9e5OqHJf9eWH2pjJ+Ue7fnxx//LBrm0b7R/tb9nty5u35uuzOmuwb/YNk796pimSOzpo1o//07rxz6JUs1pok3d+rLcmXh1jL0bvnnqFXcCTs+WSt/v0+/vjRbDlaKzKSzz333Nx4441DL2Oydu4cfStjz54kz4zuW7s2uf/+ZMOGQZc2lez35D1nz58a3feCtckNN0zVntcMfidiuWb2jh2js22rwhNPjC4Zembvt+574YuSG29MTj55qFUdkX37Vtm/T+35ZE3Bfq9fn2zceOhjDjWzV2Qkz6QNG5IPfzi59NLRf0H79o1uT1E8rCj2e/LsOYexcePh/0JbOU5OPvreg1/P56+OeFid7Plk2e8VeU3y1q1b2/bt24dexjB27kwefDDZtEk8TIL9nrwp3/Oqmrlrks3sB6f29bwi2fPJmvL9PtTMdiZ5pdmwYSpfhCuW/Z48e8408XqePHs+WTO8334FHAAAdEQyAAB0RDIAAHREMgAAdEQyAAB0RDIAAHREMgAAdEQyAAB0RDIAAHREMgAAdEQyAAB0RDIAAHREMgAAdEQyAAB0RDIAAHREMgAAdKq1NvQaDlJVO5M8NPQ6BnRqkseHXsQMsd+TN817fnZrbcPQi5gkM3uqX88rlT2frGne7+ed2SsykmddVW1vrW0deh2zwn5Pnj1nmng9T549n6xZ3W+XWwAAQEckAwBARySvTNcMvYAZY78nz54zTbyeJ8+eT9ZM7rdrkgEAoONMMgAAdETyClNV26rq7qq6r6p+fuj1TLOq+khV7aiqvxh6LbOgqs6sqj+qqq9W1V1V9c+HXhMslZk9Web25JjZLrdYUarq+CT3JLkwySNJbkvyttbaVwZd2JSqqu9P8nSSj7XWvnfo9Uy7qjotyWmttS9V1boktyd5q9c3q5WZPXnm9uSY2c4krzTnJbmvtXZ/a+0bST6R5C0Dr2lqtdb+OMkTQ69jVrTWHmutfWn8/lNJvprk9GFXBUtiZk+YuT05ZrZIXmlOT/LwvNuPZMZekMyGqtqU5PuSfHHYlcCSmNnMhFmd2SJ5ZakF7nM9DFOlqk5M8l+S/IvW2u6h1wNLYGYz9WZ5ZovkleWRJGfOu31GkkcHWgssu6o6IaNhe21r7TNDrweWyMxmqs36zBbJK8ttSbZU1eaqekGSi5J8buA1wbKoqkry4SRfba29f+j1wDIws5laZrZIXlFaa88m+dkkf5jRBfKfaq3dNeyqpldVXZfkliTnVtUjVXXp0Guacn83yTuS/P2qumP89sahFwVHy8yePHN7omZ+ZvsVcAAA0HEmGQAAOiIZAAA6IhkAADoiGQAAOiIZAAA6IhkAADoiGQAAOiIZAAA6/w8dZIEaE4YnvAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 864x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# nbi:hide_in\n",
    "# library\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "plt.rcParams[\"figure.figsize\"] = (12, 5)\n",
    "\n",
    "range_x = np.array([0, 1, 2])\n",
    "pmf_values = np.array([1/4, 1/2, 1/4])\n",
    "\n",
    "fig, [ax1, ax2] = plt.subplots(1,2, num=1, clear=True)\n",
    "\n",
    "ax1.set_ylim(-0.05,0.6) \n",
    "ax1.set_xlim(-0.7, 2.6)\n",
    "ax1.axhline(y=0, color='k')\n",
    "ax1.set_xticks(range_x)\n",
    "ax1.set_yticks(pmf_values)\n",
    "\n",
    "ax2.set_ylim(-0.05, 0.6) \n",
    "ax2.set_xlim(-0.7, 2.6)\n",
    "ax2.axhline(y=0, color='k')\n",
    "ax2.set_xticks(range_x)\n",
    "ax2.set_yticks(pmf_values)\n",
    "\n",
    "\n",
    "# Plotting line graphs using plt.stem with stems removed\n",
    "ax1.scatter(range_x,np.zeros(range_x.shape), color =\"red\", s=20)\n",
    "markers,stems,base = ax1.stem(range_x, pmf_values, markerfmt=' ', linefmt=\"blue\", basefmt=\"black\", use_line_collection=True)\n",
    "stems.set_linewidth(1.3)\n",
    "ax1.set_title(\"Line graph\")\n",
    "\n",
    "\n",
    "# PLotting with plt.bar instead of plt.hist works better when f(x) are knowwn\n",
    "ax2.scatter(range_x,np.zeros(range_x.shape), color =\"red\", s=20)\n",
    "ax2.bar(range_x, pmf_values, width=1, color=(1, 1, 1, 0), edgecolor='blue', linewidth=1.3, label=\"Histogran\")\n",
    "ax2.set_title(\"Histogram\")\n",
    "\n",
    "plt.show();\n",
    "                       "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "### Example 2\n",
    "\n",
    "If Range$(X)=\\{x_1,\\dots,x_k\\}$ and $P(x_1)=\\cdots=P(x_k)=\\frac{1}{k}$ then we say that $X$ has uniform distribution.\n",
    "\n",
    "in the example below, the range of $X$ is \n",
    "\n",
    "$$ \\text{Range}(X)=\\{-3, -6, 5, 9, 2, 1, 3, 8, 7, 10\\}$$\n",
    "\n",
    "with $f(x)=0.1$ for every $x\\in \\text{Range}(X).$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# nbi:hide_in\n",
    "# library\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "plt.rcParams[\"figure.figsize\"] = (12, 5)\n",
    "\n",
    "K=10\n",
    "range_x=np.array([-3, -6, 5, 9, 2, 1, 3, 8, 7, 10])\n",
    "pmf_values=np.ones(range_x.size)/range_x.size\n",
    "\n",
    "# Sort x and y according to order in x\n",
    "sortargs = range_x.argsort()\n",
    "range_x = range_x[sortargs]\n",
    "pmf_values = pmf_values[sortargs]\n",
    "\n",
    "#cdf values using cumsum function with padding\n",
    "cdf_values = np.cumsum(pmf_values)\n",
    "\n",
    "def padding(cdf_values, range_x):\n",
    "    edge = (range_x[0]-2, range_x[-1]+2)\n",
    "    range_x_padded = np.pad(range_x, (1,1), 'constant', constant_values=edge)\n",
    "    cdf_values_padded = np.pad(cdf_values, (1,1), 'constant', constant_values=(0, 1))\n",
    "    return cdf_values_padded, range_x_padded, edge\n",
    "\n",
    "cdf_values_padded, range_x_padded, edge = padding(cdf_values, range_x)\n",
    "# setting up the figure\n",
    "fig, [ax1, ax2] = plt.subplots(1,2, num=1, clear=True)\n",
    "\n",
    "ax1.set_ylim(-0.07,1.1)\n",
    "ax1.axhline(y=0, color='k')\n",
    "ax1.set_xlim(edge)\n",
    "ax1.set_xticks(range_x)\n",
    "ax1.set_yticks(cdf_values_padded)\n",
    "\n",
    "ax2.set_ylim(-0.07,1.1) \n",
    "ax2.axhline(y=0, color='k')\n",
    "ax2.set_xlim(edge)\n",
    "ax2.set_xticks(range_x)\n",
    "ax2.set_yticks(cdf_values_padded)\n",
    "\n",
    "\n",
    "# plot line grapghs\n",
    "ax1.scatter(range_x,np.zeros(range_x.shape), color =\"red\", s=20)\n",
    "markers,stems,base = ax1.stem(range_x, pmf_values, markerfmt=' ',\n",
    "                              linefmt=\"blue\", basefmt=\"black\", use_line_collection=True)\n",
    "stems.set_linewidth(1)\n",
    "ax1.set_title(\"pmf\")\n",
    "\n",
    "# plot cdf using step function\n",
    "ax2.scatter(range_x,np.zeros(range_x.shape), color =\"red\", s=20)\n",
    "ax2.step(range_x_padded, cdf_values_padded, linewidth=1, where='post')\n",
    "ax2.set_title(\"cdf\")\n",
    "\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "### Relative frequency histogram for pmf\n",
    "\n",
    "Suppose our data   consists of values of the random variable on a sequence of trial outcomes. As mentioned earlier, probability of an event represents how frequent the experiment outcome terminates in the event, in a large number of repetitive trials. Hence,  the pmf at $x\\in\\text{Range}(X)$ can be empirically estimated using the relative frequency \n",
    "\n",
    " $$f_{\\text{emp}}(x)= \\frac{\\text{number of elements in data = }x}{ \\text{ size of data}}.$$\n",
    " \n",
    "Resulting relative frequency histogram will approximate the pmf histogram.\n",
    "\n",
    "In the example below, we consider the following experiment: a dice is tossed twice and the random variable is the maximum of the two tosses: \n",
    "\n",
    "$$S=\\{(i,j):\\; 1\\leq i\\leq n, 1\\leq j\\leq 6\\}$$\n",
    "\n",
    "and for any $s=(i,j)$,\n",
    "      \n",
    " $$X(i,j)=\\max\\{i,j\\}.$$\n",
    " \n",
    " It can be seen that in this case, $\\text{range}(X)=\\{1,\\dots, n\\}$ and, for any $x\\in \\text{range}(X)$,\n",
    " \n",
    " $$f(x)=\\frac{2x-1}{n^2}.$$\n",
    "\n",
    "The data consist of values of $X$ on 1000 random pairs of tosses."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 287,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# nbi:hide_in\n",
    "# library\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "plt.rcParams[\"figure.figsize\"] = (8, 5)\n",
    "\n",
    "n = 6\n",
    "num_samples=1000\n",
    "\n",
    "range_x = np.arange(1,n+1)\n",
    "pmf_values = np.array([(2*i-1)/n**2 for i in range(1,n+1)])\n",
    "\n",
    "# generate data\n",
    "toss = np.random.randint(1,n+1,(2,num_samples))\n",
    "data = np.amax(toss, axis=0).squeeze()\n",
    "\n",
    "# compute empirical pmf\n",
    "def epmf(data):\n",
    "    erange_x, counts = np.unique(data, return_counts=True)\n",
    "    epmf_values = counts/data.size\n",
    "    return epmf_values, erange_x\n",
    "\n",
    "epmf_values, erange_x = epmf(data)\n",
    "\n",
    "# plot \n",
    "plt.ylim(-0.02,0.4) \n",
    "plt.axhline(y=0, color='k')\n",
    "plt.xticks(range_x)\n",
    "\n",
    "plt.scatter(range_x,np.zeros(range_x.shape), color =\"red\", s=20)\n",
    "plt.bar(range_x, pmf_values, width=1, color=(1, 1, 1, 0), edgecolor='blue', linewidth=1, label=\"True histogran\")\n",
    "plt.bar(erange_x, epmf_values, width=0.9, color=(1, 1, 1, 0), edgecolor='green', linewidth=1,linestyle=\"--\", label=\"Relative frequency histogram\")\n",
    "plt.legend()\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "### Empirical cdf\n",
    "\n",
    "Similar to empirical pmf, the empirical cfd is computed as\n",
    "\n",
    "$$F_{\\text{emp}}(x)=\\frac{\\text{number of outomes }\\leq x}{\\text{size of data}}.$$\n",
    "\n",
    "We consider the double die toss experiment. Let the random variable $X$ be the maximum of two tosses. Again,   $\\text{Range}(X)=\\{1,\\dots, 6\\}$ and it can be checked that\n",
    "     $$F(x)=\n",
    "\\begin{cases}\n",
    "0 & x<1\\\\\n",
    "\\sum\\limits_{i=1}^k\\frac{2i-1}{6^2}=\\frac{k^2}{6^2} & k\\leq x<k+1, \\text{ for }k=1,\\dots,5\\\\\n",
    "1 & 6<x.\n",
    "\\end{cases}$$\n",
    "In the numerical example below we collect 1000 pairs of tosses of the die and collect the maximum value of each pair as our data. The graph shows the real and the empirical cdf-s."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# nbi:hide_in\n",
    "# library\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "plt.rcParams[\"figure.figsize\"] = (8, 5)\n",
    "\n",
    "n = 6\n",
    "num_samples=100\n",
    "range_x = np.arange(1,n+1)\n",
    "\n",
    "# generate data\n",
    "toss = np.random.randint(1,n+1,(2,num_samples))\n",
    "data = np.amax(toss, axis=0).squeeze()\n",
    "\n",
    "# compute true cdf values\n",
    "cdf_values = np.array([i**2/n**2 for i in range(1,n+1)])\n",
    "\n",
    "# compute empirical cdf values\n",
    "def ecdf(data):\n",
    "    erange_x, counts = np.unique(data, return_counts=True)\n",
    "    cdf_emp = np.cumsum(counts)/data.size\n",
    "    return cdf_emp, erange_x\n",
    "\n",
    "ecdf_values, erange_x = ecdf(data)\n",
    "\n",
    "# add padding \n",
    "def padding(cdf_values, range_x):\n",
    "    edge = (range_x[0]-2, range_x[-1]+2)\n",
    "    range_x_padded = np.pad(range_x, (1,1), 'constant', constant_values=edge)\n",
    "    cdf_values_padded = np.pad(cdf_values, (1,1), 'constant', constant_values=(0, 1))\n",
    "    return cdf_values_padded, range_x_padded\n",
    "\n",
    "cdf_values_padded, range_x_padded = padding(cdf_values, range_x)\n",
    "ecdf_values_padded, erange_x_padded = padding(ecdf_values, erange_x)\n",
    "\n",
    "# plot setup\n",
    "plt.ylim(-0.07,1.1)\n",
    "plt.xlim(-1.2,8.2)\n",
    "plt.axhline(y=0, color='k')\n",
    "plt.xticks(range_x)\n",
    "\n",
    "# plot cdf using step function\n",
    "plt.scatter(range_x,np.zeros(range_x.shape), color =\"red\", s=20)\n",
    "plt.step(range_x_padded, cdf_values_padded, where='post', color=\"blue\", linewidth=1, label=\"True cdf\")\n",
    "plt.step(erange_x_padded, ecdf_values_padded, where='post', color=\"green\", linewidth=1, linestyle=\"--\", label='Empirical cdf')\n",
    "plt.legend()\n",
    "\n",
    "plt.show();\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "### Hypergeometric distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose  $N$ balls in an urn, $K$ of which are red, the rest are blue. $n$ balls are selected without order and without replacement.    \n",
    " Let $S$ be the set of all such selections.  Consider the following random variable:for $s\\in S$, define $X(s)$ to be the number of red balls in $s$.     \n",
    " \n",
    "The  histogram of this random variable is shown below. Red circles represent the range of $X$. You can change the values of the paramaters by playing with the slider. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "f8451098e2b847759614fae57561fa0c",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=80.0, description='N', min=1.0, readout_format='', step=1.0), FloatSli…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# nbi:hide_in\n",
    "# library\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from scipy.special import comb\n",
    "from ipywidgets import interact, FloatSlider\n",
    "\n",
    "\n",
    "\n",
    "def hypergeometric_pmf(N, K, n):\n",
    "    # the range of the hyper-geometric distribution\n",
    "    range_x = np.arange(max(0, n-(N-K)), min(n, K)+1, 1)\n",
    "    \n",
    "    # compute the pmf at value i\n",
    "    def hyper_pmf(N,K,n,i):\n",
    "        pmf_val = comb(K, i, exact=True) * comb(N-K, n-i, exact=True) / comb(N, n, exact=True)\n",
    "        return pmf_val\n",
    "    pmf_values = np.array([hyper_pmf(N,K,n,i) for i in range_x])\n",
    "\n",
    "    # plot setup\n",
    "    plt.figure(figsize=(14,7)) \n",
    "    plt.axhline(y=0, color='k')\n",
    "    plt.xlim(-2,N+2)\n",
    "    plt.xticks(np.arange(0, N+1, 5))\n",
    "\n",
    "    # plotting with plt.bar instead of plt.hist works better when f(x) are knowwn\n",
    "    plt.scatter(range_x,np.zeros(range_x.shape), color =\"red\", s=20)\n",
    "    plt.bar(range_x, pmf_values, width=1, color=(1, 1, 1, 0), edgecolor='blue', linewidth=1.3, label=\"Histogran\")\n",
    "    plt.plot();\n",
    "    \n",
    "# create interactive variables\n",
    "N = FloatSlider(min=1.0, max=100.0, step=1.0, value=80, readout_format='')\n",
    "K = FloatSlider(min=1.0, max=N.value, step=1.0, value=40, readout_format='')\n",
    "n = FloatSlider(min=1.0, max=N.value, step=1.0, value=30, readout_format='')\n",
    "\n",
    "# enforce K<=N and n<=N\n",
    "def update_range(*args):\n",
    "    K.max = N.value\n",
    "    n.max = N.value\n",
    "N.observe(update_range, 'value')\n",
    "\n",
    "# display the interactive plot\n",
    "interact(hypergeometric_pmf, N=N, K=K, n=n);\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "**Problem (1.2-15 in the textbook):**\n",
    "    \n",
    "<div class=\"alert alert-block alert-info\">\n",
    "\n",
    "\n",
    "Five cards are selected at random without replacement from a standard, thoroughly shuffled 52-card deck of playing cards. Let X equal the number of face cards (kings, queens, jacks) in the hand. Forty observations of $X$ yielded the following data:\n",
    "    \n",
    "$$\\text{Data}=\\{2, 1, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 2, 3, 0, 1, 1, 0, 3,\n",
    "\t\t\t\t\t\t\t1, 2, 0, 2, 0, 2, 0, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0.\\}$$\n",
    "    \n",
    "1.  Determine the pmf of $X$.<br>\n",
    "2. Draw a probability histogram for this distribution. <br>\n",
    "3. Determine the relative frequencies of 0, 1, 2, 3, and superimpose the relative frequency histogram on your probability histogram. \n",
    "    \n",
    "</div>\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:** \n",
    "\n",
    "1. $X$ has Hypergeometric distribution with parameters $(N=52, K=12, n=5)$.  $\\text{Range}(X)=\\{0,\\dots, 5\\}$ and \n",
    "\n",
    "$$f(x)=\\frac{{12\\choose x}{40 \\choose 5-x}}{{52 \\choose 5}}.$$\n",
    "\n",
    "2. Values of pmf are computed below.\n",
    "\n",
    "3. We compute the frequency pmf values and display the histogram.  $\\blacksquare$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True pmf =  [2.53181273e-01 4.21968788e-01 2.50900360e-01 6.60264106e-02\n",
      " 7.61843199e-03 3.04737280e-04]\n"
     ]
    }
   ],
   "source": [
    "# nbi:hide_in\n",
    "import numpy as np\n",
    "from scipy.special import comb\n",
    "\n",
    "N = 52\n",
    "K = 12\n",
    "n = 5\n",
    "\n",
    "# compute the range of X\n",
    "range_x = np.arange(max(0, n-(N-K)), min(n, K)+1, 1)\n",
    "\n",
    "def hyper_pmf(N,K,n,i):\n",
    "    pmf_val = comb(K, i, exact=True) * comb(N-K, n-i, exact=True) / comb(N, n, exact=True)\n",
    "    return pmf_val\n",
    "\n",
    "pmf_values = np.array([hyper_pmf(N,K,n,i) for i in range_x])\n",
    "print(\"True pmf = \",pmf_values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.325, 0.4  , 0.225, 0.05 ])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# nbi:hide_in\n",
    "def epmf(data):\n",
    "    erange_x, counts = np.unique(data, return_counts=True)\n",
    "    epmf_values = counts/data.size\n",
    "    return epmf_values, erange_x\n",
    "\n",
    "data = np.array([2, 1, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 2, 3, 0, 1, 1, 0, 3,\n",
    "                1, 2, 0, 2, 0, 2, 0, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0])\n",
    "\n",
    "epmf_values, erange_x = epmf(data)\n",
    "display(epmf_values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# nbi:hide_in\n",
    "# library\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.special import comb\n",
    "plt.rcParams[\"figure.figsize\"] = (10, 5)\n",
    "\n",
    "# plot setup\n",
    "plt.axhline(y=0, color='k')\n",
    "plt.xticks(range_x)\n",
    "\n",
    "plt.scatter(range_x,np.zeros(range_x.shape), color =\"red\", s=20)\n",
    "plt.bar(range_x, pmf_values, width=1, color=(1, 1, 1, 0), edgecolor='blue', linewidth=1, label=\"True histogran\")\n",
    "plt.bar(erange_x, epmf_values, width=0.9, color=(1, 1, 1, 0), edgecolor='green', linewidth=1,linestyle=\"--\", label=\"Relative frequency histogram\")\n",
    "plt.legend()\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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