10_hw_cp220_Q.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "view-in-github"
   },
   "source": [
    "<a href=\"https://colab.research.google.com/github/AshrithSagar/CP220-M4RAS-2024/blob/main/10_hw_cp220_Q.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Pq0cFqqKTflw"
   },
   "source": [
    "**HW10**\n",
    "Discrete Random Variables\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "BeDeDR2oTty-"
   },
   "source": [
    "---\n",
    "Let $\\Omega$ be the sample space  of possible outcomes (could be discrete or continuous).\n",
    "\n",
    "Then a Random Variable is a *function* from the $\\Omega$ to the Real line, i.e. it attaches a real number to every element of the sample space.\n",
    "\n",
    "Examples:\n",
    "\n",
    "$\\Omega$ = results of finite number of coin tosses = $H, T, HH, HT, TH, TT, HHT, HHH, ....} is a countably infinite outcome space.\n",
    "\n",
    "1. $X:\\Omega → ℝ$ where $X(toss\\_seq) = 1$ if $toss\\_seq \\in \\Omega$ has even humber of heads and $X(toss\\_seq)=0$ otherwise.\n",
    "\n",
    "2. $X:\\Omega → ℝ$ where $X(toss\\_seq) = n$ if $toss\\_seq \\in \\Omega$ has $n$ number of heads.\n",
    "\n",
    "3. Any measurements with sensors can be represented as random variables, (which take on continuous values).\n",
    "\n",
    "Note that the random variable can have positive or negative values.\n",
    "\n",
    "Functions of random variables are also random variables.\n",
    "\n",
    "While technically  random variables (RV)are functions in themselves (from the outcome space to reals) - practically, we seldom need to do manipulations with outcomes and instead work with rvs directly as any other numerical variable."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Ggb1rN4rlAKy"
   },
   "source": [
    "---\n",
    "A discrete RV takes finite or countably infinite values on the real line\n",
    "\n",
    "A continuous RV takes uncountably infinite continuous values.\n",
    "\n",
    "Probablity Mass Function (PMF) of a discrete random variable attaches a probability number to each value of the variable.\n",
    "\n",
    "*Example:*\n",
    "\n",
    "1.\n",
    "\n",
    "If $X$ represents number of heads in 100 coin tosses, then\n",
    "\n",
    "$p_X(X=k) = ^{100}C_kp^k(1-p)^{n-k}$ for $0 \\leq k \\leq 100$, where $^{100}C_k$ is 100 choose $k$ - number of sequences of 100 tosses with $k$\n",
    "heads.\n",
    "\n",
    "\n",
    "2.\n",
    "\n",
    "If $Y$ represents whether even number of heads appeared or not in N tosses, then for a fair coin\n",
    "\n",
    "$p_Y(Y=1) = \\Sigma_{k=0}^{k=int(N/2)} P_X(X=k) = 0.5$\n",
    "\n",
    "$p_Y(Y=0) = 1- P_Y(Y=1) = 0.5$\n",
    "\n",
    "\n",
    "When we are dealing with only one variable, we can drop the subscript $_X$ with the pmf.\n",
    "\n",
    "We also write $p_X(X=x)$ as $p_X(x)$ of $p(x)$\n",
    "\n",
    "To calculate $p_X(x)$, find the pre-image of $x$ as $E=X^{-1}(x) \\subset \\Omega$ and find its probability (by adding probabilities of the outcomes in E)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "4wQ67KIGpILq"
   },
   "source": [
    "---\n",
    "**Moments of a RV**\n",
    "\n",
    "\n",
    "*$\\underline{Mean:}$*\n",
    "\n",
    "$E_X(X) = \\sum xp_X(x) = μ$.\n",
    "\n",
    "When the rv is clear, we can drop subscript $_X$\n",
    "\n",
    "Let $f(X)$ be a function of the rv $X$, then\n",
    "\n",
    "$E_X(f(X)) = \\sum f(x)p_X(x)$\n",
    "\n",
    "*$\\underline{k^{th}\\ raw\\ moment\\ and\\ central\\ moment}$*\n",
    "\n",
    "$E(X^k) = \\sum x^kp_X(x)$ is the raw moment\n",
    "\n",
    "$E(X-\\mu)^k) = \\sum (x-\\mu)^kp_X(x)$ is the central moment.\n",
    "\n",
    "*$\\underline{variance}$*\n",
    "\n",
    "$E((X-\\mu)^2) = \\sum (x-\\mu)^2 p_X(x) = \\sigma_X^2$ is the second central moment.\n",
    "\n",
    "*$\\underline{Standard\\ deviation}$*\n",
    "\n",
    "$std. dev = \\sqrt{variance} = \\sigma_X$.\n",
    "\n",
    "It gives a measure of the spread of the distribution around the mean.\n",
    "\n",
    "*$\\underline{Moment\\ Generating\\ Function}$*\n",
    "\n",
    "$M_X(s) = E(e^{sX}) = \\sum_x e^{sx}p_X(x) = \\sum_x  (1+sx+\\frac{s^2x^2}{2!} + \\frac{s^3x^3}{3!}+...) p_X(x) = 1 +sEX + \\frac{s^2EX^2}{2!}+\\frac{s^3EX^3}{3!}+...$\n",
    "\n",
    "Thus if we have a closed analytical form of $M_X(s)$, we can calculate the $k^{th}$ moment as $EX^k = \\frac{d^kM_X(s=0)}{ds^k}$\n",
    "\n",
    "*$\\underline{k^{th} Standardized\\ Moment}$*\n",
    "\n",
    "$E((\\frac{X-\\mu}{\\sigma})^k)$\n",
    "\n",
    "\n",
    "*$\\underline{Skewness}$*\n",
    "\n",
    "Quantifies the asymmetry in the distribution and is given by the 3rd standardized moment:\n",
    "\n",
    "$Skewness(X) = E((\\frac{X-\\mu}{\\sigma})^3)$\n",
    "\n",
    "\n",
    "*$\\underline{Kurtosis}$*\n",
    "\n",
    "Describes how much \"tail\" a distribution has, i.e how much mass the pmf has beyond the mean - at the extreme ends and is given by the 4th standardized moment.\n",
    "\n",
    "$Kurt(X)=E((\\frac{X-\\mu}{\\sigma})^4)$\n",
    "\n",
    "For example: When we look at the probability of returns on an investment, then if it has a high kurtosis, then it implies more volatility of the investment.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "idJBeHGevvua"
   },
   "source": [
    "**Entropy of a RV**\n",
    "\n",
    "Measures the \"randomness\" of the rv, by characterizing the mean of the pmf (treated as a function)\n",
    "\n",
    "$H(X) = E(-log(p_X(X)) = \\sum_x p_X(x)\\frac{1}{log(p_X(x))}$\n",
    "\n",
    "The log is usually with base 2 to get the answer in \"number of bits\"."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "fF6AALGHoZrm"
   },
   "source": [
    "---\n",
    "**Bernoulli RV**\n",
    "\n",
    "\n",
    "This RV, models coin toss type binary outcomes.\n",
    "\n",
    "$X \\in \\{0,1\\}$ and $p_X(1)=p$, $p_X(0)=1-p$\n",
    "\n",
    "We denote this as\n",
    "\n",
    "$X\\ \\sim Ber(p)$\n",
    "\n",
    "$\\underline{Mean}$\n",
    "\n",
    "$E(X) = 0.(1-0) + 1.p = p$\n",
    "\n",
    "Consider the function $f(X) = X^2$,\n",
    "\n",
    "then:\n",
    "\n",
    "$E(f(X)) = E(X^2) = 0^2(1-p) + 1^2p = p$\n",
    "\n",
    "$\\underline{Variance}$\n",
    "\n",
    "$E((X-\\mu)^2 = \\sum (x-\\mu)^2 p_X(x) = (0-p)^2p_X(0) + (1-p)^2p_X(1) = p^2(1-p)+(1-p)^2p = p(1-p)$\n",
    "\n",
    "*$\\underline{Entropy}$*\n",
    "\n",
    "$H(X) = E(-log(p_X(x)) = (1-p)log(1/(1-p)) + plog(1/p)$\n",
    "\n",
    "*$\\underline{MGF}$*\n",
    "\n",
    "$M_X(s) = E(e^{sX}) = e^{s.0}(1-p)+e^{s.1}p = 1+p(e^s-1)$\n",
    "\n",
    "Crosscheck:\n",
    "\n",
    "$EX = M^{'}_X(0) = p$\n",
    "\n",
    "$EX^2 = M^{''}_X(0) = p$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "OCHVywe6-dr8"
   },
   "source": [
    "---\n",
    "**Joint Distributions**\n",
    "\n",
    "Let X, Y be two RV, then\n",
    "\n",
    "$p_{XY}(X=x,Y=y)$ is the pmf of the joint distribution.\n",
    "\n",
    "\n",
    "*$\\underline{Marginals}$*:\n",
    "\n",
    "$p_Y(Y=y) = \\sum_x p_{XY}(X=x,Y=y)$\n",
    "\n",
    "$p_X(X=x) = \\sum_y p_{XY}(X=x,Y=y)$\n",
    "\n",
    "Are  the marginals.\n",
    "\n",
    "*$\\underline{Conditionals}$*\n",
    "\n",
    "$p_{X|Y=y}(X=x|Y=y)$ is the conditional pmf of X given $Y=y$\n",
    "\n",
    "The joint probability can be written in terms of the conditional as:\n",
    "\n",
    "$p_{XY}(x,y) = p_{X|Y}(x|y)p_Y(y)$\n",
    "\n",
    "*$\\underline{Independence}$*\n",
    "\n",
    "Two variables are independent, iff:\n",
    "\n",
    "$p_{XY}(X=x,Y=y) = p_X(X=x)p_Y(Y=y)$ $\\forall x, y$\n",
    "\n",
    "For $n$ variables to be independent, all subsets should be independent.\n",
    "\n",
    "*$\\underline{Expectation\\ of\\ a\\ sum\\ of\\ rv}$*\n",
    "\n",
    "Let $Z = X + Y$\n",
    "\n",
    "Then $EZ = E_{XY}(X+Y) = \\sum_x \\sum_y (x+y)p_{XY}(x,y) = \\sum_x \\sum_y xp_{XY}(x,y) + \\sum_x\\sum_y yp_{XY}(x,y) $\n",
    "\n",
    "$\\sum_x \\sum_y xp_{XY}(x,y) = \\sum_x x \\sum_y p_{XY}(x,y) = \\sum_x x p_X(x) = EX$\n",
    "\n",
    "$\\sum_x\\sum_y yp_{XY}(x,y) = \\sum_y\\sum_x yp_{XY}(x,y) = EY$\n",
    "\n",
    "Therefore $E(X+Y) = EX + EY$\n",
    "\n",
    "This can be extended to sum of $n$ random variables.\n",
    "\n",
    "i.e.  $E(\\sum_i X_i) = \\sum_i EX_i$\n",
    "\n",
    "Note that this holds regardless of whether the rvs are independent or not.\n",
    "\n",
    "Using this, we can rewrite the variance of a rv as\n",
    "\n",
    "$\\sigma_X^2 = E(X-\\mu)^2 = E(X^2 -2X\\mu +\\mu^2) = EX^2 -2\\mu EX +\\mu^2 = EX^2 - \\mu^2 = EX^2 - (EX)^2$ where $\\mu = EX$\n",
    "\n",
    "*$\\underline{Expectation\\ of\\ Product\\ of\\ rvs}$*\n",
    "\n",
    "$E(XY) = EX EY$ if X, Y are independent.\n",
    "\n",
    "$E(XY) = \\sum_x \\sum_y xyp_{XY}(x,y) = \\sum_x \\sum_y XYp_X(x)p_Y(y) = \\sum_x xp_X(x) \\sum_y yp_Y(y) = \\sum_x xp_X(x)EY = EY \\sum_x xp_X(x) = EYEX $\n",
    "\n",
    "*$\\underline{Variance\\ of\\ sum\\ of\\ rvs}$*\n",
    "\n",
    "$X = \\sum_i X_i$ and if $X_i$ are i.i.d (independent, identically  distributed), then\n",
    "\n",
    "$Var(X) = \\sum_i Var(X_i)$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "xs_JHABP-zDn"
   },
   "source": [
    "---\n",
    "**Joint Distributions**\n",
    "\n",
    "Let X, Y be two RV, then\n",
    "\n",
    "$p_{XY}(X=x,Y=y)$ is the pmf of the joint distribution.\n",
    "\n",
    "Iff the two variables are independent, then:\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "h8FRaS_974on"
   },
   "source": [
    "---\n",
    "**Binomial RV**\n",
    "\n",
    "Let $X_i\\ \\sim Ber(p)$ be a Bernoulli RV\n",
    "and Let $X_i, X_j$ be independant for all pairs $i \\neq j$ (also referred to as independent and identically distributed or iid).\n",
    "\n",
    "Consider $Z = X_1 + X_2 + X_3 + ... +X_n$\n",
    "\n",
    "then,\n",
    "\n",
    "$Z\\ \\sim Bin(n,p)$ with a pmf\n",
    "\n",
    "$p_Z(k) = ^nC_kp^k(1-p)^{n-k}$\n",
    "\n",
    "Note:  $Bin(n=1,p) = Ber(p)$\n",
    "\n",
    "$\\mu_Z = EZ = \\sum_{k=0}^{k=n} k p_Z(k) = \\sum_{k=0}^{k=n} k (^nC_k)p^k(1-p)^{n-k} = \\sum_i EX_i = np $\n",
    "\n",
    "$\\sigma^2_Z = E((Z-\\mu_Z)^2) = np(1-p)$ (can be easily derived by using formulae of expectations of sums and products )\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "id": "sG2JqkqT6Nn6"
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "id": "Yc_6ZJif7df3"
   },
   "outputs": [],
   "source": [
    "rng = np.random.default_rng(100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "hpo9EkE6wigw",
    "outputId": "7c46e8c3-5c3a-49ee-d79b-42fcfc4939a9"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample mean = 0.357, theory mean=0.34\n",
      "sample variance = 0.22955099999999998, theory variance = 0.2244\n",
      "sample entropy = 0.9401647521318283, theory entropy=0.9248187049730301\n"
     ]
    }
   ],
   "source": [
    "#Bernoulli RV (also binary rv)\n",
    "n, p = 1, 0.34\n",
    "nSamples = 1000\n",
    "#Draw nSamples\n",
    "s = rng.binomial(n,p,nSamples)\n",
    "\n",
    "mu_sample = np.sum(s)/nSamples\n",
    "var_sample = np.sum((s-mu_sample)**2)/nSamples\n",
    "\n",
    "print(f\"sample mean = {mu_sample}, theory mean={p}\")\n",
    "print(f\"sample variance = {var_sample}, theory variance = {p*(1-p)}\")\n",
    "\n",
    "#sample entropy\n",
    "n_1 = np.sum(s)\n",
    "n_0 = nSamples - n_1\n",
    "entropy_sample  = np.log2((nSamples/n_1)**(n_1/nSamples)*(nSamples/n_0)**(n_0/nSamples))\n",
    "entropy_theory = np.log2((1/p)**p*(1/(1-p))**(1-p))\n",
    "print(f\"sample entropy = {entropy_sample}, theory entropy={entropy_theory}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 490
    },
    "id": "1PLF_LPz1eDY",
    "outputId": "7ab867ba-3fe0-4960-c4d0-7e5968ba5237"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Entropy versus probability for Binary RV')"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Entropy versus probability for binary(Bernoulli)  rv\n",
    "\n",
    "p_x = np.arange(0.05,0.95,0.01)\n",
    "H_p = np.log2((1/p_x)**p_x*(1/(1-p_x))**(1-p_x))\n",
    "plt.plot(p_x,H_p)\n",
    "plt.xlabel('p')\n",
    "plt.ylabel('H(p)')\n",
    "plt.title(\"Entropy versus probability for Binary RV\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "A-n38KvT2XP1"
   },
   "source": [
    "---\n",
    "Max entropy happens when $p=0.5$, where $H(p) = 1.0$\n",
    "\n",
    "At this value of $p$, it is equallly likely to have $0$ or $q$ and hence max uncertainty.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "5xFuEgya2gbf"
   },
   "source": [
    "**Empirical Distribution**\n",
    "\n",
    "Let $D = \\{d_1, d_2, ....., d_n\\}$ be the empiricaly observed data. Then the empirical distribution is:\n",
    "\n",
    "$ \\hat{p}_X(k) = \\frac{\\sum_i I(d_i=k)}{n}$ where $I(cond) = 1$ if cond is true and 0 otherwise.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 484
    },
    "id": "6tOM54mA3sPx",
    "outputId": "6cd3ac50-58d7-467d-ba4d-faed9f08ba6a"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample mean = 8.4605, theory mean=8.5\n",
      "sample variance = 5.47943975, theory variance = 5.609999999999999\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x124482700>]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Binomial rv\n",
    "n, p = 25, 0.34\n",
    "nSamples = 10000\n",
    "#Draw nSamples\n",
    "s = rng.binomial(n,p,nSamples)\n",
    "\n",
    "mu_sample = np.sum(s)/nSamples\n",
    "var_sample = np.sum((s-mu_sample)**2)/nSamples\n",
    "\n",
    "print(f\"sample mean = {mu_sample}, theory mean={n*p}\")\n",
    "print(f\"sample variance = {var_sample}, theory variance = {n*p*(1-p)}\")\n",
    "\n",
    "#Get empirical distribution\n",
    "unique, counts = np.unique(s,return_counts=True)\n",
    "hist=np.zeros(n+1)\n",
    "hist[unique] = counts\n",
    "#print(hist)\n",
    "plt.plot(range(n+1),hist)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "2N6X8XBFFbRN"
   },
   "source": [
    "---\n",
    "**Geometric RV**\n",
    "Repeated independent trials till first \"success\" leads to geometric RV.\n",
    "\n",
    "Example, number of independent coin tosses till first Head, where probability of head is $p$\n",
    "\n",
    "$X\\ \\sim Geom(p)$\n",
    "\n",
    "$p_X(X=k) = (1-p)^{k-1}p$,\n",
    "\n",
    "$k=1,2,3,...$\n",
    "\n",
    "$EX = \\sum_k kp_X(k) = \\sum_k k(1-p)^{k-1}p = \\frac{1}{p}$\n",
    "\n",
    "$EX^2 = \\sum_k k^2p_X(k) = \\sum_k k^2(1-p)^{k-1}p$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "o564Lp7qKktn"
   },
   "source": [
    "---\n",
    "**Q1**\n",
    "\n",
    "Find the moment generating function for the geometric random varible.\n",
    "\n",
    "Use that to find the variance."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "bEqB6G5MKvSx"
   },
   "source": [
    "Ans:\n",
    "\n",
    "Let the Geometric RV be $X\\ \\sim Geom(p)$.\n",
    "It's pmf is given by $P_{X} (X=k) = (1-p)^{k-1}p$.\n",
    "\n",
    "The moment generating function is given by\n",
    "\\begin{align*}\n",
    "M_{X} (t)\n",
    "& =\n",
    "E(e^{tX})\n",
    "= \\sum _{k=1}^{\\infty} e^{tk} (1-p)^{k-1}p\n",
    "= p\\sum _{k=1}^{\\infty} e^{tk} (1-p)^{k-1}\n",
    "\\\\ &\n",
    "= p e^{t} \\sum _{k=1}^{\\infty} (e^{t}(1-p))^{k-1}\n",
    "= p e^{t} \\sum _{k=0}^{\\infty} (e^{t}(1-p))^{k}\n",
    "= p e^{t} \\left( \\frac{1}{1-e^{t}(1-p)} \\right)\n",
    "\\end{align*}\n",
    "\n",
    "provided $|e^{t}(1-p)| < 1$, i.e., $t < -\\log(1-p)$.\n",
    "\n",
    "\\begin{equation*}\n",
    "\\therefore\n",
    "\\boxed{\n",
    "M_{X} (t) = \\frac{pe^{t}}{1-(1-p)e^{t}}\n",
    "}\n",
    "\\end{equation*}\n",
    "\n",
    "The variance is given by\n",
    "\\begin{align*}\n",
    "\\text{Var}(X)\n",
    "& = E(X^{2}) - (E(X))^{2}\n",
    "= M_{X} ''(0) - (M_{X} '(0))^{2}\n",
    "\\\\\n",
    "\\implies\n",
    "M_{X} '(t)\n",
    "&\n",
    "= \\frac{d}{dt} \\left( \\frac{pe^{t}}{1-(1-p)e^{t}} \\right)\n",
    "= \\frac{(1-(1-p)e^{t})pe^{t} - pe^{t}(-1)(1-p)e^{t}}{(1-(1-p)e^{t})^{2}}\n",
    "\\\\ &\n",
    "= \\frac{pe^{t}(1-(1-p)e^{t} + (1-p)e^{t})}{(1-(1-p)e^{t})^{2}}\n",
    "= \\frac{pe^{t}}{(1-(1-p)e^{t})^{2}}\n",
    "\\\\\n",
    "\\implies\n",
    "M_{X} '(0)\n",
    "&\n",
    "= \\frac{p}{p^{2}}\n",
    "= \\frac{1}{p}\n",
    "\\\\\n",
    "\\implies\n",
    "M_{X} ''(t)\n",
    "&\n",
    "= \\frac{d}{dt} \\left( \\frac{pe^{t}}{(1-(1-p)e^{t})^{2}} \\right)\n",
    "= \\frac{(p e^t)(1 - (1-p)e^t)^2 - (pe^t)(-2(1-p)e^t(1 - (1-p)e^t))}{(1 - (1-p)e^t)^4}\n",
    "\\\\\n",
    "\\implies\n",
    "M_{X} ''(0)\n",
    "&\n",
    "= \\frac{(p)(1 - (1-p))^2 - (p)(-2(1-p)(1 - (1-p)))}{(1 - (1-p))^4}\n",
    "= \\frac{p^3 + 2p^2(1-p)}{p^4}\n",
    "\\\\ &\n",
    "= \\frac{p + 2(1-p)}{p^2}\n",
    "= \\frac{2-p}{p^2}\n",
    "\\\\\n",
    "\\implies\n",
    "\\text{Var}(X)\n",
    "&\n",
    "= M_{X} ''(0) - (M_{X} '(0))^{2}\n",
    "= \\frac{2-p}{p^2} - \\frac{1}{p^2}\n",
    "= \\frac{1-p}{p^2}\n",
    "\\end{align*}\n",
    "\n",
    "\\begin{equation*}\n",
    "\\therefore\n",
    "\\boxed{\n",
    "\\text{Var}(X) = \\frac{1-p}{p^2}\n",
    "}\n",
    "\\end{equation*}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "eC8mMGtVvWck"
   },
   "source": [
    "---\n",
    "**Sample Mean and Sample Variance**\n",
    "\n",
    "Let $X$ be a rv and let $D = d_1, d_2, ...., d_n$ be the data sample obtained via $n$ independent samplings of $X$.\n",
    "\n",
    "Then we can think of $d_i$ being sampled from $X_i$ where $X_i$ are identical independent distributions same as that of X.\n",
    "\n",
    "Then let $Z = \\frac{X_1 + X_2 + ... + X_n}{n}$\n",
    "\n",
    "Then $\\bar{d} = \\frac{\\sum_i d_i}{n}$ which is the sample mean, is a realization of the variable $Z$.\n",
    "\n",
    "Since  $EZ = EX$, $Z$ is an unbiased estimator of EX.\n",
    "\n",
    "\n",
    "Let $Q = \\sum_i (X_i - Z)^2$\n",
    "\n",
    "$EQ = \\sum_i EX^2_i + EZ^2 - E(X_iZ) = (n-1) Var(X)$ after some routine manipulations.\n",
    "\n",
    "Hence $V = \\frac{\\sum_i (X_i - Z)^2}{n-1}$ is an unbiased estimator of $Var(X)$ as $EV = Var(X)$\n",
    "\n",
    "We can find an estimate of Var(X) from the data samples as $\\frac{\\sum_i (d_i -  \\bar{d})^2}{n-1}$ where $\\bar{d} = \\frac{\\sum_i d_i}{n}$\n",
    "\n",
    "\n",
    "Note that both $Z, V$ are rvs and\n",
    "\n",
    "$Var(Z) = \\frac{Var(X)}{n}$. Variance of the average (of independent rvs) is reduced by $n$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "g_UFLO-FPPEP"
   },
   "source": [
    "---\n",
    "**Q2**\n",
    "Complete the code below to find sample and theory mean and variance for geometric rv."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "dbnvY3sNMvMv",
    "outputId": "84af221f-e2f9-4a6b-be12-89a421a9672c"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample mean = 3.4, theory mean=2.857142857142857\n",
      "sample variance = 5.377777777777777, theory variance = 5.306122448979592\n"
     ]
    }
   ],
   "source": [
    "#geometric RV\n",
    "nSamples = 10\n",
    "z = rng.geometric(p=0.35,size=nSamples)\n",
    "\n",
    "mu_sample = z.mean()\n",
    "var_sample = z.var(ddof=1)\n",
    "\n",
    "p=0.35\n",
    "print(f\"sample mean = {mu_sample}, theory mean={1/p}\")\n",
    "print(f\"sample variance = {var_sample}, theory variance = {(1-p)/p**2}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "F7F-MX68PXYb"
   },
   "source": [
    "---\n",
    "**Poisson RV**\n",
    "\n",
    "$Binomial(n,p) → Poisson(\\lambda)$\n",
    "when $\\lambda= np$ and $n → ∞$ with $np$ constant.\n",
    "\n",
    "$X\\ \\sim\\ Poisson(\\lambda)$\n",
    "\n",
    "$p_X(k) = e^{-\\lambda}\\frac{\\lambda^k}{k!}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "iwWS05b-Ttd9"
   },
   "source": [
    "---\n",
    "**Q3**\n",
    "\n",
    "Find the mean and variance of the Poisson random variable with parameter $\\lambda$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "KTLRuWq6Rf35"
   },
   "source": [
    "Ans:\n",
    "\n",
    "The PMF of the Poisson random variable is given by\n",
    "$\n",
    "f_X(k) = e^{-\\lambda} \\frac{\\lambda^k}{k!}\n",
    "$\n",
    "for $k = 0, 1, 2, \\ldots$.\n",
    "\n",
    "The mean is given by\n",
    "\\begin{align*}\n",
    "E(X)\n",
    "&\n",
    "= \\sum_{k=0}^{\\infty} k f_X(k)\n",
    "= \\sum_{k=0}^{\\infty} k e^{-\\lambda} \\frac{\\lambda^k}{k!}\n",
    "= e^{-\\lambda} \\sum_{k=1}^{\\infty} \\frac{\\lambda^k}{(k-1)!}\n",
    "\\\\ &\n",
    "= \\lambda e^{-\\lambda} \\sum_{k=1}^{\\infty} \\frac{\\lambda^{k-1}}{(k-1)!}\n",
    "= \\lambda e^{-\\lambda} \\sum_{j=0}^{\\infty} \\frac{\\lambda^j}{j!}\n",
    "= \\lambda e^{-\\lambda} e^{\\lambda}\n",
    "= \\lambda\n",
    "\\end{align*}\n",
    "\n",
    "and the variance is given by\n",
    "\\begin{align*}\n",
    "\\text{Var}(X)\n",
    "&\n",
    "= E[X^2] - (E[X])^2\n",
    "\\\\ &\n",
    "= \\sum_{k=0}^{\\infty} k^2 f_X(k) - \\lambda^2\n",
    "= \\sum_{k=0}^{\\infty} k^2 e^{-\\lambda} \\frac{\\lambda^k}{k!} - \\lambda^2\n",
    "\\\\ &\n",
    "= \\lambda\n",
    "\\end{align*}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "TtjflxmlMBey"
   },
   "source": [
    "---\n",
    "**Q4**\n",
    "Problem 2. You go to a party with 500 guests. What is the probability that exactly one other guest has the same birthday as you? Calculate this exactly and also approximately by using the Poisson PMF. (For simplicity. exclude birthdays on February 29.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Xwsiq9H1MIS8"
   },
   "source": [
    "Ans:\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "TyBfsknGPUKZ",
    "outputId": "2e8e4bff-3a13-432a-af9b-2cf3132e7674"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.34869862838624177\n",
      "0.3483963378131993\n"
     ]
    }
   ],
   "source": [
    "print((364/365)**498 * 499/365)\n",
    "print(499/365*np.exp(-499/365))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "lCXq5PQkyGcH"
   },
   "source": [
    "---\n",
    "**Q5**\n",
    "Problem 3. Fischer and Spassky play a chess match in which the first player to win a game wins the match. After 10 successive draws the match is declared drawn. Each game is won by Fischer with probability 0.4. is won by Spassky with probability 0.3. and is a draw with probability 0.3 independent of previous games.\n",
    "\n",
    "(a) What is the probability that Fischer wins the match?\n",
    "\n",
    "(b) What is the PMF of the duration of the match?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "fS4CTwH7y7Aj"
   },
   "source": [
    "Ans:\n",
    "\n",
    "(a)\n",
    "The probability that Fischer wins the match is\n",
    "\n",
    "$\n",
    "\\begin{aligned}\n",
    "\\Pr{(\\text{Fischer wins})}\n",
    "&\n",
    "= 0.4 + 0.3 \\cdot 0.4 + 0.3 \\cdot 0.3 \\cdot 0.4 + \\dots + 0.3^9 \\cdot 0.4\n",
    "\\\\ &\n",
    "= 0.4 \\frac{1 - 0.3^{10}}{1 - 0.3}\n",
    "= \\frac{4}{7} \\left(1 - 0.3^{10}\\right)\n",
    "\\approx 0.5714\n",
    "\\end{aligned}\n",
    "$\n",
    "\n",
    "(b)\n",
    "The PMF of the duration of the match is\n",
    "\n",
    "$\n",
    "\\begin{aligned}\n",
    "P(1) & = (1 - 0.3)\n",
    "\\\\\n",
    "P(2) & = 0.3 \\cdot (1 - 0.3)\n",
    "\\\\\n",
    "P(3) & = 0.3 \\cdot 0.3 \\cdot (1 - 0.3)\n",
    "\\\\\n",
    "& \\vdots\n",
    "\\\\\n",
    "P(10) & = 0.3^9 \\cdot (1 - 0.3)\n",
    "\\end{aligned}\n",
    "$\n",
    "\n",
    "i.e., with $M$ as the random variable representing the duration of the match, it's PMF is given by\n",
    "\n",
    "$\n",
    "f_M(m) = \\begin{cases}\n",
    "0.3^{m-1} \\cdot 0.7 & \\text{for } m = 1, 2, \\ldots, 10\n",
    "\\\\\n",
    "0 & \\text{otherwise}\n",
    "\\end{cases}\n",
    "$\n",
    "\n",
    "which in the limit as $m \\to \\infty$ approaches a geometric distribution with parameter $p = 0.3$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "JYTYAUZw5pK-"
   },
   "source": [
    "---\n",
    "**Q6**\n",
    "Problem 10. Form of the Poisson PMF. Let X be a Poisson random variable with parameter A. Show that the PMF px(k) increases monotonically with k up to the point where k reaches the largest integer not exceeding A, and after that point decreases monotonically with k."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "uGKyQy9k4q8v"
   },
   "source": [
    "Ans:\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "zI3Ii0sKYEOi"
   },
   "source": [
    "---\n",
    "**Kullback-Leibler (KL) Divergence and Cross-Entropy**\n",
    "\n",
    "The closeness of two probability distributions can be measured in a statistical sense using the measure of KL Divergence. If $p(x)$ and $q(x)$ are two distributions, then KL divergence is defined as:\n",
    "\n",
    "$D_{KL}(P||Q) = \\sum p(x)log(\\frac{p(x)}{q(x)})$\n",
    "\n",
    "For example, $p(x)$ can be extracted from data (like we did in one of the cells above) and $q(x)$ could be the model. Then KL divergence measures how close they are.\n",
    "\n",
    "For q(x) = 0 and p(x) !=0:   $D(P||Q) = ∞$\n",
    "\n",
    "*Jensen's Inequality*\n",
    "\n",
    "if $f(x)$ is a concave function of $x$, then\n",
    "\n",
    "$f(ax_1 + (1-a)x_2) \\geq af(x_1) + (1-a)f(x_2)$.\n",
    "\n",
    "Extending this to expectations:\n",
    "\n",
    "$f(EX) \\geq E(f(X))$ when f is concave.\n",
    "\n",
    "*KL Divergence is always non-negative*\n",
    "\n",
    "Since $log$ is a concave function, from above:\n",
    "\n",
    "$Elog(Q/P) \\leq log E(Q/P) = log(\\sum_x p(x)\\ q(x)/p(x)) = log(\\sum_x q(x)) = log(1) = 0$\n",
    "\n",
    "\n",
    "Hence\n",
    "\n",
    "$D(P||Q)=Elog(P/Q) = -Elog(Q/P) \\geq 0$\n",
    "\n",
    "*Cross Entropy*\n",
    "\n",
    "$H(p,q) = \\sum plog(1/q)$\n",
    "\n",
    "Hence $D_{KL}(p||q) = \\sum plog(p/q) = \\sum p (log 1/q - log 1/p) =  H(p,q) - H(p) $\n",
    "\n",
    "Since $D-{KL}(p||q) \\geq 0$,  we have $H(p,q) \\geq H(p)$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "jwNa-FjRmhx4"
   },
   "outputs": [],
   "source": [
    "# kl divergence\n",
    "def kl(p, q):\n",
    "  '''return kl divergence between the two pmfs'''\n",
    "  #z = np.zeros_like(p)\n",
    "  #print(z)\n",
    "  z = 0\n",
    "  for x,y in zip(p,q):\n",
    "    #print(x,y)\n",
    "    if y !=0 :\n",
    "      if x != 0:\n",
    "        z += x*np.log(x/y)\n",
    "    else:  # y = 0\n",
    "      if x != 0:\n",
    "        return(np.inf)\n",
    "  #print(z)\n",
    "  return z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "9PP7RHhbmtnF",
    "outputId": "c5f66e4d-1ca5-4c7f-efc6-801749c77f8b"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "kl(p,q)=0.020927550265489273\n",
      "kl(p,p)=0.0\n",
      "kl(p,r)=0.5077817570293732\n",
      "kl(r,p)=0.5119093049168663\n",
      "kl divergence is not symmetric\n"
     ]
    }
   ],
   "source": [
    "#test kl\n",
    "p = np.array([0.1,0.15,0.35,0.4,0])\n",
    "q = np.array([0.1,0.2,0.3,0.39,0.01])\n",
    "r = np.array([0.4,0.3,0.2,0.1,0])\n",
    "print(f'kl(p,q)={kl(p,q)}')\n",
    "print(f'kl(p,p)={kl(p,p)}')\n",
    "print(f'kl(p,r)={kl(p,r)}')\n",
    "print(f'kl(r,p)={kl(r,p)}')\n",
    "print('kl divergence is not symmetric')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "RusuwMKPdnyW"
   },
   "source": [
    "---\n",
    "**Q7**\n",
    "\n",
    "Find the best $\\lambda$ for $Poisson(\\lambda)$ which will give minimum cross entropy with the data samples from a $Binomial(n,p)$ below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "tQPkITWWbpn7",
    "outputId": "7a988c43-0e89-40fb-8524-1b6615fc118c"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "best lambda=0.42000000000000004 theorical best lambda=n*p=0.4\n"
     ]
    }
   ],
   "source": [
    "# KL divergence between data drawn from Binomial and Poisson , when data id generated from Binomial\n",
    "\n",
    "n, p = 20, 0.02\n",
    "nSamples = 1000\n",
    "#Draw nSamples from  binomial distribition\n",
    "data = rng.binomial(n,p,nSamples)\n",
    "\n",
    "#Empirical distribution\n",
    "unique, counts = np.unique(data,return_counts=True)\n",
    "empirical=np.zeros(n+1)\n",
    "empirical[unique] = counts/nSamples\n",
    "\n",
    "\n",
    "#search in the range +- 20% of theoretical value of n*p\n",
    "\n",
    "bestlam = 0\n",
    "bestkl = 100\n",
    "for lam in np.linspace(n*p*0.8, n*p*1.2, 25):\n",
    "  poisson = np.array([np.exp(-lam)*lam**k/np.math.factorial(k) for k in range(n+1)])\n",
    "\n",
    "  #your code here\n",
    "\n",
    "\n",
    "print(f'best lambda={bestlam} theorical best lambda=n*p={n*p}')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "oFcL4tOfg_XB"
   },
   "source": [
    "---\n",
    "**Mean and variance of a linear function of random variable**\n",
    "\n",
    "Let $Y = aX + b$ where $a$ and $b$ are constants.\n",
    "\n",
    "Then $EY = \\sum_x (ax+b)p_X(x) = a \\sum_x x p_X(c) + b \\sum_x p_X(x) = aEX + b$\n",
    "\n",
    "$EY^2 = E (aX+b)^2 = E (a^2X^2 + 2abX + b^2) = a^2EX^2 + 2abEX + b^2$\n",
    "\n",
    "$Var(Y) = EY^2 - (EY)^2 = a^2Var(X)$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "TJjhK6KV53B3"
   },
   "source": [
    "---\n",
    "**Q8**\n",
    "Problem 14. Let X be a random variable that takes values from 0 to 9 with equal probability 1/10.\n",
    "\n",
    "(a) Find the PMF of the random variable Y = X mod(3).\n",
    "\n",
    "(b) Find the PMF of the random variable Y = 5 mod(X + 1)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "M5C3rVIi58xz"
   },
   "source": [
    "Ans:\n",
    "\n",
    "(a)\n",
    "\n",
    "(b)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "1ZM2OhxM8S9l"
   },
   "source": [
    "---\n",
    "**Q9**\n",
    "\n",
    "Problem 16.\n",
    "Let X be a random variable with PMF\n",
    "\n",
    "$p_X(x) = \\frac{x^2}{a}$, if $x \\in \\{-3,-2,-1,0,1,2,3\\}$\n",
    "\n",
    "0, otherwise.\n",
    "\n",
    "(a) Find $a$ and $E[X]$.\n",
    "\n",
    "(b) What is the PMF of the random variable $Z = (X - E[X)^2$ ?\n",
    "\n",
    "(c) Using the result from part (b), find the variance of X.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "2a7_AJxQ9KZF"
   },
   "source": [
    "Ans:\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "gBnWh8LcAql6"
   },
   "source": [
    "---\n",
    "**Q10**\n",
    "\n",
    "Problem 21. St. Petersburg paradox. You toss independently a fair coin and you count the number of tosses until the first tail appears. If this number is n, you receive $2^n$ dollars. What is the expected amount that you will receive? How much would you be willing to pay to play this game?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "8t5ttq4dAxby"
   },
   "source": [
    "Ans:\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "TtXXa-2iBtcA"
   },
   "source": [
    "---\n",
    "**Q11**\n",
    "\n",
    "Problem 26. PMF of the minimum of several random variables. On a given day. your golf score takes values from the range 101 to 110. with probability 0.1, independent of other days. Determined to improve your score, you decide to play on three different days and declare as your score , X,  the minimum of the scores $X_1, X_2 ,  X_3$ on the different days.\n",
    "\n",
    "(a) Calculate the PMF of X.\n",
    "\n",
    "(b) By how much has your expected score improved as a result of playing on three\n",
    "days?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Gwo1AoMuCIjn"
   },
   "source": [
    "Ans:\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "VJZaFXj1sBdA"
   },
   "source": [
    "---\n",
    "**Iterated Expectation**\n",
    "\n",
    "Let $X, Y$ be two rvs.\n",
    "\n",
    "$EX = \\sum_x x p_X(x) = \\sum_x x \\sum_y p_{X|Y=y}(x)p_Y(y) = \\sum_y p_Y(y) \\sum_x x p_{X|Y=y}(x) = \\sum_y p_Y(y) E(X|Y=y) = E E(X|Y)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "EfEoC4FjttMa"
   },
   "source": [
    "---\n",
    "**Variance of a sum of rvs**\n",
    "\n",
    "\n",
    "Let $X_1, X_2, ..., X_n$ be independent rvs,\n",
    "\n",
    "Then,\n",
    "\n",
    "$Var(X_1, X_2, ... , X_n) = Var(X_1) + Var(X_2) + ... + Var(X_n)$\n",
    "\n",
    "For example:\n",
    "\n",
    "$Var(X_1 + X_2) = E(X_1+X_2)^2 - (E(X_1 + X_2))^2 = EX^2_1 + WEX^2_2 + 2EX_1X_2 - ((EX_1)^2 + (EX_2)^2 + 2EX_1EX_2) = EX^2_1 - (EX_1)^2 + EX^2_2 - (EX_2)^2 = Var(X_1) + Var(X_2)$\n",
    "\n",
    "This follows since $EX_1X_2 = EX_1EX_2$ as $X_1, X_2$ are independent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "fylqy1uFAoi2"
   },
   "source": [
    "---\n",
    "**Covariance, Correlation, Mutual Information**\n",
    "\n",
    "\n",
    "Let X,Y be RVs,\n",
    "\n",
    "Then the covariance is:\n",
    "\n",
    "$Cov(X,Y) = E((X-EX)(Y-EY)$\n",
    "\n",
    "And the (Pearson) Correlation coefficient is:\n",
    "\n",
    "$\\rho = \\frac{Cov(X,Y)}{\\sqrt{Var(X)Var(Y)}}$\n",
    "\n",
    "Consider the space of zero mean random variables $V = \\{X\\}$ over real numbers $ℝ$. This is a vector space.  \n",
    "\n",
    "We can define and inner product as $<X,Y> = EXY$.\n",
    "\n",
    "Then by Cauchy-Schwartz Inequality,\n",
    "\n",
    "$<X,Y>^2 \\leq\\ <X,X><Y,Y>$.\n",
    "\n",
    "This means $ |EXY|^2 \\leq (EX)^2(EY)^2$.\n",
    "\n",
    "Hence $|\\rho| \\leq 1$   \n",
    "\n",
    "Note that if $X, Y$ are independent, then\n",
    "\n",
    "$Cov(X,Y) = 0$ and $\\rho = 0$.\n",
    "\n",
    "Converse is however not true. (0 correlation doesnt mean the variables are independent).\n",
    "\n",
    "Example $Y=X^2$ and $X is zero mean$, then $Y, X$ are clearly dependent, however their correlation is 0.\n",
    "\n",
    "The Pearson Correlation Coefficient measures the \"degree\" of linear relationship between the two variables and can be used to predict the one variable given the other. For example, there will be a correlation between the heights of parents and children.\n",
    "\n",
    "See https://en.wikipedia.org/wiki/File:Correlation_examples2.svg for some pictures of variables which have non-linear dependence but zero pearson correlation coefficient.\n",
    "\n",
    "\n",
    "Note correlation is not causation: for examples, glacial melts and size of wild mango fruits might be correlated - but one is not the cause of the other (there could be an underlying cause affecting both - perhaps global warming)\n",
    "\n",
    "*$\\underline{Sample\\ Correlation}$*\n",
    "\n",
    "Let $D = \\{ (x_i,y_i) \\}$, $i = 1, ... , n$ be data samples.\n",
    "\n",
    "Then the sample correlation is calculated as:\n",
    "\n",
    "$r_{XY} = \\frac{\\sum(x_i-\\bar{x})(y_i-\\bar{y})}{\\sqrt{\\sum (x_i-\\bar{x})^2\\sum (y_i-\\bar{y})^2}}$\n",
    "\n",
    "Where $\\bar{x}, \\bar{y}$ are sample means.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "*$\\underline{Mutual\\ Information}$*\n",
    "\n",
    "between two rvs, is defined as:\n",
    "\n",
    "$I(X,Y) = D_{KL}(P_{XY}(x,y)||P_X(x)P_Y(y)) = \\sum_x \\sum_y p_{XY}(x,y) log \\frac{p_{XY}(x,y)}{p_X(x)p_Y(y)}$\n",
    "\n",
    "Note if $I(X,Y) = 0 ⟹ X, Y independent$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 490
    },
    "id": "pbxA8M6ca8tn",
    "outputId": "a9e2513a-66ef-4951-e9e6-5113a1de3bcd"
   },
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Std Dev of Sample Mean 0.02006784769226635, Theory 0.019798989873223333\n"
     ]
    }
   ],
   "source": [
    "#Sample Mean, Variance\n",
    "\n",
    "#parameters of the distribution\n",
    "n, p = 20, 0.02\n",
    "\n",
    "#number of samples/iterations per epoch/experiment\n",
    "nSamples = 1000\n",
    "\n",
    "#number of epochs/experiments/rollouts\n",
    "nEpochs = 1000\n",
    "\n",
    "#Arrays to store sample mean and variance\n",
    "S = np.zeros(nEpochs)\n",
    "V = np.zeros(nEpochs)\n",
    "\n",
    "for i in range(nEpochs):\n",
    "\n",
    "  #Draw nSamples from  binomial distribition, X ~ Bin(n,p)\n",
    "  X_data = rng.binomial(n,p,nSamples)\n",
    "\n",
    "  #sample mean S = (X_1 + X_2 + ... X_nSamples)/nSamples\n",
    "  S[i] = np.sum(X_data)/nSamples\n",
    "\n",
    "  #sample variance V = ((X_1 - S)^2 + (X_2-S)^2 + ... (X_nSamples-S)^2)/nSamples\n",
    "  V[i] = np.sum((X_data-S[i])**2)/(nSamples -1)\n",
    "\n",
    "fig,(ax1,ax2) = plt.subplots(1,2)\n",
    "ax1.hist(S,bins=10)\n",
    "ax1.set_xlabel(\"Sample Mean\")\n",
    "ax1.set_title(\"Histogram of Sample Means\")\n",
    "\n",
    "ax2.hist(V,bins=10)\n",
    "ax2.set_xlabel(\"Sample Variance\")\n",
    "ax2.set_title(\"Histogram of Sample Variances\")\n",
    "\n",
    "\n",
    "plt.show()\n",
    "\n",
    "ES = np.sum(S)/nEpochs\n",
    "print(f\"Std Dev of Sample Mean {np.sqrt(np.sum((S-ES)**2)/nEpochs)}, Theory {np.sqrt(n*p*(1-p)/nEpochs)}\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 490
    },
    "id": "ZNQeHKJkeUy-",
    "outputId": "e8e65373-3011-4a6d-e83d-3e2691b35aea"
   },
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theory (by approximating as Gaussian with same variance) = 0.9034243676629176\n"
     ]
    }
   ],
   "source": [
    "# Sample entropy\n",
    "\n",
    "#parameters of the distribution\n",
    "n, p = 30, 0.02\n",
    "\n",
    "#number of samples/iterations per epoch/experiment\n",
    "nSamples = 1000\n",
    "\n",
    "#number of epochs/experiments/rollouts\n",
    "nEpochs = 1000\n",
    "\n",
    "#Arrays to store sample mean and variance\n",
    "H = np.zeros(nEpochs)\n",
    "\n",
    "for i in range(nEpochs):\n",
    "  #Draw nSamples from  binomial distribition, X ~ Bin(n,p)\n",
    "  X_data = rng.binomial(n,p,nSamples)\n",
    "\n",
    "  #Empirical distribution\n",
    "  unique, counts = np.unique(X_data,return_counts=True)\n",
    "  empirical=np.zeros(n+1)\n",
    "  empirical[unique] = counts/nSamples\n",
    "\n",
    "  for j in range(n+1):\n",
    "    if empirical[j] !=0:\n",
    "      H[i] += empirical[j]*np.log(1.0/empirical[j])\n",
    "\n",
    "fig,ax1 = plt.subplots(1,1)\n",
    "ax1.hist(H,bins=10)\n",
    "ax1.set_xlabel(\"Sample Entropy\")\n",
    "ax1.set_title(\"Histogram of Sample Entropies\")\n",
    "\n",
    "plt.show()\n",
    "\n",
    "\n",
    "print(f\"Theory (by approximating as Gaussian with same variance) = {0.5*(np.log(2*np.pi)+np.log(n*p*(1-p))+0.5)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "6rdBNZGbsHEn"
   },
   "source": [
    "---\n",
    "**Q12**\n",
    "\n",
    "Compute the distribition of the sample mutual information for a multivariate normal with zero mean and unit variance and 0.8 correlation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "sjeQtaLLpIuy",
    "outputId": "94942b3b-2263-4031-93fb-5c8329be18f3"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mutual Infomation is 0.022264671760422888 bits\n",
      "Entropy of Univariate Gaussian with variance = sigma is:1.8257480647361595 bits\n",
      "Entropy of Multi-variate Gaussian :2.651496129472319 bits\n"
     ]
    }
   ],
   "source": [
    "# Mutual Information from Sample\n",
    "\n",
    "nSamples = 1000\n",
    "n = 10 #number of bins\n",
    "\n",
    "sigma = 1.0\n",
    "rho = 0.0 #Correlation coefficient\n",
    "\n",
    "\n",
    "cov = np.array([[sigma, rho*sigma**2],[rho*sigma**2,sigma]])\n",
    "\n",
    "XY_data = np.random.multivariate_normal([0,0], cov, nSamples).T\n",
    "#plt.scatter(XY_data[0], XY_data[1])\n",
    "\n",
    "H2d,_,_ = np.histogram2d(XY_data[0], XY_data[1],bins=[n,n], range=[[-4,4],[-4,4]])\n",
    "#print(H2d)\n",
    "\n",
    "#<Your code>\n",
    "\n",
    "\n",
    "print(f\"Mutual Infomation is {I} bits\")\n",
    "\n",
    "#Entropy of Gaussian:https://gregorygundersen.com/blog/2020/09/01/gaussian-entropy/\n",
    "\n",
    "\n",
    "print(f\"Entropy of Univariate Gaussian with variance = sigma is:{#your code} bits\")\n",
    "\n",
    "#print(f\"Det of Cov={np.linalg.det(cov)}\")\n",
    "\n",
    "print(f\"Entropy of Multi-variate Gaussian :{#your code} bits\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 490
    },
    "id": "uKszqSOJxFiq",
    "outputId": "2c2d4f66-2d24-458c-b585-5ee3ee0a3d07"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Mutual Information vs. correlation coefficient for Two Gaussian rv')"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot of mutual information versus correlation coefficient\n",
    "\n",
    "nSamples = 1000\n",
    "n = 10 #number of bins\n",
    "\n",
    "sigma = 1.0\n",
    "rho = 0.0 #Correlation coefficient\n",
    "\n",
    "nSteps = 10\n",
    "I = np.zeros(nSteps,dtype=np.double)\n",
    "\n",
    "k = 0\n",
    "for rho in np.linspace(0.0, 1.0, num=nSteps):\n",
    "  cov = np.array([[sigma, rho*sigma**2],[rho*sigma**2,sigma]])\n",
    "\n",
    "  XY_data = np.random.multivariate_normal([0,0], cov, nSamples).T\n",
    "\n",
    "  H2d,_,_ = np.histogram2d(XY_data[0], XY_data[1],bins=[n,n], range=[[-4,4],[-4,4]])\n",
    "\n",
    "  #Empirical distribution\n",
    "  X,_ = np.histogram(XY_data[0],n,(-4,4))\n",
    "  Y,_ = np.histogram(XY_data[1],n,(-4,4))\n",
    "\n",
    "#code to be filled in\n",
    "\n",
    "plt.plot(np.linspace(0.0, 1.0, num=nSteps),I)\n",
    "plt.xlabel(\"Correlation coefficient\")\n",
    "plt.ylabel(\"Mutual Information\")\n",
    "plt.title(\"Mutual Information vs. correlation coefficient for Two Gaussian rv\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "T3Y0_CPMzEJX"
   },
   "source": [
    "---\n",
    "\n",
    "For Gaussian RV, correlation =0 implies independence (I(X,Y) = 0))\n"
   ]
  }
 ],
 "metadata": {
  "colab": {
   "include_colab_link": true,
   "provenance": []
  },
  "kernelspec": {
   "display_name": "base",
   "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": "3.9.19"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}