2018/07/28/distributions.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import bernoulli, binom, uniform, norm\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bernoulli\n",
    "***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = 0.3\n",
    "rv1 = bernoulli(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0,\n",
       "       0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0,\n",
       "       0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0,\n",
       "       1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1,\n",
       "       0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv1.rvs(100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array(0.3), array(0.21), array(0.87287156), array(-1.23809524))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv1.stats('mvsk')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.7, 0.3])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv1.pmf([0,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.7, 1. ])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv1.cdf([0,1]) # Cumulative distribution function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Binomial\n",
    "***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# A new pill is created that cures cancer, however, it has downsides. If you take this pill\n",
    "# it has a 95% chance of curing you, otherwise the pill is fatal due to side effects.\n",
    "# Let's say you distribute a pill to 40 different individuals that have cancer. Each person\n",
    "# takes the pill.\n",
    "# What is the probability that between 1 and 3 people will die?\n",
    "# If each death results in a $7M settlement, what is the expected payout?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 40\n",
    "p = 0.05\n",
    "rv2 = binom(n, p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array(2.), array(1.9), array(0.65292863), array(0.37631579))"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv2.stats('mvsk')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 3, 1, 2, 4, 0, 2, 0, 2, 1, 1, 1, 3, 3, 2, 0, 2, 3, 4, 1, 1, 5,\n",
       "       2, 0, 1, 6, 1, 2, 1, 3, 1, 4, 2, 1, 1, 3, 1, 1, 3, 1])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# get 40 samples from the distribution\n",
    "# each value represents the number of people out of 40 that will die from the drug\n",
    "rv2.rvs(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.28512157e-01, 2.70551909e-01, 2.77671696e-01, 1.85114464e-01,\n",
       "       9.01215152e-02, 3.41513110e-02, 1.04850516e-02, 2.68038914e-03,\n",
       "       5.81926590e-04, 1.08898543e-04, 1.77676570e-05, 2.55038139e-06,\n",
       "       3.24390616e-07, 3.67730253e-08, 3.73260031e-09, 3.40517923e-10,\n",
       "       2.80031186e-11, 2.08072708e-12, 1.39931938e-13, 8.52770815e-15,\n",
       "       4.71268082e-16, 2.36224603e-17, 1.07374819e-18, 4.42276144e-20,\n",
       "       1.64883650e-21, 5.55397558e-23, 1.68642983e-24, 4.60234262e-26,\n",
       "       1.12463260e-27, 2.44929059e-29, 4.72670114e-31, 8.02495949e-33,\n",
       "       1.18790519e-34, 1.51566850e-36, 1.64236525e-38, 1.48183331e-40,\n",
       "       1.08321148e-42, 6.16336547e-45, 2.56095518e-47, 6.91215973e-50,\n",
       "       9.09494702e-53])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv2.pmf(range(n+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.12851216, 0.39906407, 0.67673576, 0.86185022, 0.95197174,\n",
       "       0.98612305, 0.9966081 , 0.99928849, 0.99987042, 0.99997932,\n",
       "       0.99999708, 0.99999963, 0.99999996, 1.        , 1.        ,\n",
       "       1.        , 1.        , 1.        , 1.        , 1.        ,\n",
       "       1.        , 1.        , 1.        , 1.        , 1.        ,\n",
       "       1.        , 1.        , 1.        , 1.        , 1.        ,\n",
       "       1.        , 1.        , 1.        , 1.        , 1.        ,\n",
       "       1.        , 1.        , 1.        , 1.        , 1.        ,\n",
       "       1.        ])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv2.cdf(range(n+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.27055191, 0.2776717 , 0.18511446])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "probs = rv2.pmf([1, 2, 3]) # probability of death for 1 to 3 people\n",
    "probs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7333380679338366"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the probability that between 1 and 3people will die from the drug\n",
    "probs.sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(range(n+1), rv2.pmf(range(n+1)), marker='+', alpha=0.3, color='#ff33cc', label='pmf')\n",
    "plt.plot(range(n+1), rv2.cdf(range(n+1)), marker='*', alpha=0.3, color='#33ccff', label='cdf')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "14000000.0"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# expected payout\n",
    "rv2.stats('m') * 7_000_000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Normal\n",
    "***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# you read on the internet that the average height of an adult\n",
    "# female is 5 feet 4 inches, with a standard deviation of 1/2 foot.\n",
    "# your company sells pants to shorter women. what is the probability \n",
    "# of randomly selecting a woman between 4'9\" and 5'2\" to be in your\n",
    "# commercial?\n",
    "# you company does not want to sell to the tallest, 10% of women.\n",
    "# what height is the cutoff where you will not offer your services?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# mean = 60 + 4 = 64\n",
    "# std = 6\n",
    "# lower = 48 + 9 = 57\n",
    "# upper = 60 + 2 = 62\n",
    "\n",
    "rv3 = norm(64, 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array(64.), array(36.), array(0.), array(0.))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv3.stats('mvsk')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(range(54, 74), rv3.pdf(range(54, 74)), marker='+', alpha=0.3, color='#ff33cc', label='pmf')\n",
    "plt.plot(range(54, 74), rv3.cdf(range(54, 74)), marker='*', alpha=0.3, color='#33ccff', label='cdf')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.03366645, 0.03433614, 0.03500902, 0.03568477, 0.03636304,\n",
       "       0.03704349, 0.03772576, 0.03840949, 0.03909429, 0.0397798 ,\n",
       "       0.04046563, 0.04115137, 0.04183663, 0.042521  , 0.04320408,\n",
       "       0.04388542, 0.04456463, 0.04524125, 0.04591487, 0.04658505,\n",
       "       0.04725133, 0.04791329, 0.04857047, 0.04922242, 0.0498687 ,\n",
       "       0.05050886, 0.05114243, 0.05176898, 0.05238805, 0.0529992 ,\n",
       "       0.05360196, 0.05419591, 0.05478059, 0.05535556, 0.0559204 ,\n",
       "       0.05647466, 0.05701792, 0.05754975, 0.05806975, 0.05857751,\n",
       "       0.05907261, 0.05955468, 0.06002331, 0.06047814, 0.06091879,\n",
       "       0.06134491, 0.06175614, 0.06215215, 0.06253261, 0.0628972 ])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# cannot do because the space is NOT discrete\n",
    "rv3.pdf(np.linspace(57, 62))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.4763904629507554"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# wrong\n",
    "rv3.pdf(np.linspace(57, 62)).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.12167250457438122, 0.36944134018176367)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# subtracting 2 areas\n",
    "a, b = rv3.cdf([57, 62])\n",
    "a, b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.24776883560738244"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# probabilty of selecting a woman between 57 and 62 inches\n",
    "b - a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "71.6893093932676"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# almost 6 feet is the max height of your customer\n",
    "rv3.ppf(.90)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}