Second_driveative_poisson_distribution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Is the density function of distributions from the one-parameter Exponential family always convex?\n",
    "\n",
    "#### Answer: \n",
    "##### No. \n",
    "##### We will explain the answer by providing an CounterExample."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Using the Sympay Library in Python, we define the density function of the Poisson distribution as a one-parameter distribution of the exponential family, then we obtain the second derivative of the density function of this distribution, and if this density function is a convex function, it must always be in The support of the variable should be greater than zero, which we show is not the case"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\lambda^{x} e^{- \\lambda}}{x!}$"
      ],
      "text/plain": [
       "lambda**x*exp(-lambda)/factorial(x)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## load library\n",
    "import sympy\n",
    "from sympy import symbols, sqrt, Rational, exp, diff, factorial, lambdify, Abs\n",
    "\n",
    "# define parameters and variable as symbols \n",
    "la, x = symbols('lambda x') \n",
    "\n",
    "## define density function of poisson distribution\n",
    "expr =  exp(-la) * la ** x / factorial(x)\n",
    "expr\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\lambda^{x} \\left(- \\frac{\\left(\\operatorname{polygamma}^{2}{\\left(0,x + 1 \\right)} + \\operatorname{polygamma}{\\left(1,x + 1 \\right)} - \\frac{2 \\Gamma\\left(x + 1\\right) \\operatorname{polygamma}^{2}{\\left(0,x + 1 \\right)}}{x!}\\right) \\Gamma\\left(x + 1\\right)}{x!} + \\log{\\left(\\lambda \\right)}^{2} - \\frac{2 \\log{\\left(\\lambda \\right)} \\Gamma\\left(x + 1\\right) \\operatorname{polygamma}{\\left(0,x + 1 \\right)}}{x!}\\right) e^{- \\lambda}}{x!}$"
      ],
      "text/plain": [
       "lambda**x*(-(polygamma(0, x + 1)**2 + polygamma(1, x + 1) - 2*gamma(x + 1)*polygamma(0, x + 1)**2/factorial(x))*gamma(x + 1)/factorial(x) + log(lambda)**2 - 2*log(lambda)*gamma(x + 1)*polygamma(0, x + 1)/factorial(x))*exp(-lambda)/factorial(x)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## Get second driveative of density function \n",
    "df2 = diff(expr, x, 2)\n",
    "df2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "## convert expression to a function \n",
    "f = lambdify([la, x], df2)\n",
    "## Here we create a Poisson distribution with mean 1 from the above function by assigning the second derivative to our mean\n",
    "def fl(x):\n",
    "    return f(1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Here we plot the second derivative of a Poisson distribution with mean 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 900x700 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "plt.rcParams[\"figure.figsize\"] = [9, 7]\n",
    "plt.rcParams[\"figure.autolayout\"] = True\n",
    "\n",
    "\n",
    "x = np.linspace(0, 10, 1000)\n",
    "\n",
    "plt.plot(x, fl(x), color='red')\n",
    "\n",
    "plt.show()"
   ]
  }
 ],
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