study/textbooks/000_Introduction to Probability, Statistics, and Random Processes/ch3/notes.ipynb

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   "source": [
    "import os\n",
    "import sys\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import seaborn as sns\n",
    "\n",
    "sns.set_theme(style=\"whitegrid\", context=\"notebook\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6732353-51d5-4478-9cf8-5834e57e5a4e",
   "metadata": {},
   "source": [
    "# Chapter 3 Notes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f0046c2",
   "metadata": {},
   "source": [
    "## Random Variables (3.1.1 - 3.1.3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c73b89f",
   "metadata": {},
   "source": [
    "***Definition.*** Random Variables: \\\n",
    "A random variable $X$ is a function from the sample space to the real numbers. ie\n",
    "$$X : S \\to \\mathbb{R}$$\n",
    "\n",
    "Each outcome ($\\omega$) in the sample space must have a $X(\\omega)$ defined. \n",
    "\n",
    "> Note that $X$ is a deterministic function. The randomness comes from the fact that we dont know the inputs to $X$, ie the outcome of the random experiment"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "afa12aa6",
   "metadata": {},
   "source": [
    "***Definition.*** $X$ is a discrete random variable, if its range is countable"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "abd4a3ea",
   "metadata": {},
   "source": [
    "### More on $X$\n",
    "\n",
    "Let $\\Omega$ be a sample space and let $X$ be a random variable on $\\Omega$\n",
    "\n",
    "- $X$ is a function, and $\\forall \\omega \\in \\Omega, X(\\omega) \\in \\mathbb{R}\\quad$ (ie $X(\\omega)$ is defined on all **outcomes**) \n",
    "- $P_x(1)$ is asking: For event $A = \\{\\omega \\in \\Omega \\mid X(\\omega) = 1 \\}$, what is $P(A)$?\n",
    "- $X$ induces a partition of $\\Omega$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a7dab8e8",
   "metadata": {},
   "source": [
    "***Definition.*** Let $X$ be a discrete random variable with range $R_X = \\{x_1, x_2, x_3, \\dots\\}$ (finite or countably infinite). The function\n",
    "\n",
    "$$P_X(x_k) = P(X = x_k), \\text{for} k = 1,2,3,\\dots,$$\n",
    "\n",
    "is called the *probability mass function (PMF)* of $X$. (also called the probability distribution)\n",
    "\n",
    "Note that if $x \\notin R_X$, then $P_X(x) = 0$ "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84ca67da",
   "metadata": {},
   "source": [
    "## Properties of PMF\n",
    "- $0 \\geq P_X(x) \\geq 1, \\forall x$\n",
    "- $\\sum_{x \\in R_X}P_X(x) = 1$ \n",
    "- for any set $A \\subset R_X, P(X \\in A) = \\sum_{x \\in A} P_X(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "57323d97",
   "metadata": {},
   "source": [
    "## Independent Random Variables (3.1.4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6261f7d",
   "metadata": {},
   "source": [
    "***Definition.*** Consider two discrete random variables $X$ and $Y$. We say that $X$ and $Y$ are independent if:\n",
    "\n",
    "$$P(X=x, Y=y) = P(X=x)P(Y=y)$$\n",
    "\n",
    "In general, if two random variables are independent, then you can write\n",
    "\n",
    "$$P(X \\in A, Y \\in B) = P(X \\in A)(Y \\in B)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b0362c1",
   "metadata": {},
   "source": [
    "***Definition.*** Consider $n$ discrete random variables $X_1, X_2, X_3, \\dots, X_n$. We say that $X_1, X_2, X_3, \\dots, X_n$ are independent if:\n",
    "\n",
    "$$P(X_1 = x_1, X_2 = x_2, X_3 = x_3, \\dots, X_n = x_n) = P(X_1 = x_1)P(X_2 = x_2) \\dots P(X_n = x_n)$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e50c7c48",
   "metadata": {},
   "source": [
    "## Special Distributions (3.1.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d0bc1818",
   "metadata": {},
   "source": [
    "### Bernoulli Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "853e9acb",
   "metadata": {},
   "source": [
    "***Definition.*** A random variable $X$ is said to be a *Bernoulli* random variable with parameter *p*, shown as $X \\sim \\text{Bernoulli}(p),$ if its PMF is given by\n",
    "\n",
    "$$P_X(x) = \\begin{cases} \n",
    "p     & \\text{for } x = 1, \\\\ \n",
    "1 - p & \\text{for } x = 0, \\\\ \n",
    "0     & \\text{otherwise.} \n",
    "\\end{cases}$$\n",
    "\n",
    "where $0 \\lt p \\lt 1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "d2c94117",
   "metadata": {},
   "outputs": [
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",
      "text/plain": [
       "<Figure size 500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = 0.7\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(5, 3))\n",
    "ax.bar([0, 1], [1 - p, p], width=0.1, color=[\"steelblue\", \"steelblue\"])\n",
    "ax.set_xticks([0, 1])\n",
    "ax.set_xticklabels([\"0\", \"1\"])\n",
    "ax.set_ylim(0, 1)\n",
    "ax.set_xlabel(\"x\")\n",
    "ax.set_ylabel(\"$P_X(x)$\")\n",
    "ax.set_title(f\"Bernoulli(p={p}) PMF\")\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9e1caee1",
   "metadata": {},
   "source": [
    "The Bernoulli random variable is also called the *indicator* random variable. ie\n",
    "\n",
    "The indicator random variable $I_A$ for an event $A$ is defined by\n",
    "\n",
    "$$I_A = \\begin{cases} \n",
    "1     & \\text{if the event } A \\text{ occurs}, \\\\ \n",
    "0     & \\text{otherwise.} \n",
    "\\end{cases}$$\n",
    "\n",
    "$p = P(A)$, so we can write\n",
    "\n",
    "$I_A \\sim Bernoulli(P(A))$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4adc3d0e",
   "metadata": {},
   "source": [
    "### Geometric Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90768a90",
   "metadata": {},
   "source": [
    "***Definition.*** A random variable $X$ is said to be a *geometric* random variable with parameter *p*, shown as $X \\sim \\text{Geometric}(p),$ if its PMF is given by\n",
    "\n",
    "$$P_X(k) = \\begin{cases} \n",
    "p(1-p)^{k-1}     & \\text{for } k = 1, 2,3, \\dots, \\\\  \n",
    "0     & \\text{otherwise.} \n",
    "\\end{cases}$$\n",
    "\n",
    "where $0 \\lt p \\lt 1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "f8b684b2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 700x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = 0.3\n",
    "k = np.arange(1, 20)\n",
    "pmf = p * (1 - p) ** (k - 1)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(7, 3))\n",
    "ax.bar(k, pmf, width=0.4, color=\"steelblue\")\n",
    "ax.set_xlabel(\"k\")\n",
    "ax.set_ylabel(\"$P_X(k)$\")\n",
    "ax.set_title(r\"$X \\sim Geometric(p=$\" + f\"{p}\" + r\"$)$\")\n",
    "ax.set_xticks(k)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d61240ca",
   "metadata": {},
   "source": [
    "### Binomial Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5a1475ef",
   "metadata": {},
   "source": [
    "***Definition.*** A random variable $X$ is said to be a *binomial* random variable with parameter $n$ and $p$, shown as $X \\sim \\text{Binomial}(n,p),$ if its PMF is given by\n",
    "\n",
    "$$P_X(k) = \\begin{cases} \n",
    "\\binom{n}{k}p^k(1-p)^{n-k}     & \\text{for } k = 1, 2,3, \\dots, n \\\\  \n",
    "0     & \\text{otherwise} \n",
    "\\end{cases}$$\n",
    "\n",
    "where $0 \\lt p \\lt 1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "1c029592",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x300 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from math import comb\n",
    "\n",
    "def binomial_pmf(k, n, p):\n",
    "    return comb(n, k) * p**k * (1 - p)**(n - k)\n",
    "\n",
    "configs = [(20, 0.6), (20, 0.4)]\n",
    "\n",
    "fig, axes = plt.subplots(1, 2, figsize=(10, 3))\n",
    "for ax, (n, p) in zip(axes, configs):\n",
    "    k = np.arange(0, n + 1)\n",
    "    pmf = [binomial_pmf(ki, n, p) for ki in k]\n",
    "    ax.bar(k, pmf, width=0.4, color=\"steelblue\")\n",
    "    ax.set_xlabel(\"k\")\n",
    "    ax.set_ylabel(\"$P_X(k)$\")\n",
    "    ax.set_title(f\"Binomial(n={n}, p={p}) PMF\")\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2084d145",
   "metadata": {},
   "source": [
    "***Lemma*** If $X_1, X_2, \\dots, X_n$ are independent $Bernoulli(p)$ random variables, then the random variable $X$ defined by $X = X_1 + X_2 + \\dots + X_n$ has $Binomial(n,p)$ distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5d2d483f",
   "metadata": {},
   "source": [
    "### Negative Binomial (Pascal) Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de41c0a4",
   "metadata": {},
   "source": [
    "***Definition.*** A random variable $X$ is said to be a *pascal* random variable with parameter $m$ and $p$, shown as $X \\sim \\text{Pascal}(m,p),$ if its PMF is given by\n",
    "\n",
    "$$P_X(k) = \\begin{cases} \n",
    "\\binom{k-1}{m-1}p^m(1-p)^{k-m}     & \\text{for } k = m, m+1,m+1, \\dots \\\\  \n",
    "0     & \\text{otherwise} \n",
    "\\end{cases}$$\n",
    "\n",
    "where $0 \\lt p \\lt 1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "30067c10",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 900x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def pascal_pmf(k, m, p):\n",
    "    return comb(k - 1, m - 1) * p**m * (1 - p)**(k - m)\n",
    "\n",
    "m, p = 4, 0.3\n",
    "k = np.arange(m, m + 30)\n",
    "pmf = [pascal_pmf(ki, m, p) for ki in k]\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(9, 3))\n",
    "ax.bar(k, pmf, width=0.4, color=\"steelblue\")\n",
    "ax.set_xlabel(\"k\")\n",
    "ax.set_ylabel(\"$P_X(k)$\")\n",
    "ax.set_title(f\"Pascal(m={m}, p={p}) PMF\")\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71d6da5d",
   "metadata": {},
   "source": [
    "### Hypergeometric Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2e1643c0",
   "metadata": {},
   "source": [
    "***Definition.*** A random variable $X$ is said to be a *Hypergeometric* random variable with parameter $b$, $r$ and $k$, shown as $X \\sim \\text{Hypergeometric}(b,r,k),$ if its range is $R_X = \\{\\text{max}(0,k-r),\\text{max}(0,k-r)+1,\\text{max}(0,k-r)+2,\\text{min}(k,b)\\},$ and its PMF is given by\n",
    "\n",
    "$$P_X(x) = \\begin{cases} \n",
    "\\frac{\\binom{b}{x}\\binom{r}{k-x}}{\\binom{b+r}{k}}     & \\text{for } x \\in R_X \\\\  \n",
    "0     & \\text{otherwise} \n",
    "\\end{cases}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7fd3014",
   "metadata": {},
   "source": [
    "### Poisson Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c043ad82",
   "metadata": {},
   "source": [
    "***Definition.*** A random variable $X$ is said to be a *Poisson* random variable with parameter with $\\lambda$, shown as $X \\sim Poisson(\\lambda)$, if its range is $R_X = \\{ 0, 1, 2, 3, \\dots \\}$, and its PMF is given by\n",
    "\n",
    "$$P_X(k) = \\begin{cases} \n",
    "\\frac{e^{-\\lambda}\\lambda^k}{k!}     & \\text{for } x \\in R_X \\\\  \n",
    "0     & \\text{otherwise} \n",
    "\\end{cases}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "21d22a91",
   "metadata": {},
   "source": [
    "***Theorem*** Let $X \\sim Binomial(n, p = \\frac{\\lambda}{n})$, where $\\lambda \\gt 0$ is fixed. Then for any $k \\in \\mathbb{N} \\cup \\{0\\}$, we have\n",
    "\n",
    "$$\\lim_{x \\to \\infty}P_X(k) = \\frac{e^{-\\lambda}\\lambda^k}{k!} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05a5e06b",
   "metadata": {},
   "source": [
    "## Cumulative Distribution Function (3.2.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "043abe94",
   "metadata": {},
   "source": [
    "***Definition*** The cumulative distribution function (CDF) of random variable $X$ is defined as\n",
    "$$F_X(x) = P(X \\leq x) \\text{ for all } x \\in \\mathbb{R}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9768e347",
   "metadata": {},
   "source": [
    "NOTE: If we have the PMF, we can find the CDF from it. In particular, if $R_X = \\{ x_1, x_2, x_3, \\dots \\}$, we can write\n",
    "\n",
    "$$F_X(x) = \\sum_{x_k \\leq x}P_X(x_k)$$\n",
    "\n",
    "NOTE: For all $a \\leq b$, we have\n",
    "$$P(a \\lt X \\leq b) = F_X(b) - F_X(a)$$\n",
    "\n",
    "NOTE: To find $P(X \\lt x)$, for a discrete random variable, we can simply write\n",
    "\n",
    "$$P(X \\lt x) = P(X \\leq x) - P(X = x) = F_X(x) - P_X(x)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0fd28226",
   "metadata": {},
   "source": [
    "## Expectation (3.2.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d9ad903b",
   "metadata": {},
   "source": [
    "***Definition*** Let $X$ be a discrete random variable with range $R_X = \\{ x_1, x_2, x_3, \\dots \\}$ (finite or countably infinte). The *expected* value of $X$, denoted by $E[X]$ is defined as\n",
    "\n",
    "$$E[X] = \\sum_{x_k \\in R_X}x_kP(X=x_k) = \\sum_{x_k \\in R_X}x_kP_X(x_k)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e4c5500d",
   "metadata": {},
   "source": [
    "***Theorem*** Expectation is linear. We have\n",
    "- $E[aX +b] = aE[X] + b, \\forall a,b \\in \\mathbb{R}$\n",
    "- $E[X_1 + X_2 + \\dots + X_n] = E[X_1] + E[X_2] + \\dots + E[X_n]$, for any set of random variables $X_1, X_2, \\dots, X_n$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "110d9ecf",
   "metadata": {},
   "source": [
    "## Functions of Random Variables (3.2.3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ed681ae",
   "metadata": {},
   "source": [
    "Law of the unconscious statistician (LOTUS) for discrete random variables:\n",
    "\n",
    "$$E[g(X)] = \\sum_{x_k \\in R_X}g(x_k)P_X(x_k)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9c9c6a44",
   "metadata": {},
   "source": [
    "## Variance (3.2.4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5d7e781f",
   "metadata": {},
   "source": [
    "***Definition*** The **variance** of a random variable $X$ is defined as\n",
    "\n",
    "$$\\text{Var}(X) = E[(X - E[X])^2] = \\sum_{x_k \\in R_X} (x_k - E[X])^2P_X(x)$$\n",
    "\n",
    "Or\n",
    "\n",
    "$$\\text{Var}(X) = E[X^2] - (E[X])^2 = \\sum_{x_k \\in R_X}x_k^2P_X(x_k) - (E[X])^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1def8444",
   "metadata": {},
   "source": [
    "***Definition*** The *standard deviation* of a random variable $X$ is defined as\n",
    "\n",
    "$$\\text{SD}(X) = \\sigma_X = \\sqrt{\\text{Var}(X)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "def50a00",
   "metadata": {},
   "source": [
    "***Theorem*** For a random variable $X$ and real numbers $a$ and $b$,\n",
    "\n",
    "$$\\text{Var}(aX + b) = a^2\\text{Var}(X)$$\n",
    "\n",
    "$$\\text{SD}(aX + b) = |a|\\text{SD}(X)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e32be7d1",
   "metadata": {},
   "source": [
    "***Theorem*** If $X_1, X_2, \\dots, X_n$ are independent random variables and $X = X_1 + X_2 + \\dots + X_n$, then\n",
    "\n",
    "$$\\text{Var}(X) = \\text{Var}(X_1) + \\text{Var}(X_2) + \\dots + \\text{Var}(X_n) $$"
   ]
  }
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