1110_HW2_Q2.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 2. using inverse transform algorithm\n",
    "# (a)\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "np.random.seed(0) \n",
    "N = int(1e5)\n",
    "U = np.random.rand(N) # uniform한 난수 생성 \n",
    "X_Inv = 1 - np.sqrt(1-U)  # Fx(u) inverse function (0 <= u <= 1)\n",
    "\n",
    "plt.hist(X_Inv, 100) \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "95% CI => [ 0.33237382115153424 , 0.33529623057342717 ]\n"
     ]
    }
   ],
   "source": [
    "# 2. using inverse transform algorithm\n",
    "# (b) estimate the mean & 95% CI\n",
    "from scipy.stats import norm\n",
    "\n",
    "X_mean = np.mean(X) \n",
    "X_std = np.std(X)\n",
    "\n",
    "alpha_95 = norm.ppf(0.975)\n",
    "\n",
    "print(\"95% CI => [\", X_mean-alpha_95*X_std/np.sqrt(N) ,\",\", X_mean+alpha_95*X_std/np.sqrt(N),\"]\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "95% CI => [ 0.33237382115153424 , 0.33529623057342717 ]\n"
     ]
    }
   ],
   "source": [
    "# 2-(c) using Acceptance-Rejection, repeat (a) & (b)\n",
    "c = 2\n",
    "idx = 0\n",
    "X = np.zeros(N)\n",
    "\n",
    "while(idx<N) :\n",
    "    U1 = np.random.rand() \n",
    "    U2 = np.random.rand() \n",
    "    \n",
    "    if(c*U2 <= 2*(1-U1)):\n",
    "        X[idx] = U1 \n",
    "        idx = idx + 1\n",
    "        \n",
    "plt.hist(X,100) \n",
    "plt.show()\n",
    "\n",
    "X_mean = np.mean(X) \n",
    "X_std = np.std(X)\n",
    "\n",
    "alpha_95 = norm.ppf(0.975)\n",
    "print(\"95% CI => [\", X_mean-alpha_95*X_std/np.sqrt(N) ,\",\", X_mean+alpha_95*X_std/np.sqrt(N),\"]\")\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}