Bayesian Statistics The Fun Way - Python Solutions # CHAPTER 15.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "import scipy.stats\n",
    "from scipy.stats import beta\n",
    "from scipy.stats import binom\n",
    "from scipy.stats import norm\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import pandas as pd\n",
    "import random\n",
    "%matplotlib inline\n",
    "\n",
    "# this will make plots a little bit prettier\n",
    "sns.set(style='darkgrid')\n",
    "sns.set_context('talk')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# From Parameter Estimation To Hypothesis Testing: Building A Bayesian A/B Test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Monte Carlo Simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_trials = 100000\n",
    "A = beta.rvs(39, 121, size=n_trials)\n",
    "B = beta.rvs(53, 107, size=n_trials)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.95978"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(B > A)/n_trials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "ratios = B/A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x23ef04b91c0>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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hRP7Yxz7W6tpxxx1XaD/11FMd1tnmvajTmXm1VwcAAAAAoCfY42F0nz598qlPfSpJsnz58vz6179us19zc3N+9rOfFb4+44wzWl0fPnx4jjzyyCTJk08+2e7q6Pr6+kyfPj1JUlZWlnHjxrW6fuyxx2bIkCFJkjlz5hS283inNWvWZM6cOUmSwYMH5/jjj291/ZRTTkl5eXmS5N57783mzZvbrPPiiy/mySefTJIcfvjh+cAHPtBmPwAAAACA7myPh9FJ8sUvfrGw5cY3v/nNvPLKK62uNzU15aabbsrzzz+fJPnkJz/Z5nYWkyZNKvS/5pprUl1d3er65s2bc/3112f16tVJkvPOO2+7AwN79eqVL33pS0mSurq6XH311amtbX3aZm1tba666qrCXtCXXXZZysrKWvXZb7/9cv755yfZuvf01772te0C6TfeeCPXXnttYbX35MmT2/2MALaprCzt8f8rLe2d3qW907tXl/gxBQAAAOwGpXt6AEnyoQ99KJ///Oczbdq0VFdX55xzzsl5552XD3/4w9mwYUNmz56dP/7xj0mS/v3751vf+labdcaOHZtPf/rT+cUvfpGXX34548ePz8SJEzNq1KhUV1dn1qxZWbZsWZLkoIMOyjXXXNNmnYsvvjgPP/xwnnvuuTzzzDOFOiNGjMif//znzJw5s3Do4FFHHZVLLrmkzTpXX3115s2blxUrVuSRRx7J0qVL87nPfS5DhgzJ0qVLM2PGjNTU1CRJxo0bl1NPPfVdfY5Az1Zbtzlvb2rIxrq2/6ZFT1LVpzwNjVv29DAAAACA3aikeduy3CI4+eSTs2LFiiQdHwbY1NSUG264ITNmzGi3z/7775+pU6fmgx/8YLt96uvrc+2112bevHnt9jnkkEPyox/9KCNGjGi3z/r163P55Zdn0aJF7fY5+uijM2XKlAwYMKDdPq+++momTZq03WrvlsaOHZvvfOc7qaysbLdPMTU2bklNTd0euTfQOYMHV6W6ZmPeqKnLxk0Ne3o4RTd4wD5paNiS2rrGvPxazZ4eTlF99PAD09Borj1NR3OtqqpISUrSnObU1tbvgRHuPu8f1i9D+lVmUL/KbNrUuKeHA++5wYOrUlJSkubm5qxZU9vxG4AuzTMNPYtnujgGDOiT0tLeu/TeLrEyOtm6PcYNN9yQM844IzNmzMjvf//7rFmzJmVlZRk5cmTGjh2biy66KH379t1hnYqKivzgBz/I448/ngcffDDPPvtsampqUllZmVGjRuXUU0/N+eefn4qKih3W6d+/f6ZPn57Zs2dnzpw5Wbx4cTZs2JCqqqqMGTMm48ePz1lnnZXevXf8wY8YMSIPPfRQ7rvvvjz66KNZunRpamtr069fvxxxxBE555xzcsopp+z05wXsnZqbm7N85YY9PYyiG9x/nz09BAAAAGA3K+rKaLoHK6Oh69u2Mnr1urfz4n+0fbBqT2IFbc9krv/NymjoOay4gp7FMw09i2e6ON7NymgnQwEAAAAAUHTCaAAAAAAAik4YDQAAAABA0QmjAQAAAAAoOmE0AAAAAABFJ4wGAAAAAKDohNEAAAAAABSdMBoAAAAAgKITRgMAAAAAUHTCaAAAAAAAik4YDQAAAABA0QmjAQAAAAAoOmE0AAAAAABFJ4wGAAAAAKDohNEAAAAAABSdMBoAAAAAgKITRgMAAAAAUHTCaAAAAAAAik4YDQAAAABA0QmjAQAAAAAoOmE0AAAAAABFJ4wGAAAAAKDohNEAAAAAABSdMBoAAAAAgKITRgMAAAAAUHTCaAAAAAAAik4YDQAAAABA0QmjAQAAAAAoOmE0AAAAAABFJ4wGAAAAAKDoSvf0AAAA6L4G9dsnVX3KUlraO5WVe3o0743m5qS+vnFPDwMAALodYTQAALuspCTZWN+YjZu3ZEvjlj09nKLbp6I0fSr9Cg0AALvCb9IAALwrGzc35u2NjWnc0rSnh1JUJSXJ4L6VwmgAANhFfpMGAOBda25Olq/csKeHUVTvH9ZvTw8BAAC6NQcYAgAAAABQdMJoAAAAAACKThgNAAAAAEDRCaMBAAAAACg6YTQAAAAAAEUnjAYAAAAAoOiE0QAAAAAAFJ0wGgAAAACAohNGAwAAAABQdMJoAAAAAACKThgNAAAAAEDRCaMBAAAAACg6YTQAAAAAAEUnjAYAAAAAoOiE0QAAAAAAFJ0wGgAAAACAohNGAwAAAABQdMJoAAAAAACKrnRPDyBJvv/97+fOO+/cpffedNNNOfvsswtfr1q1KieddFKn3/93f/d3ueiii9q9Pm/evMyaNSvPP/981q9fn/79+2fkyJE544wzcvbZZ6esrKzDezQ1NWX27Nl56KGHsnjx4tTW1mbgwIEZPXp0JkyYkNNOOy0lJSWdHjMAAAAAQHfTJcLod6NXr9aLuxcvXrxb6tbX1+erX/1qHnvssVavV1dXp7q6OgsWLMiMGTNy55135n3ve1+7ddavX58rrrgiCxcubPX66tWrs3r16vzmN7/JrFmzcvvtt6dfv367ZewAAAAAAF1NlwijTzvttIwZM6ZTfWfPnl0IiA8//PB8+tOfbnW9ZRj9jW98I8OHD99hvdGjR7f5essg+oADDsjEiRMzcuTIrFq1Kg888ECWLVuWF154IZMmTcrMmTNTVVW1XY2GhoZcfvnlWbRoUZJk5MiROffcczN06NAsX748s2bNysqVKzN//vxceeWVmTZtWkpLu8S3BAAAAABgt+oSyecHPvCBfOADH+iw37PPPptf/epXSZJ+/frlrrvuSp8+fVr1efHFF5NsXTE9ceLE7LPPPjs9nkcffbQQRI8ZMyZ33313+vbtW7h+0UUX5frrr8/cuXOzbNmyTJ06Ndddd912de6+++5CEP2JT3wid911VyoqKgrXL7nkkkyePDkLFy7MggULMmvWrFxwwQU7PV4AAAAAgK6u2xxgWFdXl+uuuy4NDQ1Jkm9/+9sZNmzYdv22rYw++OCDdymITpKpU6cmSUpKSnLTTTe1CqKTpLy8PLfeemuGDh2aJLnnnnvy5ptvtuqzefPm/OQnP0mSVFZW5sYbb2wVRCdJVVVV7rjjjkKgPmXKlDQ1Ne3SmAEAAAAAurJuE0Z/97vfzX/+538mScaNG5dTTz11uz61tbV59dVXkySHHXbYLt1n2bJlhUD7mGOOaXf7kPLy8lx44YVJkk2bNuXxxx9vdX3+/PlZu3ZtYbz7779/m3UGDRqUM888M8nWfaSfeeaZXRo3AAAAAEBX1i3C6MWLF2f69OlJkn333Tf/63/9rzb7vfjii2lubk6y62H0008/XWifcMIJO+z78Y9/vNB+4oknWl176qmnOl2n5fV31gEAAAAA6Am6xJ7RHbnpppuyZcuWJMnkyZPbXWW8bb/o5L/D6DfffDNLlizJ22+/nYEDB2b06NEpLy9v914vvfRSod3e4YbbHHrooSkpKUlzc3OrgxOTZMmSJZ2uM2rUqEL7nXUAAAAAAHqCLh9G/9u//VthtfKQIUNy6aWXttu3ZZD71ltv5Ytf/GKefPLJVvsw9+nTJ2eeeWauvPLKNkPtFStWFNrDhw/f4djKy8szaNCgrFmzJitXrkxzc3NKSkqSJK+99lqn67Tc+7rl/QEAAAAAeoouv03HP//zPxfaX/jCF7Y7BLClliujr7vuuvz2t7/d7kDAurq6zJw5M5/97Gfz7LPPbldj2z7PSTJgwIAOx9e/f/8kSWNjY2prawuvr1u3LklSVlaWqqqqHdaorKwsHLa4fv36Du8JAAAAANDddOmV0YsXLy6siu7Xr18mTpzYbt8tW7Zk6dKlha+rqqoyefLkjBs3LkOHDs1bb72Vp556KlOmTMnSpUuzZs2afPnLX87999+fESNGFN63cePGQruysrLDMbYMx+vq6rLffvu1qtOZGtvqbNy4sdX93yu9e/fK4ME7DsyBrqOqqv3/KNdTlJX2TnNJUlbW1OPna649087Mtbt/FnvT97WyojT79ClPVVVlOlhrwF7K79TQs3imoWfxTHcNXTqM/ulPf1po/83f/M0OVxivXLkyvXv3TrJ1O4977723Vcg8aNCgnH766Tn55JMzadKkLFiwIOvXr8/NN9+cu+66q9Bv297UydZVzR1puf90y/dua3emRst+jY2Nneq/O23bWgToDkpSkr3jmd06z71jvubaM3Vmrj3lc9h7vq9b5+h3J9ri3wvoWTzT0LN4pruOLhtG19TUZO7cuUmS0tLSXHzxxTvsP2LEiCxatChr165Nc3NzBg8e3Ga/ffbZJ7feemtOOeWUNDQ05PHHH8/q1atzwAEHJGm9krmhoaGwfUZ7Nm/eXGi3DJ4rKytTV1eXhoaGHU+0xb3eWeO90tzc/J7fE9hVzWnO3vHMbp3n3jFfc+2ZOpprSUp6zOew93xft87R706807ZDzYGewTMNPYtnujh2NeDvsmH0vHnzCgHtcccdl4EDB3bqfYMGDeqwz9ChQ3Pcccfl3/7t35IkCxYsyJlnnplk6wGH29TX13dYa9OmTYV2y+C6T58+qaur61SNlnVa3v+9smVLU2pq6t7z+wKd1/KvE9XWdu7Ple6soXFLGhu3pKGhqcfP11x7po7m2nI7i+7+WexN39dNfSuzsW5zast6ZdOm9/5vs9E1tfwZvWZN7Q56At2BZxp6Fs90cQwY0Celpb136b1d9gDDX/7yl4X26aefvtvrjxo1qtBevXp1od23b99CuzOHCW7YsCHJ1j2fW24jsq3O5s2bU1e346B306ZNhTC6s6E7AAAAAEB30iXD6Nra2jz55JNJtm5bMXbs2N1+j9LStheFH3LIIYX2ypUrd1ijvr4+a9euTZIMGzas3Tqvv/76Duu0vM876wAAAAAA9ARdMox+6qmnCnsxH3/88a1WK7fnt7/9bf75n/85t9xyS1566aUO+69atarQ3n///QvtQw89tNBesmTJDmssWbKksOfMYYcd1upayzodjaflfd5ZBwAAAACgJ+iSYfTChQsL7Y9+9KOdes+8efPyj//4j/npT3+aX/ziFzvs29jYmKeeeqrw9Uc+8pFC+7jjjiu0W/Zpy+9+97tC+2Mf+1ira7urDgAAAABAT9Alw+jnnnuu0D766KM79Z4TTzyx0H7ggQdaHSz4TjNmzMgbb7yRJPnLv/zLDB8+vHBt+PDhOfLII5MkTz75ZLuro+vr6zN9+vQkW7cSGTduXKvrxx57bIYMGZIkmTNnTmE7j3das2ZN5syZkyQZPHhwjj/++B3OEwAAAACgO+qSYfS2bS169eqVD3/4w516z0knnVTYp/n111/PN77xjcJWHy398pe/zC233JIk6d27d77yla9s12fSpElJkqamplxzzTWprq5udX3z5s25/vrrCwcfnnfeedsdPNirV6986UtfSpLU1dXl6quvTm1t61M7a2trc9VVV2Xjxo1JkssuuyxlZWWdmi8AAAAAQHfS9il+e9DatWvz1ltvJdm6Unifffbp1PtKS0tz88035/Of/3w2btyYRx55JC+88ELOOeecjBgxIuvXr8+//uu/5te//nXhPddff32OOuqo7WqNHTs2n/70p/OLX/wiL7/8csaPH5+JEydm1KhRqa6uzqxZs7Js2bIkyUEHHZRrrrmmzTFdfPHFefjhh/Pcc8/lmWeeKdQZMWJE/vznP2fmzJmFwwuPOuqoXHLJJTv1WQEAAAAAdBddLozetto4SYYNG7ZT7z3yyCPz4x//OF/5yleyatWqLF++PN/5zne269enT598/etfz8SJE9ut9Y//+I+59tprM2/evKxbty5TpkzZrs8hhxySH/3oR+0esNi7d+/8+Mc/zuWXX55FixZlxYoV+e53v7tdv6OPPjpTpkyxKhoAAAAA6LG6XBjdciuLAw88cKff/9GPfjSPPPJI/u///b95/PHH89JLL+Wtt95KVVVVhg8fnk9+8pM577zzcsABB+ywTkVFRX7wgx/k8ccfz4MPPphnn302NTU1qayszKhRo3Lqqafm/PPPT0VFxQ7r9O/fP9OnT8/s2bMzZ86cLF68OBs2bEhVVVXGjBmT8ePH56yzzkrv3r13eq4AAAAAAN1Flwuj//Iv/7KwZ/Su2nfffXPxxRfn4osvftfjGTt2bMaOHfuuavTq1SsTJkzIhAkT3vV4AAAAAAC6oy55gCEAAAAAAD2LMBoAAAAAgKITRgMAAAAAUHTCaAAAAAAAik4YDQAAAABA0QmjAQAAAAAoOmE0AAAAAABFJ4wGAAAAAKDohNEAAAAAABSdMBoAAAAAgKITRgMAAAAAUHTCaAAAAAAAik4YDQAAAABA0QmjAQAAAAAoOmE0AAAAAABFJ4wGAAAAAKDohNEAAAAAABSdMBoAAAAAgKITRgMAAAAAUHTCaAAAAAAAik4YDQAAAABA0QmjAQAAAAAoOmE0AAAAAABFJ4wGAAAAAKDohNEAAAAAABSdMBoAAAAAgKITRgMAAAAAUHTCaAAAAAAAik4YDQAAAABA0QmjAQAAAAAoOmE0AAAAAABFJ4wGAAAAAKDohNEAAAAAABSdMBoAAAAAgKITRgMAAAAAUHTCaAAAAAAAik4YDQAAAABA0QmjAQAAAAAoOmE0AAAAAABFJ4wGAAAAAKDohNEAAAAAABSdMBoAAAAAgKITRgMAAAAAUHTCaAAAAAAAik4YDQAAAABA0QmjAQAAAAAoOmE0AAAAAABFJ4wGAAAAAKDohNEAAAAAABSdMBoAAAAAgKITRgMAAAAAUHTCaAAAAAAAiq50Tw+gpcbGxhx99NHZvHlzp/pfdtll+drXvtbmtWeeeSbTp0/PH/7wh6xduzZVVVU56KCD8pnPfCbnnXde9t13307dY968eZk1a1aef/75rF+/Pv3798/IkSNzxhln5Oyzz05ZWVmHNZqamjJ79uw89NBDWbx4cWprazNw4MCMHj06EyZMyGmnnZaSkpJOjQcAAAAAoDvqUmH0K6+80ukguj1NTU258cYbc88997R6vaamJjU1NfnTn/6Un//85/n+97+fMWPGtFunvr4+X/3qV/PYY4+1er26ujrV1dVZsGBBZsyYkTvvvDPve9/72q2zfv36XHHFFVm4cGGr11evXp3Vq1fnN7/5TWbNmpXbb789/fr124UZAwAAAAB0fV0qjF68eHGhPXny5BxxxBE77H/wwQdv99ptt91WCKL79euX888/P6NHj05NTU1mz56dP/3pT3n11Vfz5S9/Offff38OPPDANmu3DKIPOOCATJw4MSNHjsyqVavywAMPZNmyZXnhhRcyadKkzJw5M1VVVdvVaGhoyOWXX55FixYlSUaOHJlzzz03Q4cOzfLlyzNr1qysXLky8+fPz5VXXplp06altLRLfUsAAAAAAHaLLpV8vvjii4X2hAkTcsghh+zU+5999tlMmzYtSTJs2LBMnz49Q4cOLVy/8MILc8stt+RnP/tZqqurc8stt+R73/vednUeffTRQhA9ZsyY3H333enbt2/h+kUXXZTrr78+c+fOzbJlyzJ16tRcd91129W5++67C0H0Jz7xidx1112pqKgoXL/kkksyefLkLFy4MAsWLMisWbNywQUX7NScAQAAAAC6gy51gOG2ldGVlZV5//vfv9Pv/+EPf5jm5uYkyTe/+c1WQXSS9OrVK1//+tcLK67nzp2bl19+ebs6U6dOTZKUlJTkpptuahVEJ0l5eXluvfXWQv177rknb775Zqs+mzdvzk9+8pPCfG688cZWQXSSVFVV5Y477kifPn2SJFOmTElTU9NOzxsAAAAAoKvrUmH0tpXRhx56aHr12rmh1dbW5oknnkiydVX02LFj2+xXUlKSSy+9NEnS3NycRx55pNX1ZcuWFULxY445pt19pcvLy3PhhRcmSTZt2pTHH3+81fX58+dn7dq1SZJx48Zl//33b7POoEGDcuaZZybZuo/0M88808FMAQAAAAC6ny4La1VTAAAgAElEQVQTRq9atSo1NTVJksMOO2yn379w4cI0NjYmST7+8Y/vsO/xxx+fkpKSJCkE2Ns8/fTThfYJJ5ywwzot7/POOk899VSn67S8/s46AAAAAAA9QZfZM7rl4YXbwui6urq8+OKLeeutt9KvX78ceuihhS0t3umll14qtEePHr3Dew0cODCDBw9OdXV1XnrppTQ1NRVWYu9MnUMPPTQlJSVpbm5uNf4kWbJkSafrjBo1qtB+Zx0AAAAAgJ6gS4bRSXLVVVflV7/6VRoaGgqvlZeX56//+q9z7bXX5uCDD27Vf8WKFYX28OHDO7zfsGHDUl1dnYaGhrzxxhs58MADd7pOeXl5Bg0alDVr1mTlypVpbm4urLh+7bXXOl1n2LBhbc4DAAAAAKCn6DLbdGzbLzpJvv3tb+exxx5rFUQnWw8FnDt3bj772c9ut53Ftv2Zk2TAgAEd3q9///6F9vr16991ncbGxtTW1hZeX7duXZKkrKwsVVVVO6xRWVmZffbZZ7uxAAAAAAD0FF1mZXTLMLq8vDyXXnppxo8fn4MOOiibNm3KH/7wh/zwhz/MokWL8vbbb+eqq67K//k//ydHHHFEkq2HCG5TWVnZ4f0qKioK7bq6ukJ748aN76rOfvvt16pOZ2psq7Nx48ZW93+v9O7dK4MH7zgwB7qOqqqKjjt1c2WlvdNckpSVNfX4+Zprz7Qzc+3un8Xe9H2trCjNPn3KU1VVmQ7WGrCX8js19CyeaehZPNNdQ5cIo+vr6wsrgvv06ZN/+Zd/KYTMydag9lOf+lT+6q/+Ktdff30efvjh1NfX53//7/+dBx54ICUlJYXDC5Otq5E7Ul5eXmhv2bKlzfbuqNOZGi37tZzHe2Xb1iJAd1CSkuwdz+zWee4d8zXXnqkzc+0pn8Pe833dOke/O9EW/15Az+KZhp7FM911dIkwuqKiIgsWLMiGDRuyadOmHHDAAW326927d/7+7/8+v/vd71JTU5N///d/z6JFi/KRj3yk1Qrkd27v0ZbNmzcX2i0D43fW2bZ9xq7Uqaur69RYWo65s+H17tTc3Pye3xPYVc1pzt7xzG6d594xX3PtmTqaa0lKesznsPd8X7fO0e9OvNO2Q82BnsEzDT2LZ7o4djXg7xJh9Db9+vVLv379dtinT58+GTduXGbMmJEkWbBgQT7ykY+kT58+hT719fUd3qvlth4tA+fdWaeurq5TNVrWaXn/98qWLU2pqanruCOwx7T860S1tZ37c6U7a2jcksbGLWloaOrx8zXXnqmjubbczqK7fxZ70/d1U9/KbKzbnNqyXtm06b3/22x0TS1/Rq9ZU7uDnkB34JmGnsUzXRwDBvRJaWnvXXpvlznAcGeMGjWq0F69enWSpG/fvoXXOnMI4IYNGwrtQYMGFdq7WqeioqLVQYXb6mzevLnVntRt2bRpUyGMHjhwYIf3BAAAAADobrplGF1auv2C7kMOOaTQXrlyZYc1tvWprKxsFQDvTJ36+vqsXbs2STJs2LB2x/P66693aixt1QEAAAAA6Am6RBi9aNGi/PSnP81tt92Wp59+usP+q1atKrT333//JMmhhx5aeG3JkiU7fP+6deuyZs2awvt69frvj2Fn6ixZsqSw58xhhx3W6lrLOi+99FKHdbZ5Zx0AAAAAgJ6gS4TRv//973PLLbfkxz/+cR588MEO+//ud78rtI855pgkyZFHHlnYs3n+/Pkdvn9biPyxj32s1bXjjjuu0H7qqac6PY5i1QEAAAAA6Am6RBh94oknFtqPPfZYYdVyW/71X/81zz77bJJkxIgR+ehHP5pk68F/n/rUp5Iky5cvz69//es239/c3Jyf/exnha/POOOMVteHDx+eI488Mkny5JNPtrs6ur6+PtOnT0+SlJWVZdy4ca2uH3vssRkyZEiSZM6cOYXtPN5pzZo1mTNnTpJk8ODBOf7449ueOAAAe9Sgfvukqk95Skt7p7KydK/5X0VFlzr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      "text/plain": [
       "<Figure size 1680x840 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(12,6), dpi=140)\n",
    "sns.distplot(ratios, kde=False, bins=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In Python you need to create ECDF function yourself, so let's do that"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ecdf(data):\n",
    "    \"\"\" Compute ECDF \"\"\"\n",
    "    x = np.sort(data)\n",
    "    n = x.size\n",
    "    y = np.arange(1, n+1) / n\n",
    "    return(x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = ecdf(ratios)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Please note that our plot can differ from the book due to the random nature of Monte Carlo Simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x23ef04e8e20>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.lineplot(x, y) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. Suppose a director of marketing with many years of experience tells you he believes very strongly that the variant without images (B) won’t perform any differently than the original variant. \n",
    "### How could you account for this in our model? Implement this change and see how your final conclusions change as well"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "prior_a = 300\n",
    "prior_b = 700\n",
    "A = beta.rvs(36 + prior_a, 114 + prior_b, size=n_trials)\n",
    "B = beta.rvs(50 +  prior_a, 100 +  prior_b, size=n_trials)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.73872"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(B > A)/n_trials"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. The lead designer sees your results and insists that there’s no way that variant B should perform better with no images. She feels that you should assume the conversion rate for variant B is closer to 20 percent than 30 percent. Implement a solution for this and again review the results of our analysis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "a_prior_a = 30\n",
    "a_prior_b = 70\n",
    "\n",
    "b_prior_a = 20\n",
    "b_prior_b = 80"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = beta.rvs(36 + a_prior_a, 114 + a_prior_b, size=n_trials)\n",
    "B = beta.rvs(50 +  b_prior_a, 100 +  b_prior_b, size=n_trials)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.65724"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(B > A)/n_trials"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. Assume that being 95 percent certain means that you’re more or less “convinced” of a hypothesis. Also assume that there’s no longer any limit to the number of emails you can send in your test. If the true conversion for A is 0.25 and for B is 0.3, explore how many samples it would take to convince the director of marketing that B was in fact superior. Explore the same for the lead designer.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "a_true_rate = 0.25\n",
    "b_true_rate = 0.3\n",
    "\n",
    "prior_a = 300\n",
    "prior_b = 700\n",
    "\n",
    "p_b_sup = 0\n",
    "n_samples = 0\n",
    "n_trials = 100000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "def b_superior(B, A, n_trials):\n",
    "    return sum(B > A)/n_trials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "def draw_results(n_samples, rate):\n",
    "    results = []\n",
    "    for i in range(int(n_samples/2)):\n",
    "        draw = random.random()\n",
    "        results.append(draw)\n",
    "    results = np.array(results)\n",
    "    a = sum(results <= rate)\n",
    "    b = sum(results > rate)\n",
    "    return a, b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100\n",
      "200\n",
      "300\n",
      "400\n",
      "500\n",
      "600\n",
      "700\n",
      "800\n",
      "900\n",
      "1000\n",
      "1100\n",
      "1200\n"
     ]
    }
   ],
   "source": [
    "while p_b_sup < 0.95:\n",
    "    n_samples += 100\n",
    "    a_samples_a, a_samples_b = draw_results(n_samples, a_true_rate)\n",
    "    b_samples_a, b_samples_b = draw_results(n_samples, b_true_rate)\n",
    "    a_samples = beta.rvs(a_samples_a + prior_a, a_samples_b + prior_b, size=n_trials)\n",
    "    b_samples = beta.rvs(b_samples_a + prior_a, b_samples_b + prior_b, size=n_trials)\n",
    "    p_b_sup = b_superior(b_samples, a_samples, n_trials)\n",
    "    print(n_samples)"
   ]
  }
 ],
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