exercises/exercises_21_discrete_RVs.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "eea5d985-784c-4445-8187-deabc1e28ccc",
   "metadata": {},
   "source": [
    "# Exercises for Section 2.1 Discrete random variables \n",
    "\n",
    "This notebook contains the solutions to the exercises from [Section 2.1 Discrete random variables]() of the **No Bullshit Guide to Statistics**."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "735dd137-e844-435b-a4f7-7b6a98279e2f",
   "metadata": {
    "tags": []
   },
   "source": [
    "### Notebooks setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "18365c68-a281-4bc1-921b-6a3d6e08471e",
   "metadata": {
    "id": "VFzPCGo6I1NJ"
   },
   "outputs": [],
   "source": [
    "import os\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "73ca00fa-0820-416b-959f-74126de30439",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Pandas setup\n",
    "pd.set_option(\"display.precision\", 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "13b33328-fdd3-4a55-9275-5db87fb55e81",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Figures setup\n",
    "DESTDIR = \"figures/prob\"  # where to save figures\n",
    "\n",
    "%config InlineBackend.figure_format = 'retina'"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5131f2bb-7233-44a8-bc3a-eb99c6767b0c",
   "metadata": {},
   "source": [
    "## Python functions defined in the text"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "65ba89d4-5eae-46bd-bcd5-ecd6a0ef0209",
   "metadata": {},
   "outputs": [],
   "source": [
    "# define the probability mass function for the random variable D\n",
    "def fD(d):\n",
    "    if d in {1,2,3,4,5,6}:\n",
    "        return 1/6\n",
    "    else:\n",
    "        return 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "cde3d156-cdf7-4aee-85b0-10dfbfefdfee",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "def fH(h):\n",
    "    lam = 20\n",
    "    return lam**h * np.exp(-lam) / np.math.factorial(h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "724b5396-14b3-40ce-b19d-72b0e04e6579",
   "metadata": {},
   "outputs": [],
   "source": [
    "def FH(b):\n",
    "    return sum([fH(h) for h in range(0,b+1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "29a8567f-3c23-478f-832a-64fe1aa54189",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "2c01b315-47f3-4789-8b0f-fce90afebf30",
   "metadata": {},
   "source": [
    "## Exercies 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4c988c0c-fdca-45db-9064-7ca894aee994",
   "metadata": {},
   "source": [
    "#### E2.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "9fd05121-0e80-490f-afd1-1a07edea0fe0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6666666666666666"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum([fD(d) for d in range(2,5+1)])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15eb5c67-4212-4380-a8d1-3e46a8e6811f",
   "metadata": {},
   "source": [
    "#### E2.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "d46b2b74-90d9-44c0-8cb8-9bc51457d0bd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.25, 0.25, 0.25, 0.25]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define the probability mass function for the random variable D4\n",
    "def fD4(d):\n",
    "    if d in {1,2,3,4}:\n",
    "        return 1/4\n",
    "    else:\n",
    "        return 0\n",
    "\n",
    "[fD4(d) for d in range(1,4+1)]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a1aba98-d84c-460f-89b8-04c9903f6dda",
   "metadata": {},
   "source": [
    "#### E2.3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "cd80ce88-5e4b-453d-8292-dc74ce85465e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, '$f_Y$')"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 261,
       "width": 377
      },
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ys = [1,2,3]\n",
    "fYs = [0.2, 0.5, 0.3]\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "plt.stem(ys, fYs, basefmt=\" \")\n",
    "ax.set_xticks([1,2,3])\n",
    "ax.set_xlim(0,4)\n",
    "ax.set_yticks([0])\n",
    "ax.set_xlabel('$y$')\n",
    "ax.set_ylabel('$f_Y$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "049faa1c-8047-4f0c-9a96-b8eb238a6e11",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.2, 0.5, 0.3]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# fY(1), fY(2), fY(3) are\n",
    "fYs"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "693e87e6-0a05-446d-9adc-b5c83d2eede7",
   "metadata": {},
   "source": [
    "#### E2.4\n",
    "\n",
    "Define $A$ as the set of outcomes $\\{5,6\\}$,\n",
    "which is a subset of the sample space $\\{1,2,3,4,5,6\\}$ for the random variable $D$.\n",
    "The complement of $A$ is $A^c= \\{1,2,3,4\\}$,\n",
    "and we know $ \\Pr(A^c)= \\frac{4}{6}$.\n",
    "Applying the complement rule,\n",
    "we obtain $\\Pr(A) = 1 - \\Pr(A^c)  = 1 - \\frac{4}{6} = \\frac{2}{6}$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3ace9745-cff9-4934-95fd-dda239bba74e",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e442adfa-b76f-4a73-8d7e-95e2ef9b4679",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e73a7427-4ae7-4b8a-aa5a-b06d7d763d5c",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8db0c5ac-39d0-4a9a-8f91-f27d51f848d4",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "61744e95-25a6-46a8-9af7-4b87a6ea65f3",
   "metadata": {},
   "source": [
    "## Exercies 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "555d8e29-c63f-4cbb-81bd-85d67ba908ff",
   "metadata": {},
   "source": [
    "#### E2.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "829a384f-8309-433b-9f9f-eba7f5315a24",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9815299064117703"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FH(30) - FH(10-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "48f9d3ba-b8fa-493a-a1dd-b34e9e3b5cae",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "e2e21acd-b3ec-4ee1-ba3e-ed0e38d410dc",
   "metadata": {},
   "source": [
    "#### E2.6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "b52e61f4-4c41-44d9-9700-b62aa03a6929",
   "metadata": {},
   "outputs": [],
   "source": [
    "# define the probability mass function for the random variable D\n",
    "def fD(d):\n",
    "    if d in {1,2,3,4,5,6}:\n",
    "        return 1/6\n",
    "    else:\n",
    "        return 0\n",
    "\n",
    "def FD(b):\n",
    "    return sum([fD(d) for d in range(0,b+1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "42e477cd-718f-4fcf-b757-36ac1774fd54",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.16666666666666666,\n",
       " 0.3333333333333333,\n",
       " 0.5,\n",
       " 0.6666666666666666,\n",
       " 0.8333333333333333,\n",
       " 0.9999999999999999)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FD(1), FD(2), FD(3), FD(4), FD(5), FD(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7344105c-a62c-4887-88cd-dd816f9ec396",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "d20b4e0b-1023-4df8-ba22-cefca7f7714c",
   "metadata": {},
   "source": [
    "#### E2.7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8ab8ee5c-7708-473b-bbf0-7194d1a2d451",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, '$F_D$')"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 261,
       "width": 386
      },
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import randint\n",
    "\n",
    "rvD = randint(1, 6+1)\n",
    "ds = np.linspace(0, 7, 400)\n",
    "FDs = rvD.cdf(ds)\n",
    "\n",
    "ax = sns.lineplot(x=ds, y=FDs)\n",
    "ax.set_xlabel('b')\n",
    "ax.set_ylabel('$F_D$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2fa057a7-7d69-48ec-88a3-70583f5a1ea3",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "b20d9c85-a069-4dfb-a2e4-15ed8aa087f0",
   "metadata": {},
   "source": [
    "#### E2.8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "b2e8319b-cbc8-4dd3-a1c7-74a072210bb9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.3814219494471548,\n",
       " 0.47025726683924,\n",
       " 0.5590925842313252,\n",
       " 0.6436976484142636,\n",
       " 0.7206113431260257)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FH(18), FH(19), FH(20), FH(21), FH(22)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6c182b7c-9d66-4e4c-ab9b-4821b26523be",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "15ab6383-5f9e-43ac-9c59-3966bfba4aaf",
   "metadata": {},
   "source": [
    "#### E2.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "a5be55b0-5d96-4629-95e6-55abb34cf492",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.22107420155444096, 0.29702839792467384, 0.3814219494471548)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FH(16), FH(17), FH(18)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "af40ef23-5272-4c73-ad86-a2c886d38da0",
   "metadata": {},
   "source": [
    "So $F_H^{-1}(0.25) = 17$.\n",
    "This is the smallest value $b$ for which $F_H(b)$ is greater than $0.25$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "5b4428fa-3827-4466-895f-295734a54f66",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "17.0"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ALT. by calling the inverse-CDF function on rvH object\n",
    "from scipy.stats import poisson\n",
    "rvH = poisson(20)\n",
    "rvH.ppf(0.25)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1a58bcc0-4c4c-41b2-9a7a-c135418cba3e",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "216898fb-7226-4647-b60b-7f3ac9b3e98f",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1b7403d8-ab9b-4a17-90eb-b2d41057c2f9",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "812b1b34-1a1c-4e52-92e0-fa16724444c5",
   "metadata": {},
   "source": [
    "## Exercies 3\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8749355-10db-4d3b-94f2-b14c706a10c5",
   "metadata": {},
   "source": [
    "#### E2.10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "91e760ff-33ae-46da-a94a-1becc7b4f1f1",
   "metadata": {},
   "outputs": [],
   "source": [
    "# define the probability mass function for the random variable C\n",
    "def fC(c):\n",
    "    if c in {\"heads\", \"tails\"}:\n",
    "        return 1/2\n",
    "    else:\n",
    "        return 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "29ccab7f-0eb9-4c99-b803-593fbd089bd9",
   "metadata": {},
   "outputs": [],
   "source": [
    "def w(c):\n",
    "    if c == \"heads\":\n",
    "        return 1.90\n",
    "    else:\n",
    "        return 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "d3935db8-0f81-4242-ac64-a01c044696e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.95"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum([w(c) * fC(c) for c in {\"heads\", \"tails\"}])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cb4656c2-b6a2-4835-ace1-d4d9b456a553",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "f5b4b749-f3b1-4f68-9f36-456d9bb5dce3",
   "metadata": {},
   "source": [
    "#### E2.11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "2866dd82-b243-4397-a2d2-b9a18a87625f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1666666666666665"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def u(d):\n",
    "    if d == 6:\n",
    "        return 4\n",
    "    elif d == 5:\n",
    "        return 3\n",
    "    else:\n",
    "        return 0\n",
    "\n",
    "sum([u(d)*rvD.pmf(d) for d in range(1,6+1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "090e07e7-4a06-43b2-9c35-c587cf5f600a",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "d8a02dea-d71f-40a2-8ccd-39db92025fa2",
   "metadata": {},
   "source": [
    "#### E2.12\n",
    "\n",
    "See derivation in the book."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "44a2a8b8-5008-44b3-b210-4d2348f9dc04",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "314b3124-1487-4eb6-9fca-13787460440e",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "adad89e9-ffaf-4b54-bf92-627c0459e113",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e065272f-c7c8-473a-8e0d-c5dd6b62a125",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "9f25ccf9-1215-4b12-9307-f038999c6c9b",
   "metadata": {},
   "source": [
    "#### E2.13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "1b99e67b-0108-4094-9477-43e24def09f9",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fD20(d):\n",
    "    if d in range(1,20+1):\n",
    "        return 1/20\n",
    "    else:\n",
    "        return 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "c4cc00a6-1599-40c8-b01d-46f4cb9d4b37",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10.5"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Expected value of the random variable D20\n",
    "sum([d*fD20(d) for d in range(1,20+1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6a932263-0d13-4fdb-a2e2-b394ffd0b82e",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f930627c-0918-4e07-a2f6-cabec5167d62",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "057dcbc1-3edd-43dc-9777-8ee5d5cb2abb",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "0df9abee-0ef6-4692-bfd6-d05d0fced5af",
   "metadata": {},
   "source": [
    "## Exercies 4"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76304f9d-be50-4d8c-a96e-bd3daa201c22",
   "metadata": {},
   "source": [
    "#### E2.14"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "be03149b-a7e1-4f93-bdb8-9d31eb6efbb6",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import randint\n",
    "rvD = randint(1,6+1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "ce5147af-33f6-4bbf-8338-4d3443e10071",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.16666666666666666,\n",
       " 0.16666666666666666,\n",
       " 0.16666666666666666,\n",
       " 0.16666666666666666,\n",
       " 0.16666666666666666,\n",
       " 0.16666666666666666]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[rvD.pmf(d) for d in range(1,6+1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "a8c338f9-778f-4b0f-90be-24743bc54b43",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9999999999999999"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum([rvD.pmf(d) for d in range(1,6+1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "d7c6c1dc-9bad-4f69-9770-10632eea3883",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6666666666666666"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# F_D(4) = \n",
    "sum([rvD.pmf(d) for d in range(1,4+1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "adaa1ba8-8117-4091-a182-370d7533c211",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6666666666666666"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# F_D(4) = \n",
    "rvD.cdf(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "4e4a5c4f-3a73-4ca5-a1cc-bfa6366023b0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3.5, 2.9166666666666665)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# mu =      sigma^2 =\n",
    "rvD.mean(), rvD.var()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "fec63594-20c4-4c2e-94b1-1099ccd424d3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8333333333333333"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def w(d):\n",
    "    if d == 6:\n",
    "        return 5\n",
    "    else:\n",
    "        return 0\n",
    "\n",
    "sum([w(d)*rvD.pmf(d) for d in range(1,6+1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "adde68cc-2136-44d3-9d65-6511dd096362",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Calling:\n",
    "# rvD.expect(w)\n",
    "\n",
    "# Raises an error:\n",
    "# ValueError: The truth value of an array is ambiguous.\n",
    "#             Use a.any() or a.all()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f9315027-bea3-4ac4-8829-6d21e3fb466a",
   "metadata": {},
   "source": [
    "This error occurs because the Python function `w` chokes\n",
    "if we pass it an `array` of inputs to evaluate,\n",
    "which is what the method `rvD.expect` does (for efficiency reasons).\n",
    "\n",
    "We can easily solve this by \"vectorizing\" the function using the NumPy helper method `vectorize`, which makes any Python function work with vector inputs (arrays of numbers)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "0a1d3a39-da7c-4bcf-b0e0-2de5b85b9718",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8333333333333333"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create a \"vectorized\" version of the function w\n",
    "vw = np.vectorize(w)\n",
    "rvD.expect(vw)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "71119ac7-240c-48db-9893-aba4b03d1b76",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "a51aea65-17e2-4d1c-827e-cc74c7602a63",
   "metadata": {},
   "source": [
    "#### E2.15"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "e3c95eeb-c82a-4a22-b1ae-440bc4de2fdd",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import randint\n",
    "rvD20 = randint(1, 20+1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "80225c96-5db6-4674-8661-c802504d63df",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.05"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rvD20.pmf(7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "fd62057a-af0c-4f3d-9083-4c9724320836",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum([rvD20.pmf(d) for d in range(1,4+1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "18572a46-0a6c-465a-8fca-9449c7e2b7d5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rvD20.cdf(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "89f86c78-338d-49aa-879c-68948c5d582c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(10.5, 33.25, 5.766281297335398)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rvD20.mean(), rvD20.var(), rvD20.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ee48ccf4-7f33-4852-b8b7-5cce99ecbcfb",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "8ee69d86-36ef-4208-a3c0-b28eb0fb811c",
   "metadata": {},
   "source": [
    "#### E2.16"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "d311b146-c5fd-4586-b0bf-52ef4441fd09",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import poisson\n",
    "rvM = poisson(40)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "f0bd9797-ae1c-4c71-a7a1-1a65a781ee9c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(40.0, 40.0, 6.324555320336759)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rvM.mean(), rvM.var(), rvM.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "d38dc295-c16f-4ea1-a0f5-7df7d503c097",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6503813622782281"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum([rvM.pmf(m) for m in range(33,44+1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "ccd6fdeb-822d-4210-a578-cf2d8aab3138",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6503813622782224"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rvM.cdf(44) - rvM.cdf(33-1) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "1c29b964-5c20-46dc-a314-08fdf0add7ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "51.0"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rvM.ppf(0.95)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "7adee4fa-6eac-45a0-9e3c-d4b1733dce53",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([40, 42, 37, 35, 44, 37, 48, 38, 46, 39])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rvM.rvs(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "872439a2-b973-41e2-9679-1999fed9e3a2",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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