Bayesian Assignment.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Poisson Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Prior Distribution \n",
    "#Parameters for gamma prior distribution\n",
    "k <- 8\n",
    "rate <- 1\n",
    "\n",
    "Prior <- rgamma(n=500, shape = k, rate = rate)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Data Observed\n",
    "X <- rpois(500, lambda = 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Posterior Distribution is Gamma distribution (Conjugate prior)\n",
    "k1 <- k + sum(X)\n",
    "rate1 <- rate + length(X)\n",
    "\n",
    "Post <- rgamma(n = 500,shape = k1, rate = rate1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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9IcVO9ad6gebUSageo96xDN441IM9C8Y12ieMARaQaK96tT9I44Ik1H7171i9ox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\nQmCC",
      "text/plain": [
       "Plot with title \"density.default(x = Post)\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(density(Post), col = 'green')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Binomial Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Prior Distribution \n",
    "#Parameters for Beta prior distribution\n",
    "a <- 1\n",
    "b <- 1\n",
    "\n",
    "Prior <- rbeta(n=500, shape1 = a, shape2 = b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Data Observed\n",
    "m = 100\n",
    "X <- rbinom(500, size = m, prob =0.45)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Posterior Distribution is Beta distribution (Conjugate prior)\n",
    "a1 <- a + sum(X)\n",
    "b1 <- b + (length(X) * m) - sum(X)\n",
    "\n",
    "Post <- rbeta(n = 500,shape1 = a1, shape2 = b1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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fQO3sxDIiNfyp3PeYdERu6Uulmac0jcHPn0r0RMsw1JwTOmSCf72Z1rSFTk3b+8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\ngJAAAYQECCAkQAAhAQL+B5iPrgSR4oW2AAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"density.default(x = Post)\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(density(Post), col = 'green')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Normal Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### a) Unknown Mean and Known Precision "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Normal Distribution\n",
    "# Parameters for Normal prior distribution\n",
    "mu <- 3\n",
    "r <- 0.7\n",
    "\n",
    "Prior <- rnorm(50, mean = mu, sd =  1/ sqrt(r))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Data Observed\n",
    "X <- rnorm(50, mean = 7, sd =  2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Posterior Distribution is Normal distribution (Conjugate prior)\n",
    "mu1 <- (r*mu +length(X)*r*(sum(X)/length(X)))/(r+length(X)*r)\n",
    "r1 <- r + length(X)*r\n",
    "\n",
    "Post <- rnorm(50, mean = mu1, sd =  1/sqrt(r1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Zd4WZgg8feLkAjEr4Tf2bxDIiLvkn2Dsw6JiHKQ5rucb0jcGOUiyXc615CoKCvx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\nTkSuQmCC",
      "text/plain": [
       "Plot with title \"density.default(x = Post)\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(density(Post), col = 'green')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### b) Known Mean and Unknown Precision"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Parameters for Gamma prior distribution\n",
    "\n",
    "alpha <- 3\n",
    "beta <- 1.5\n",
    "\n",
    "Prior <- rgamma(50, shape = alpha,rate = beta )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Data Observed\n",
    "m <- 7\n",
    "X <- rnorm(50, mean = m, sd =  2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Posterior Distribution is Gamma distribution (Conjugate prior)\n",
    "alpha1 <- alpha + length(X)/2\n",
    "\n",
    "beta1 <- beta + (1/2 * sum(X - m)^2)\n",
    "\n",
    "Post <- rgamma(50, shape = alpha1 ,rate = beta1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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GYuMfcHYQBfjBqxhCFV4XB7+RCqSUvM/0kywEg//QCjhCGF8O4vl9fc/Q8wJl1I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\n8cISAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"density.default(x = Post)\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(density(Post), col = 'green')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exponential Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Parameters for Gamma prior distribution\n",
    "a <- 2\n",
    "b <- 0.5\n",
    "\n",
    "Prior <- rgamma(500,shape =  a, rate = b )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Data Observed\n",
    "X <- rexp(500, rate = 7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Posterior Distribution is Gamma distribution (Conjugate prior)\n",
    "a1 <- a + length(X)\n",
    "b1 <- b + sum(X)\n",
    "\n",
    "Post <- rgamma(500,a1, b1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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+NmjzSUxpo8MzbvtOdI+2N7y+I5ksKSnLN16S7l5e9l76rdov72lQpHqLAk70yN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\nAQIIEiCAIAECCBIggCABAggSIOB/qoINkpPsy3cAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"density.default(x = Post)\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(density(Post))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Uniform Distribution "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "library(actuar)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Parameters for Pareto prior distribution\n",
    "w <- 4\n",
    "alpha <- 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Data Observed\n",
    "X <- runif(50, min = 0, max = 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Posterior Distribution is Pareto distribution (Conjugate prior)\n",
    "alpha.new <- alpha + length(X)\n",
    "w.new <- max(w, max(X))\n",
    "\n",
    "Post <- rpareto(50, scale = w.new, shape = alpha.new)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Vh5jkRIlgQ+2XDfu7wgJCQn+Fm7k8uu\nhITUhD/9fdtNPwcKuwv6mkZIhmzxfaSDkZAoyRBeIkRIELAypN3pLraUgRH+ERJWWxlS+XTH\nR0uJhERJdqwMqbgcfLRESIiMwHOkf6eddEtBd0CPwwjJDJmTDbfqZXTn9asZGeEPIWE9kZCu\n+/oFcnuB9QyN8IiQsN76kIpT+XC0uxZlTbnMmtIJiZLMWBvSv+pkw/HWvENstyEkRGbt95HK\nB6Nz8XpHJrGi7xHeERLWW/t9pPwqtpSBEd4REtZb+30ksYUMjvCOkLDe6lc2PC9kYod13yO8\nIySsJxTSXfanw0Puf35nUZIRK0K6fvyWhd3Gq1I6i5CMWPOItGt39G/jVSmdRUhGSD1HkkVI\niAw/2EdIELAipOrRyNOvoiMkRIaQPM+iJBs4tCMkCCAkQoKAtSGdd4/HfSd89puQEJuVIdW/\nNLX+Vyb4PtJGtw8VVoa0d5fHze0eF8Efj30QEqIj8A3ZmztKf2c24M7nfxQlWSAQUu6uhLTp\nBGxv9aHd7Vr9YGy8h3aEBAnrTzY4d6oekER/UpaQEJnVp7+z6hnSY3cRWk/PCL8S+64YNmL+\nG7JpbQ22QkhJjMDWCCmJEdja2pBOu8hf/U1IkLAypFP0P0YRYhQlpW9lSJnkv0HRP8IzQoIE\n87+zgZAgYWVIufPyu1bD7XlBJhFS+laGdM/2sj+J1B3hFyFBxOpDu8hPNoSZREnJI6R0pmBD\n1r8hS0gQQUjpTMGGVod0zesf7rsLradvhE+BJlFS6taGtG+eHrlMtCRCQmRWhnR2+6IK6ewO\nYkt6JBgSJaVu9UuEiubVDZGetUtvELYh8BIhQtI0CNtYGdLu+Yh0i/Rf7EtvELYh8xzpKvwq\n8AT3b0pK29qzdvnzdQ2iv42LkBAbke8juVz2lwgREmJj+5UNif3cE7ZDSKFQUtLWhXQ9VL/7\nZH+U/pkkQkJk1oR0379/hmIf52vtCAlCVoRUZG53rX7S/H7ZVb9IXxAhITIrQjq2znnvq9+k\nLyfFkCgpaStC2rm/47l7nP+sCyFByIqQPl5eF+dr7cLu25SUMNMhBd6zCSlhhBQQJaWLkAIi\npHStCunDxqtSPGazeQiHkEKipGSZfq0dIUEKIYVESMkipKAoKVWEFBQhpYqQgiKkVBFSWJSU\nKEIKi5AStUFI58ztJn55V5jdbZOdmpLSFDKkW+6y8+M049d3ERIiEzCkW13Q0R2Kxz0f/4WS\nCYdESWkKGNLBHasfq61/KL0Y/xXHhITIBAypeTmey1tvfL7b0wv3hhcUZIqWsfAqeEiX5phu\n/LelEBIiE/TQ7lA8Lxb1YZ78iN8QEsQEDKnI3odsbuLXdyUdEiWlKOj3kY6vfLLRxyNCQnQs\nv7Jhsx2aktJDSBsgpPQYDmnD3ZmSkkNIm6Ck1BDSJggpNYS0DUpKDCFtg5ASQ0jbIKTEENJG\nKCkthLQRQkoLIW2FkpJiN6TNd+TNFwBBhLSd7VcAMYS0ne1XADGEtCEFS4AQQtqQgiVACCFt\nScMaIMJsSCr2YRWLgARC2pKKRUACIW1KxyqwHiFtS8kysBYhbUzLOrAOIW1MyzqwDiFtTc1C\nsAYhbU3NQrCG1ZAU7b6KloLFCGlzipaCxQhpe5rWgoUISQFVi8EihKSAqsVgEULSQNdqsAAh\naaBrNViAkFRQthz8jJBUULYc/IyQdNC2HvzIaEjq9lt1C8JvCEkJfSvCLwhJC4VLwnyEpIXC\nJWE+QlJD45owFyGpoXFNmIuQ9FC5KMxDSHqoXBTmISRFdK4KcxCSIjpXhTlshqR1j9W6Lkwi\nJE20rguTCEkVtQvDBELSRe/KMIqQdNG7MowiJGUULw0jCEkZxUvDCEJSRvHSMMJkSKp3VtWL\nwxBC0kb14jCEkLRRvTgMISR1dK8O/QhJH+XLQx+LIWnfUbWvDz0ISSH1C0QHISmkfoHoICSN\n9K8QXwhJI/0rxBdC0kj/CvGFkFSKYIn4YDCkGHbSGNaINkLSKYpF4g8h6RTFIvGHkJSKY5V4\nISSl4lglXghJq0iWiQYhaRXJMtEgJLViWScqIUMqDs7tr88bGb0VQnrEs05UAoZUZK6SNzdC\nSJOiWSiChnR057Kmc7avb2SzkOLZPeNZKUKGlDWfeM92d0KaI56VImRIr3aK/b4vJNe2cMSs\nZXi8bWERLdW8gCHtXPG6tOcRaY6IlmpewJDO7vC8dHd7QpojprUaF/L09/Fdz3Xi6I2QGjGt\n1big35C95a9L9wMhzRHVYk2z98qGqPbNqBZrGiHpFtdqDTMXUmR7ZmTLtYuQlIttvVYRknKx\nrdcqQlIutvVaRUjaRbdgmwhJvfhWbBEhqRffii2yFlKMe2WMazaHkPSLcc3mEFIEoly0MYQU\ngSgXbYyxkOLcJeNctS2EFINIl20JIcUg0mVbQkhRiHXddhBSHKJduBWEFIdoF24FIUUi3pXb\nYCukiPfGiJduAiHFIua1G0BIsYh57QYQUjSiXnzyCCkaUS8+eYQUj7hXnzhCikjky08aIUUk\n8uUnzVRI0e+I0W9AuggpJtFvQLoIKSrxb0GqCCkuCWxCmggpMilsQ4oIKTZJbER6CCk2SWxE\neggpOmlsRWoshZTIHpjIZiSGkOKTynYkhZDik8p2JIWQIpTMhiSEkGLk0tmUVBBSlBLalEQY\nCimpnS+pjUkBIUUqra2JHyHFKrHNiR0hRSu17YmbnZDS2+/S26KIEVLEEtykaBFSzFLcpkgR\nUtSS3KgoEVLc0tyqCJkJKdU9LtXtig0hxS7ZDYsLIUUv3S2LCSHFL+FNiwchJYCfqtgeISUh\n7a2LASGlIfHN089KSMnvaMlvoHKElArHM6UtEVJCTGykUoSUEhtbqRIhJYXDu60QUmLMbKgy\nhJQaO1uqipGQLO1dlrZVD0JKD0+UNkBIKTK2uRoQUpKsbe/2CClJ1rZ3ezZCsrdf2dvijRFS\nojjjEBYhJcvkRm/GREhGdymjm70NQkqY1e3eAiGlzOyGhxcwJPfJx4gwNxcRu1seWsCQzuMh\nza7sZ5b3JsvbHlTIQ7tbtvc9IsCtRcb0xgcU9DnSzR19j/B/a5ExvfEBhT3ZcHY33yO831ps\nbG99MBbO2hnflXiNQwiElD7r2x+EgZDYj7gH/CMkC7gLvCMkC7gLvCMkE7gPfCMkGzh15xkh\nGcG94Ff6IbEHNbgfvCIkM7gjfCIkM7gjfCIkO7gnPEo+JPaeP9wX/hCSJdwZ3qQeErvOB+4O\nXwjJFO4OXwjJFu4PTwjJFu4PTxIPif3mG/eIH4RkDS9f9YKQzOE+8YGQ7OFO8SDtkNhl+nCv\neEBIBnG3yCMki7hfxBGSSdwx0pIOid1lEHeNMEKyibtGGCEZxX0jK+WQ2FfGcO+IIiSruHdE\nJRwSe8o47h9JhGQXL18VREiGcQ/JSTck9pJp3EdiCMk07iQphGRa7HeSa9l4JUE+ZYsRse8j\nYcR7L3Xi2TamZEOKdw8Ja+uv5IsMNrNdTIRkXmz31FQr26RESObFdE/Ne8TZYotSDSmmvWNr\n0dxXsx9rNjjAIyREcWf9+uwn9DYlGlIMu4Yi6s84LFhg4EclQkJF8R22/ExcyI1KMyTFu4VW\nOu+ylaezA24UIaGh7T4TebVCuI1KMiRt+0QcFD1Rkvu+arCNIiS8Kbjf5F82FyglQsKfTe84\nb688DbJVKYZER4tt9UI1v6+RC/GgREhoC/6agDA/AaFygvaQ6GidcM/PA75W2/scQkJHgP07\n+M87+J6WXkh0JGFoR3c9Ftz0Bv+TPM9MLiQ6EjM3ml+K2vKnWL1OTi0kOtrQWE8afrGCz+mE\nBGl9j2Q6/r/4PMUe5FOCjdDx/wta+SuakGCLp10krZDoCJP87CRJhaTkSBy6edlNUgqJjDCP\nhz0loZDoCHPJ7yuEBIvET9+lExId4SeyO0wyIdERfiT6oKRzl/19BB3hd4IpJRISHWERsZTS\nCImOsJTQvpNASFpeEYk4yew/8YdERVhLIKXYQ+LRCBJW/6RH0JD+nfL6Z1Py4z+RERzUQdCq\nmAKGVOxaP+e1Xz+CiiBucUwBQzq67HKrL92vmTuuGqHmZy6RnkU/0RswpMzd3pdvLls0QtUP\nLiNhv/6QfMCQPpbUXV/7B/z/AyITLiSJRyRAp7DPka73+tL650iALiFPf+9bB2+7wssIYBth\nv490rL+PlOUnme8jAVrE/soGQAVCAgQQEiCAkAABhAQIICRAACEBAggJEEBIgABCAgQQEiCA\nkAABhAQIICRAACEBAggJEEBIgABCAgQoDQmIzIK9XD4cBSP9j2AjtIxQcsBDSFonsBFaJsxC\nSFonsBFaJsxCSFonsBFaJsxCSFonsBFaJsxCSFonsBFaJsxCSFonsBFaJsxCSFonsBFaJsxC\nSFonsBFaJsxCSFonsBFaJsxCSFonsBFaJsxCSFonsBFaJsyiZBlA3AgJEEBIgABCAgQQEiCA\nkAABhAQIICRAACEBAggJEEBIgABCAgQQEiCAkAABhAQIICRAQMCQjpnLjsXHVWfJ8Z3b7xko\nPEF4C/pGnHeeN6I4OHe4CQ7ov+P/id5R3xMW/+57MeFm7+tN3bWvuklueuf2ewYKTxDegr4R\nx/qKTK6kzoSsvkKypL47vsgk76jvCTdDIf1z2e1xy9y/v6vKt+TGd26/Z6DwBOEt6Btxc4ei\netg7eJtwrG776HKpAQN3fC55R/XcTYLrXyZYSEd3Lf+8uNP7mrPbC967ndvvDpSeILwFfSPy\n5ublpnQmZK4QHdB/x19EHy46E85y/5eXChZS7u6Pzy8d7ij5/69z+92B0hOEt6B3xHOQ2JSB\nCS6TGtA74i77Facz4ezOcre+TLCQXOdr6030C2Hn9rsDpScIb0HviFrh9n4nHCV3xJ4Re3eX\nvKM6E3J3PbjsKDfhdxuG1HlL9vYDhCR684Mjqi+4V58TyuMuyX2wO+LkLqJ3VE9INbEvNwsQ\n0vIJsjc/OOJxz+SeSvdNOOeZ5HOMzoj6GMxrSK4s9VGIPq7+ipCWT5C9+cERRSb4lXbgXjkI\n7oOdEbvq7L3XkBqF5Pc6fuU9pNcJ/szzbti5/d6BohNkb35wxF5y9xi4VwrBsw3fIw71gank\nHTX0v3bLbyQFC6k503L/PF0kftbu/n3W7i591s7fFvSPuO/2d78Tah7/T7g3XxNekg7p5VR/\nXbp+Pq0V3PLO7fcOFJ1Qkf1/1x1xFX4G3ZnQfB/pLnhU9D1CPqTBjdjw27LBQur9fnfsr2yQ\nDqkz4i59Jqr/lQ1FLvgcqf+O9/rKhmPVVHGUO7n5u3APhru/M5TvO1Xy3u3cfusKTxMe4kcT\n3yMO0l/MuxuRiZ857rufZO+o7wlFsxFbfiMpXEhF/YrdZqiP3bBz+60rPE14iIf0PUL8qKhn\nI8ordqLnjfvuJ9k7qvf/texG/GrDp2dAOggJEEBIgABCAgQQEiCAkAABhAQIICRAACEBAggJ\nEEBIgABCAgQQEiCAkAABhAQIICRAACEBAggJEEBIgABCAgQQEiCAkAABhAQIICRAACEBAggJ\nEEBIgABCAgQQEiCAkAABhAQIICRAACEBAghpIff6B0tH/ym69r+3V/8rc8XYx+0n/73bnmGt\nq+qL18lFPXXW83dFcXDucPu+trUx0v+OYPy4LxZyLntdGP6gW2uH29eXBv7x8Ne/cHmbnDp2\nVXVx56YW9dRZT+uK7G8xf9e2Nqa9XahxXyxU7kan54XhD7r9/YP14//IenMjx8l/E3l8363e\n62aGNPavwNf/0vmxXnvr2tbGtC6iQUgLlV+l3b25MPxB52dsj2q/rI65Ln9XfN1a+6+xqVPv\nnRtSZz2tKzJXvG6jdW1rY84Dm2EYIS1UHvk0X5bHQ3r/S9t5nd3Ql/JXSPXh4jV3z3+z27l7\n7rJmpy2fqxzLK/5VjxblxzRP0Q7l26713sfziKv87/j6xH6d9XSuqBfTura1Ma2LaBDSQuWu\neqiPi/5Ccm+va3J3PbybeDweg9m9Du2q/fPU3Maxvrp+tlIFUT9XKQsrHy6qDz245laz5lPf\n732HlFcXzn+3/720znq+r2gW07q2tTGti2gQ0kLlrlXUz8HHQ2rOxj2mQ3rHU16+VAdTrr64\nL8ov/7vq7ea5iis7uzxe5zqqQ67qA//e+z60e33i0NImQrq8F/O+trUxrYtoENJC1a5VH+GM\nHfTyPoMAAAJnSURBVNrVTRTH1oeNh7S/ta6p/3g95uX1pWt58V7tv//KR5xb9UB0/3rvO6TP\nB8v+9T+GQzrnWf1A2Lr2Y2PeF9EgpIXqPWxXPiuffl5fP3LNObS7Pk+i3a+n/TOk1ztbn74v\nhx7LZ2inpqmv97ZPNqwI6VEdPfb1X/w9zBVDJ/MtIqSF6l2reuY/dmjX+thsRkjlk/rqaGn/\nvpHekK5lQtnusds1R3mzQuourbOezhVFdfTYXXb/RfO4KxZqdqLyAGtmSM3pr/v4Wbv674Pb\nna/34ZAebvevfAJzdEX1gLg0pM56ugvsXzYh9eKuWKjZie5uN7Y3Nd+QqXfDU32++ur6T3U9\nb6R+EKgvf4eU15/enOs+ukP51rX889B979xDu856Wle8lr3rvTb/uIgGIS303ElPoy+UOVY7\nYFF/U3POKxuKffXx1YmC2/dzpOvfebkqmOahqN7Jv97r6oeQ6ZAmX9lQ5NVzpI9r3xvTuogG\nIS302kmzsb21yP7Oau/Gzhi/DryyotpLG/8+eqjPOB+aK3b1s/x98+3br/fu3PtRbfzQq7We\n162+r8j+Lv5d29qY9nahRkgLvXbS6+jeWhyz8hnP++Lg9zCfGTWvsj5UrwO/VsdN7R5Oz9cu\nVBfrXfjU/k7P+73/djNDaq3n+XjYWmDvsr82ZsfJ7xZCAgQQEiCAkAABhAQIICRAACEBAggJ\nEEBIgABCAgQQEiCAkAABhAQIICRAACEBAggJEEBIgABCAgQQEiCAkAABhAQIICRAACEBAggJ\nEEBIgABCAgQQEiCAkAABhAQIICRAACEBAggJEPA/S0AQQWM6GRIAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"density.default(x = Post)\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(density(Post))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}