PyData Workshop.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 333,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: pylab import has clobbered these variables: ['random']\n",
      "`%matplotlib` prevents importing * from pylab and numpy\n"
     ]
    }
   ],
   "source": [
    "%pylab inline\n",
    "import numpy as np\n",
    "import random\n",
    "import scipy\n",
    "from statsmodels.stats import weightstats"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Statistics made simple through simulation\n",
    "\n",
    "## The 5 Step Modeling Approach\n",
    "\n",
    "1. Ask the question\n",
    "2. Select the modeling approach\n",
    "3. Formulate the model\n",
    "4. Solve the model\n",
    "5. Answer the question\n",
    "\n",
    "\n",
    "## Monte Carlo Simulation\n",
    "\n",
    "Pros\n",
    "* General method that can give approximate solutions to any probability modeling problem. (Mathematical Modeling, Meerschaert)\n",
    "* Intuitive models that are fairly straight-forward to grasp without much mathematical formalism.\n",
    "\n",
    "Cons\n",
    "* Solves via brute force and that can get very time intensive.\n",
    "* The simulation itself introduces extra variation into your solution\n",
    "\n",
    "To the last point while this is true, Meerschaert says:\n",
    "\n",
    "\"As a practical matter, there are many sources of error and variation in a modeling problem, and the additional variation produced by Monte Carlo simulation is not typically the most serious of these.\"\n",
    "\n",
    "He goes on to say that sensitivity analysis is your friend.\n",
    "\n",
    "We'll be using this term to be synonomous with Analytical Simulation. The core difference is a pure Monte Carlo simulation works in simplest terms with no higher level mathematical concepts. An analytical simulation may use probability distributions or other mathematical tooling as a shortcut.\n",
    "\n",
    "## Enough theory\n",
    "\n",
    "Easiest to explain with a simple example. Suppose you're in a room facing a large wall and large open window with a bowl full of marbles on a table behind you. \n",
    "\n",
    "How might you answer the question: \"What percentage of that wall is the window taking up?\"\n",
    "\n",
    "Process\n",
    "1. Count how many marbles are in the bowl.\n",
    "2. Close your eyes and throw marbles at the wall in a purely random way.\n",
    "3. Find the percentage of marbles that went through the window.\n",
    "\n",
    "The answer you get from step 3 will approximate how much of the wall the window takes up.\n",
    "\n",
    "Let's simulate this and see how it works."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "wall_width = 20\n",
    "wall_height = 10\n",
    "\n",
    "window_lower_left = (5, 2.5)\n",
    "window_upper_right = (15, 7.5)\n",
    "\n",
    "number_of_marbles = 1000\n",
    "\n",
    "sample_widths = np.random.random(number_of_marbles) * wall_width\n",
    "sample_heights = np.random.random(number_of_marbles) * wall_height\n",
    "\n",
    "random_marbles_thrown = zip(sample_widths, sample_heights)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We pick an X and a Y to form a coordinate that we throw a marble at. We throw all of our marbles this way. Then we gather all of the marbles that hit the wall and didn't go through the window."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x11635f110>"
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     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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Njj1Vx406I6PL3U3PLHwtlWM5EOZ+dMQsVy1QcnknyTMSdSSQ7vmZBmbV+xahnHI9p7m8\nIjReTv3iMf3Mwb1tmAfs+JaAefhpJgUS5kPE7yQ2WqaqQ8nfpXEwmTpAHsI8L3v3kSNuW4HJ95ZD\n14IdmOLpD6JZFO1s5fr77h0MaIEtOx/tUfUcRQqFcCqE8JZyunfgvnjMVciaNGuTSSltWyEHKAnz\noZDUYFUdR+XsEduaJTuYbKetWTWdNHZzu+vFNTibe/vblQULtXpcvYrTdzymy3GiG6TFrEKduyUN\nqjoKM0uDy6YMWeuOvtt0OxLZCE4bc5GwpavNjuqFdZMdgqiDhPkIkCTcbB2DImPgbh5oYiJPStLA\nYOqINk4y13q5bpoQFo71+k4ubzyhOtdku45q6PHUBM3mgVw2BFaVodGYyy2TnywoGZvxns8qt9n3\nNTrjKcbWrNK6GYvvoKQyg4Xz7QQL62hxkA4S5iNAkoC1FeZCgB/k0Wl2dLcaW604D/ujeUZhlxBJ\nvgZj6rwwyeft5o3GHG+3lwYhie32Ep+bu4rL0Sevf/3rYxEtaepvGlDy0izlgTGYTagXdako0tYs\nnoF6Gz9TGYBdscHU972Q2UVN6cIcwPUA/hDAowDuUfxefK1HEFOjszGzBDHE8cYe7ji2wjyvyAAb\nu77puvHj44LhkkvUkSH+zEIMAJ2BsIuufp3nwtQyzYMd7X0na7qIlvD9s0QD2QghdZTOsvM7yyL0\nktqueLZxZUQmuqPRvLKduLSfSXOIlirMAcwA+DqAnQBqAD4JYEU6pox6jx1JDtBAOMnCLhqlYNvA\nswhzk03eVbNSRV1Ec53PDoSvyuSiFnaLITPKtOKZ3c/z2NxB9Sxsz7EVQur6icErHA4ql8EluVfW\nsopUwfrZorouaqds1meTl/loFClbmO8A8A0AL/cE+jkAPyYdU0a9K4nYaacZ6jjRXXBMnVsmrVaj\niuaQN1N2sXmqyiGEQ9TBp8sxYhbmi5FzswjzPKfzrtqnaiFOs7lfK/zW1tZ40ibUtnWzKavwU+zm\n/iyIsWnl/VQDtypc0iayy/U5VoFhmFl+CsAWgO8BuEvxe/G1rijxaI94pIZLYzZFuei+V0UfyItz\nTNEIpo7qL0tXOSd1wjzu8AzMLIGDTU7eFN3ZSFX38I5GeS8jdxVCqoU4fhST6lqqbJm6jaPT+HNU\nZbVxeqdVIHQbTpOZpVjNfK+nmb8SwB4AfwDgZ6Rj+IkTJwafjY2NEh5DNbCxL2fVTHQdJO7ACpI4\nycLc1sYZXhS0trYW6bBhgetHOqiiJPzrqqJyogImutRdN1hF6zg7OC/P5+wqhHTauc7G7CLMk4R1\n3mY71xmOqe5prjdObGxsRGRl2cL8ZwB8NPT3LwG4Wzqm+KdQUeLCUG3SyNLAdZ1SZ85QbQCtmxrr\nriGE9su4nyQrCL+c4SJue54zNpPK9uvyLNTlS06IlQbXd6ReJr+kzIipMrOsra1Z11kWlP59TDO4\nojaP0LWZSYxFL1uYXwrgK54j9KUAzgJ4nXRMGfWuBDa2Q9XfZS0E8sPidOWML+uOd3ghnOLJo4D5\ngQDJOwLDpe5+ZsJh7wKfNJDLueqFQA8ieFxzs+hMGz6yWaWo+HDVO2GsOZGx6MOwmb8VwHkAfwrg\nTgA16fcSqj3+pBXKWZ1CuqyAruVRlUPO5R7ETKti6mdSd9Ckspp8Aiozix8qOmxN0EUTdnWyygpB\nkmlDvaLTTVsO39fW9FWrTfO5uR/ktimdqwQtGhpT0grlrMJcnK/O153VZBE2t0SXwseXwLda11iX\n2eUZqAR9WJAEC44CB2ga4V20LTfpPauEvW3WTp1pw7+++N19f1n9DDIalSWHoMpC3zTQVBkS5mNK\nWqFclJnFFVPe73BZfZtsvb5H25ldMdUh/lv+JpQyoiyS7qHKe67LcyMTfUbBfpz++wsGfHMudlNZ\no6mP9QnkbPwuw9gnYBiQMB8T8rR9Z9EK8xJEQjOM74hTRJlV19LVIS4M8ndu5jUgJmF6ZqYZVtK5\nwfNL2rjbLmWDzt8SfBd37tbrlyk3tM4y4xh3SJiPAaZwwCQHaFHlyXoPVW4ZXXhcEdjaxfMOO+R8\nNBazmAa0Xi8am6+aCfV66i3c8jK5BRtW+FFLYT/FywYDRZLfZVKcn5yTMB8LbDt/GdP3vMgyzS8a\n2Qab9zMdlfekE7i22m1eg5JJWVEnlDvEdeaXsLllksISOSdhPhbYdppR0PhsEdpfdIPlUe14RQiH\nURM4YcdzrbaLR1fxdni9flniLCbLoKR7Hjpna2Cvd9tUo8qQMB8DkqbEQThafL/MooV5Vvv7KAm0\nSUUXcin+XeWq/D7hc03vMMn2nvT+5bIFi9D8HO5N3mpdQ22IkzAfKr2eeo9M3bEq+7gpqVXRmsqo\nmAuIbKi132MhwZ4+Ht2khNi2HdnsJXbU2hU5dxTi/IcNCfMhEZgZknfN0aFzHKVp1Gm05HEy6xB6\nTMJclcdFZZdWCWchdNWLd3QRLEkKjS7NczgtctbQ1XGFhPmQSLOwQn2N7GFYaTVsEubVQGdm8TVe\n1SKq8HeM7fbs7PM8bGcPa8/y4h2dLdwuP3ly+KifAniSIGE+JPIQ5nIIGXApZ2zamPlPXxb31Xqq\nON9J60BVIewAlVe22izMEW15loskaD1l2w4v3okPIEGWTV2kVhDZIqcsju9kBOybuPZIwnxI5GFm\n4TycLW+JA75gjS5/lpMsybgI87idfrcyYx5RXcw2dhGdpM7Nshy5TiCgo/nv5banXqQUpCwWm5XI\nidiWJm6mSMJ8iLg4QHWop55qzalWm+Zra2tWjlRd3hEyrRAmrdrfxck2Pt/GxBdtc73IPfzZRL2+\nczCQBLODyWqbJMzHDL3jKTyt1adq1YWZhafZpogYEuYE58GGH2Kz7FXu52ep1V46mKklmfpMpp0w\n5hWigQLSai3EcrdP0myRhPkYodOg2+0lryEHu+ZEk1jt5To7piyIk4Q1hSNONrIyIfadDW/WHN1y\nzzZ1bVJoojkRV9BOJ3ntAgnzMULveIp2nPi00+9gu5UdIH4PfQImzmmxz6RiK1TDm2EHtvOkhGbm\nvC42TthJnyG6CvM6iBHjpQDWcPEicP78WXQ6N+DNb/5lvPjiAoA7AKwAOA2xD8jrANw4OHPHjtvQ\n6ZyOXG15+TV4+OF3A/gN75sbsbz8K8YS9Pt9rK+fAgB0OjdgZWUlj4oRI8b6+ilcvHg3gDUAwMWL\nwNNP32k8Z3v7VZHj19dPGdtHv9/HddetefcBPv3pN+Hqqw/i5Ml34qGHPh459pFHRLsH1G2ZSMBF\n8qf5gDTzGN1ul8/MtHizuZ/Xas3QlHaWi1heEdEyN3dAkZi/F9KW7uet1kJkNZ3sgHU1s5S96pQY\nHja267CZRbTFjlb7VplZVM5OYJ8ydw/NEKOAzCyjid9QxSq6S6XOMs/Dq+qCTY/jZhTfMSXv06gL\njUyz3ViQ+Iimu1VDXkqflIpZ3pXJdLxqI+igfcmx5MlZNXXCfVKEPgnzIWCTnCjoBHGnpSw8A/u2\n6th9HNjJG405RVRK/PhGY864sw4J88lBpT275kBJWvYvr1PQL9k357vXafqT5LAnYV4y7nG1KuEZ\nNqX4mx7Ht+oKm1nkBRs6Ye5r+f6CDNXUlswsk0ERTsaog32JhzeeCEdn7djx8ti9TakqdGWdJEep\nqzAnB2hGVE4ks1PoBgBvCf19I9bWrsMzz5zF5ubzePLJF7G19SyAVwD4AIDrAfwagH0Qjk/hAJ2d\n3RO5aqdzA86ffyO2tm4JfXsLgAcArGB7ewGzs2dj5VpZWcGZM6dDDs8PD+ol/j5NDlAiQthB/rWv\nPQXgMwDeC+BeAO+D3xe2toDHH78XwNswNXUT6vUOXnxRXGNq6lacPPnh8gtfZVwkf5oPKq6Z22gK\nsvZbr+/izeb+wao3+VjZXmmbMyW8ArXZPMB1zipiMsnDRCFfQ8Sk++1MNetshWaTS9YmHTKzkJml\nNEzJqVR2yKxOG9fzJ6nRE/ZkbYdmH0t0p6BgU4xohsWsZSUHaEHCHMBOiPn/4wD+AsCi9HsJ1S4X\nk9OniD0ms5QzqdGrHFqT0FGIdKiEeXghEWMNfskle7m8TR0wkzpfkcyktNFhCPPTAH7R+38dwC7p\n9+JrXTIm00paB03aBpqlYasGJUqBW13yEIKqGd/a2ppn1gvndQnHlue3r+ckzThLFeYAdgH4WsIx\nRde5dOICu8NnZloDs4u8SMJ+xxW3Bpq1YcfrkS0fOzG65GUvl+PJ5ZmoMKl0QwI83zZF0SzFCfNr\nAPxvAPcD+HOI8Isd0jFl1LtUggU6ixw4xIGXhbTb8Aa1Zo3E7xxBkn63Bpq1YZMwnxyythVdjLqq\n7fphtfX6Zdpt6oZVj3HCVZhnDU2sA/jnALoAfgnAbwP4NwA+FD7ojjvuGPz/8OHDOHz4cMbbjgIv\nAfA27/+3ALgCwAq2toB2+z48/fQn8MIL74UuZDGas+KZ8osPEc4YzocxNfUUgFuxtSX+pvwY1WZz\n8zkcPboKwJyDp9/v481v/uVYCO673tXxcrXIPAPgU3jxxesBfAPh/EFTU7cOwl9V90nKCSS32Sq1\n0XPnzuHcuXPpL+Ai+eUPhAT7TujvnwLwEemY4oewklF79I9FNIWkLHLRHVnS2RVd047aRAb42pYq\nbJIYX6JtpcMZa3LGgj09dbtimVZwiu+6PLzorVab5o3GnDcz7XGRkuJS7mf8rNd3WaySNs9kVWkD\nqgiG4AD9IwD/AkANwH8D8B+k30uodrnow7OS42Hl74NVnYHd3WVZtW4jAJscHKprTopzaRLx1yGI\n9A5x4axakanLrSKuscr9sENxvelBWG6g4NiZ7lwTwk1C2xyGMP8hAH8M4CsAzgDYKf1eQrXLRbUE\nPkmgxhMPRQcCl8aZ1LDl303Z7sJMkj1yUokK2miee1WulGib6Hna9WV8bW1NuS+o39YDbV4vzMP9\nI01CuKq3TVdhnnk5P+f8LwEsZr3OOKFaAq+y762srFgthZ+Z+Q4efNB+2XxSCgH59+1tQCy1NrO5\n+Zx33FmItAPEuCPboQNeAyCa5356el/sfNlGDXwFL764ho997AG84hX78dWvRo/f3Hwe6+unMD8/\nj7/92+/ga1/7P+DcTzFxAYzdh83NBdx11124667fHOQ5n5q6CVNT5K/JhIvkT/NBBTXzLOQxXUzS\nUpIWdqg0+XZ7SdoubFaZc5oYH4K2Jswgtdoevra2prWBt1rXaK8T9fH4ZpmlxCRtfo79RmMu0r5U\ns0XfFi7y8S8ZZ5tkZinAzJJ4AxLmMfJY2u9iZrHbu9HOhkqMD0FGw6jJrdvtKkMG6/XLtG1Sp0BE\nTSXqfTx158upluWNMVTtehJWfvqQMJ8Qkhq2bcOP2lAnyyZZdYRw3ad8r+I3OY/KvFbrtdGMXVdG\ny7NF02AwibgKc0qBO6bY2uPtuQG+jR0AarWbsbzcyfH6RJn0+308+eRfAmgpfz958p249to3YmvL\n96VsAbgHwIoyjXPcTxT38ZhiwFW/3X77zTh//uzgev61iZS4SP40H5BmngsuU0zX+PPg2FUu4oLF\nhhaTYJesKrqQQnmrwWAFcj7pkk3t1GY2SWaWAJCZZbRJ0yBdnT+uYVxZ0woQo4cqpFCXI2iUnIu+\nMz68KfmolbEsSJgPEZWgTrN4R8ZVOKeNyZ3EWN6q4jo7UwnQYaCKwIkuRJqctknCfEjoIkjSLN6R\n0S2oMC3RT7PMf5RysRPZsZkFjprGq4vAmUTnKAnzIaHSHAKzhb9B86IXMWDfIIMMjUHkwdTUXt7t\ndkOZGxdjMeFpO7Lrbu3EeKNqt67bu6VpL7rzRHnUYbKjNOiUAQnzIaEX5lEtQyQd6lg3yKgjSzR0\nf8dzObSs3V7KXOaqaztElHgb6HgzSFfnub0yYJoN9Ho9Y5qASVI0SJgPCV3DVjVMl4RaOoGrclaq\ncmukuTZRfcKJ2sKrNlXtVaepu640Np0Xbnei39gNKFWGhPkQUWkOpgRCWWyaquu6rtgcNXspkS+2\nPpVworh4u9Jr6jarOtMmdJs0LVwFCfMRQycw0zgp5e26wvt16vJR25Rv0jtNFTG1L5Mwlc/TmTzU\nx9o5+NMbF5i/AAAfhUlEQVQ66CetfZIwH0FUDVJnYydnJZEHrkvrw0I3mm9FP7OUj3WJhhrHSJuy\nIWE+woQbcKt10JuSHuPCuWnOa05TU8IFF+3bvKvPciwToqlt5dkGJ92nQ8J8RIl2oA4XUS3hJEe7\nB0I9jZ1x0rUYIkqWpfEmm3qZbYqEOQnzkSTaMFWOo2WlBmU7hZ30hk/ESasl68Jhy2bSFRRXYU5Z\nE0eG5wG8FrXal7G8fDP6/T6uu25tsBPL+fM3Yf/+q/Dtb78LnH8fr3rVK1PfyWYXdGL8yZZZ8wKA\n2wCI9vfEE6JNFtlWVO0yKVMjEcJF8qf5gDRzzrnZzFKv7+GMNSIaSHT5co/Hc09Hsxp2u13JdHMp\n73a7CeWYPG1nksiyOtMUxZLHPVTXoXYZBWRmGT6m+N6w2SSIGIjnnbjkkr082KZLZZY5FulgQU6L\nYIPetDG+xPiTVTgmRbHkcY8w1C7juApzMrPkjGweeeSRNZw5c3ow5Q1PE2+/Xfx79Ohq7Dr/+I8t\nAG8D8BYAV1jefQHAe73/nwbw9XSVIMaepE2/kzh58rjXjsXfqg2Ws96DyBcS5jmTpoF3Ojfg059+\nE7a3/W9uAfAAAHFOo3EcW1vBzuXi9zVMTd2Kzc0fwtGjq1hefg0eeeS2QeebmroJm5tX4+jR1Yhd\n3LQbDEH4lG2vpnaZAy5qfJoPJszMkna6KEwtixxQ7/oSjvv1V4KqdkL3Y9gZm+G6HYMoHr36iGyb\n2VcIJ99Dn7kzzfXCbXzS2yeGYTMHcAmAxwH8vuK3wis9SrgsyAg31uC8TsTZqTtfJNrybepRoR/O\npSEyNnaMq0uJ6mEjaLMM6n5+IKE0dAYDRrfb5e32srfRxVKq65IjVDAsYf6fAPxPAGcVvxVe6VHD\nZUGGak/GcA6WpPOFsA4WG+mTHy1SCoAJoshFZqY2KIR7EHXlqq2TIzSgdGEOYB+ATwH4SdLMk8na\nWHXC2u+M6t+nvc5mn6uaGG+S2lmWdqhuY8d4oDhE22bW9k3CvLxolvcDuBXApTlci0jBzMx38OCD\ngYPq/Pmfl5ylvmf1c9jefj8o+qD6LC+/Bg8/fGPomxuxvPwrBd7xGdRqN2N7+xe9v/sA7gXwLWxu\nXmJ9FXKEpieTMGeM/SsA3+acP84YO6w77o477hj8//Dhwzh8WHto5cnaWFXnP/igON8Pcdy//wp8\n9av3Ang5RFTMswDuQK325VDEjGBz8/nBebQatDqcP/8YgOsBnPW+uR7nzz82CIfN0g7lc2u1m3H1\n1QexutrBr//6f/UUidPww2SffPJW69Wjk7zq89y5czh37lz6C7io8fIHwH8G8E2IgOa/BvD3AD4k\nHVPChGS8yBpNIp9vzkHd86a+e/jc3FWxCAeXjHjE+JBnlk2dw163MK7ZPJDJVELRVgIMawUogGWQ\nzbxwVOFb8RWkndBGAeE0ALt5rfbSgXN1Enc8nxRUG0eo0jvYX6fDgUVeq+1JvE5au7eIkFkiv47H\nsIU5RbMUSDyKYFYS3EHnabeXlPuEAjODDHjkbKo2wR606vUGNgRpIqIDQ5Im7xopE5yzqGyTk6it\nD02Ya29AwjwTSbu++FEEqo10dZEv/sbPFNNbbfIwtYhrqAWsCVfhGwwa8fUT7fbSRLZTV2FOy/kL\nIpzOc3n5NZ5Dys3JKOd5qdU62mOvvvoQZmfPevcInEaf+tQbIcZUQKQ0fQuuvPLPAEy2s4kw5xHy\nEakm/n3McR6+hiqdsmv63c3N5wB8BkFuobcAWMOOHQ8AmKccMDa4SP40H0ygZh43h1zK/VVyLlpF\nXLPqSKs746lwZY2o2+1GlvYXsaybGE2SZl62ZjZhronbsbPsZiSjmnU2mwd4t9tVmgsnwRwIMrOU\nj9xo9Ysq3Bqi6jqNxpzn/Iznr9B1rkm0NxICXdplXTvV2adV35kGA1cTnupa7faydYqLKkLCvGRU\njVZv23YT5jqHZ5ZNn4nJRNVOVVsRJm1PGL6eSWN2bYvqfuRHW/W42FZxH2805iZCkHNOwrx01BrF\nktHMIpIRLXnJiJYTIwN0SbVsymIbEkaae7VRtY2ZmZa1th7GJimcSqFpt5eNZVTPcN0iaaoECfOS\nsZmqhjtMt9v1FuoEnSBsx3adzobJFhI2WVPYSSMpp4/pOLmtRY8Ri9LkrJxCq45udei6KbTIAGq3\nfV0VIWFeMulsg/pYWp3N2zZvdLqQsLjGRlr6+KKzeZuybYbPTWrPNlvKCWE+z0Wo4TLXbWOYVP5W\n6xoS5iTMy8NFgJqEuUnLL2qjAZ3GRlr6eJKUYtnGZGdqzyI6apoD0TzmchRLuL26pMKVyz81tTty\nrUlqkyTMR5xAy46bWXRaso0mlKU8cSer2TZPjC5JZpIsZjVh9giHxk5zYD5mPlGZYZrNA6lnin7U\n1qTNFl2FOS0aKpmVlRWcPftRHD9+J55++k5ceeU+nDz54cECiHA2OuAWvPDCGv7mb+7P7f6qRR7+\nwqFHH30CL7ywBn/vUaJ6uCwU6/f7OH78JJ5++lu48sorANQjKZQF92J29nLN3fresXfje98Drrsu\nvihJvt+jjz4B4BmITczFcbOzl+Ohhz7uXNeJw0Xyp/mANHMn1FPhfDaVsFnkQc7Q8cZmrYHNblMq\nU4kwr0S15lptT+waSXlWksvd8cw4hzLvKzrOgMws5ZMmtM90TlFTTRtzDYUpjj/yO0yzIlm9VmI+\nolSYMjEmxaHrF9r1eDgUMSnSq8qQMC+ZPMMB/cbabi/nnmd80sO8JhnXFcm6tgIc4o3GnNX6CP86\nunYup9VttQ6GymUf6VVlSJiXTJqVbiqNJVi6HGgk8nJ9/3wh8PWbPuvLObkLMCYZV2GuaivALg68\nzFmY6tdNRK/P2O5Q+K19pFeVcRXm5AAtkSBL3Stivz311Je8788CuAFbW+/B7OzZiOMnOP8tCGeY\nU2W7U7MAsZ3XKQDP4OqrD1LmuQmg07lB2hf2Rogt5U4btouLtpVaDdje/i24Zi7UZ0/8HIAgEyLn\nwKtffR+AF/HEEzcPsjT65fMdtoSe2rALMO50Ojdgx47bIBr+aUxN3TTYV7Pf70eOXV8/5aXyvAMi\nHe1p73MLLl78WwBLAK6FaOAXYvcKzv86hCBfAyBSmIYbe7/fx9Gjq5EyBOV8FsC12LHj6zh58p15\nPgpihJmaqgP4NQD3AfgVAJ/DzMydSiWg07kBU1M3QbTTZzA19RSuvvpQLuXo9/teutsvxX6bnd2D\nxx57BJ/85Edw5MhZHDlydlC+5eXXQAxCfp+50fuOGOCixqf5oOJmFs6jpg/TAgdV/K1YIdeLTX3D\nUQL+9QPzTLpsdcN0IE2a82pUsF35GUYsDAp8NlNTe/na2lrirkW6d6z2Ba1y4Yh1WTnd8dq++D+Z\nWchmXhg2Czbi+ciDBTphwe4nJYqHbPmRCOoER3nYFvMWvDYhkUUI+nEbQIoorykniypEUef8DO9k\npYpiSXZ2qtp8xyp1hO3K1apBwnyI2AjSYF/GQxHNxE9vKy99jl+z40UTqB2gWYV5EVEDpjIVFaUg\nX7dWm+bt9lIpg4iuPHK4oCl8MK/noFtVrEt1K44/pBwA7JNvBceoB5MlT7teTEy+pR4M0u1nOm6Q\nMB8i8kILXQ6VsFmm2TwQ0zjCqULTRMu4CAV9vG96zV7GdM2iZhImjbRI4akrnynfyNTUXt5ozOX+\n3FX3Nu0P67dJoWSEMx7u4vKG4emFeYcDURNOsnklPhhVXZBzTsJ8qAhhbpfd0Ccp73PaOHYbjVN1\n7WBDgPyEiqkOWYR5r9fj7faScnWsPhwvLLQWve/MNuSsGny8LPHwO2BfIcJcV369MPfbY4/72vPc\n3A8mtkFbM4vYwtC+npMYkuhDwnyIpGl4NnmfXYSJy7Gq8srx7nmaPXTOsTT3C87TpxMOD6xCG/T9\nEUuS70IsYFG9qyzl8+sbHyBVwvwaDpgdjHniqrH7A6dpwZDOlBTe4tA1adwkLhbyIWE+RNII8zy9\n9K4N39Rxy7Qlt9vLAz+A7f2CstulDfZTtuq29VPlGInex/6dxs0qeyMremUziyjb7tDfu7TL5PNE\n9Z5tNWwb853NdVS+DJtyTgKlCnMA+wFsAHgSwDkAb1UcU0K1y8elIyRdJ4vmEdUAy9N68uhgWe4f\nCNloLg+ThunbWnUzEvN9XAfo+PV1WitjTe09hiHIXMwyLs8gXKd2e9kLBDDniJlkyhbmVwC4xvv/\nLMSKlB+Rjimh2uViEkJpOl/aDhvXctxzr6Qtbx5T36z28qAMqxyY4c3mgYFGm1cETZq6utZLNwiP\nkonBdQaZ9AyS3o+uTU6Slj5UMwuA3wfwOum7outcOqPilFFFCuSRKjfNfdNsNZf1OZocoN1ul0dD\nPy+NmC6K8kP4x+dhkhiVdsY5Nz7PNLNUk4nPpCiNyuBWBkMT5gBeCeBrAHZK3xde6bIZlU6mns4X\nvytL9L49Hnbg1uu7eLO5fxDLbCKPzqmPyvBD7HwH6KXOGwpnISzgbPOHZzVtFEla4WvSsO0GsEBR\nKCLSapQZijAH0ADwZwB+VvEbP3HixOCzsbFR9DMonLxtzWm1RN3Cj6KJ1j8cmREV7L72lmXanPS7\nTsioMlPOzLQS6+VaTpvyZ2kr8rlpB4as6J5zVlOZ7EeIrvSMticxAzPHu48zGxsbEVlZujAH8BKI\n/aFu0vxe+EMYBnnZml0Ecl6dOy/nZVxoxjt2ozGXqzCz1fCS4vdd7qX7zaZ8WYVdONLHpq0U5dSO\n+2emB20vq7asMpcJIR5fiRpOK0BmlhyFOQAG4EMA3mc4pvBKjypy51A1fNNuLDJpOo7cCeWwOMZ2\n80ZjjrdaC7zVWnAOEdRr6aJ89fplqTu7bX11sx15laVp4DOZrHTvyKZ8aYVd8FxVGzgUeT+9oAxS\nUQSx8LYDTPIgYdd25KigKlO2MP9xANsAPg/gce/zBumYMuo9cqg6h0pbjAoKkUlRXq7c7Xb5zEzL\na9yrTh01bt+OrzQUHWmWh5N4uey96HfWVmuBy06yVuuawoV5UrlsTFJmZ3Jc0PgLaJLKl1ZTFuVR\nbRBhs6zefd1CloEpjW08fk31YDpJDk8ZWjQ0Iug0Pb2ZRZ0JURVFIAS6a5icH48tC6YOF9kaFzkw\nH+pUi1Zav9yJ/YHHd4DmbTNO05kDW6xeWMXNCOEwz6jtNpjdRN+ZbgFMGrOWeG/xQSTJzJAUxWO+\nn9ks5TK4Rs1w+sEl3j7jpqxJ0cRlSJgPGbUt2azFmISN6vt6/TLrxh2fxoY7jZ9S1+/4u7nIaJcs\nzF0EbZYOmbUzm6bxKi3a17hFArSwFhyOqgjPsPy89A0ubLx2OXlsyq1aN5BkZshiZomuSo3PzsQx\nybmHom1jnpvSVUSP7fBabY/V/qKTAAnzISI3TPfk+/lEZajKpTLnANOxawuBlGxmMdmYkwRvmdpW\nkuYnlys8QOl2sVeZZKLL8XfzVutg5rILG7XbuoEspqmk5GMqP0TygLIcK4+s8YcHURLkASTMh4iq\nk9supok7KsWGzsLp5T5tTrq+ED77Yx2tXr/MygEat/93IrvT6ARPXqYT+Zp2jk21T0J9rHgejcYc\nbzTmeLN5YCBo4s9SNSjms1m268Dn+nzD1w/iuIOMiWEt2nagiA4KybHhRbSJKkDCfIjk5bCLbq91\nP6/Xd/Jm84DVQhwTwp46zf1oBCHQZ1J1oni2R7vUpnmEsoVJEgSq38NRLeH/xxeldDhjDR61l++N\n2HKFH+TlXM5Jb2OmKsr0ZHttlQJRr+9U1pdzu3fX6/V4vb6LB4u1XsaBndyf4ai0edN1yWZOwnwo\n5KVh5CXwbEIjgflUS/GF4Jvnwnm6zFUxwaq4bhtHmwu2AiYsvAO77zwPz3rk7IZBGJ5e0EQ1dH+3\nKH1KXdV5w3AK656d8BXY1Ve12lNssjErPZN5T4mYV67CVZXDtBvSpEDCfMjIgiONVpGHpmIbGqlL\n/ZpUx3p9T6jDzvB6faf0ndruHtXcDnGgwRmL7ylpWw7XvSGjMwpVyGHgYBTPSy/M1YPjPp6UjzzL\nYJ3nzEYnRE3X1znwg7amCn31r6merahNgPOcsbj5KsssbtwgYT4k5EZehPblck2b0EjVxrymOvmo\nBoVW6xqr3Xt6vV7Etu5rs7Vaso1ZHihVWrHJVn/kyDFpIYpZMAbRG/ZmB9+J12od1Dr04nb8ec9X\nkez8cxXmpsFfZ4JyzbUSLVN8XYHvBDUpDr1ejzeb+7mfd15nussyixs3SJgPAbUWnC0pkKrjuJgU\nXEIjbevkH6+6drN5wKp8am32WOLzkcujCtvTOTaj54bNQdH4cV2Eiy7SQicMgwFArN5kbCaWsTFY\nWxDdbzNpT0yXAd3mWJ2mbda+ddE9PS606rCZZXowWKsUh/C94vugdrhYKGWe7VUVEuZDIM101YWo\ngNavAIx2OHVopK0wNwlm1VZ3jDWtbJxphXn8vOS4cfW5PR52+tbruwamlTTORLVfwncwR2dB8kAQ\nmIjc2kke79CVJNNffC2DeA6Nxpzx2YXbi5ixRdv3jh17ue1erVWDhPkQ0Dn18lq9aBP7rC5HNDQy\nq5km3HlFlEd4f82Oleavcxq6x+FHtTa3utiFjKYxlQXCXG2PVx+bTuAmPeuyhLlfFlcFRj2wB3H1\nIjx3Obc6jBskzIeAycadZn/LMDqtX9WBkzqcS+dOEmQuu9urrt1uL/Nmcz9vNOasbMXxWcduLhxr\n87xW25No+0+fH8XOrOXb8cU2cNM8WEQUnCvbjJNs8qp72Jg9stY77TN0vZ9amB/yPjO81VqY6IgW\nEuZDIm2HS7qWi2aSdL+inWdZHbw254lBJGxHNYcB2tRFh402qjNridjqSxPLabLJm56VeA6HvAFt\niQMd5QrcNPXO8gxd7hefpe3mwBqXV+qmjQobd0iYjxA2wiBpAJBT1rqs6CtCU5M10TSdLMv0X63N\nJScFS4Pb4Bgvl8req4rGML0ztQlvSRER1PBMX/m827KEZpBWd94bnKa5a2bQquIqzOsgSmVz8zkc\nPbqKzc3n8eSTT2Br6x4AwCOPrOHMmdNYXz+FixfvBrAGANjaAtrtD2B29iw2N58H8Eqsr58CAKys\nrESu3e/3B791OjfEfl9ZWRncQxxzGisrK+j3+zh+/E48/fSzuPLKfTh58njsXP/61177RmxtzQMA\nNjb6WFj4UczO7snt+aShVvsyOp07cr+u7nnZUq//EwBfAfA2iL3ObwHww5Fj+v0+rrtuzXvnQTvQ\n3+cCLlz4Mji/B34bEdwLoTuJ7y5eBNbXT2mvI7cVAMZyJLWttJw//xi2t98K4AEAd3vf3gSx300+\n95gYXCR/mg8mWDNP0rKFRhVd7KLTWpO0xCymCxu7LeeqKBa9M9bluWQ5NylWvkiSoofE8zLnFk9n\nyok7TQOnazpznMmcl+Z9uUXc6OozWTZyGZCZpRxsnZtJ9m/RyZOFdp7OzTAuERWqaAW5/K7PrtVa\nGNh6XQS6i122SNOByexkIwRdnKxiUY0f+hdVCur1XZFUBK6RSqZIFF20lumZuMTCq9YMNBpzAz8C\n2cxJmBeG0GbDnWmGM9ZMjMrQ2XvlBq8SQHkIc/117YS5zWBk8+xsQy1t0QnsrDMAWTCbBIqpDEnn\nBQuIFrWROXGh1+PAoUiWSzdtOG6H1814VMerInMCpcVtwZyc6jc+g83eRsYREuYlYCuUZeJml73W\nWmnSuWnNMC5mlvggFnQym+X4wbOLmh6CAcE+ZXBSvYJ7uc9WVM/apPVmdSzb5C0Pnpt6IVKWKBL/\nfqo9Pv02Et1sORqZozJ9JW1vpyqTywx2EiBhXgJqYW7X2LJM+/1z5RS5U1O7ebu97H3Ug4NJsAmz\nh93mAN1u18tvssiBLlflvTahWj0qQuuSl9arMNUrm+lJNVgnzYrUecCz1MHmHnnZtE3lUK0r8E0t\nOs097eCWpX9VCRLmJRDXUOOOzCzXThL20cauFoLydbKEA4bLFpgEku+pQq11HeJpV0ImDVLZFgsF\nglOEzkXv4dc3SLOgX75veqY2mR+LmIG4PkvdtnI6m7rcFmwVmSJMceMICfOS8J14zeb+zPG94Wva\nCJ9o54mbLVSpBPJYSRcfRIJde+TObrelmOj4/gbQeZhEVCYQ15mQyvQkC5To8wxvGxcVaC5lt838\nKNclSZinFaK2q35t2m3cdLXbOicOOUBJmJdGFtNJGJtOGR9A4hvmxrPPuWVMTFM+lcatWyCjs93L\ndlfbFAh5Pf8wqvqE7fnxZxHfnCMpV7xuYEtjflMJSj8Vr84er3pupmepc5z6pj+T/8dmNqmq1yQK\ncR8S5mOMSVh2u13O2AwPMvKJCJodO+KCOwhhc9NykzBpYC4bT+s6qT9YCXvrcKfVOkGrF+Zyutbk\nNAOqe9Trl6XaHjDwpyxJ5pB4JkJbbVp1j7Srk+OzSXP7zOpUrgKlC3MAPwHgMQBfAPB2xe/F17oi\nmLRWVTSBb+aQTR/N5gHrmOM0ZVQJYpVj08UJ6JOX/de1/PL3SXZbldlKbL5tn3ws6oPws0+uDu7X\nai1kNIVx7/9x522WSB9T1InuGtHnmewfKbodjAOlCnMAl0CsV74KwEsAfB7Aj0jHlFHvymAbZeCv\n+Avig6NOSZewx7zKHeytuZhqEwFbZ2CWMtqYeGSHri6PfNIAYDOIymGJYqDuDd5xPpEgM7Ey5SEs\nXa+hmz2kXUxVdcoW5q8F0Av9/Q4A75COKb7WFUfXQX2BGQjB4Tb+LDZOV2dgGnQCIklwuAgW12eg\nfrfHIv+6vMf4c7ycixBSMWML28bzSLqW16KsPK9dFcoW5v8awAdCf78FwG9KxxRf64oT76C7eKt1\n0CmiYdTJyxnoeg8bYV6kYNHPugIN3fU92s5w8nAwFumkJAdoucJ81UaYnzhxYvDZ2Ngo/CFUkapr\nMmUMRq5mFvncIgSLaqAGGjyPFAfj3B4mkY2NjYisLFuYL0pmluMAbpOOKf4pEJzz8dZkyhI+tg7Q\nMlHZ3vMoyzi3B8JdM2finHQwxuoAvgTgdQCeAfAnAN7EOf9i6Bie5R7E5FBUzmyCGEcYY+CcM+vj\nswpaxtgygHsA1CFMLr8h/U7CnCAIwpHShXniDUiYEwRBOOMqzGtFFoYgCIIoBxLmBEEQFYCEOUEQ\nRAUgYU4QBFEBSJgTBEFUABLmBEEQFYCEOUEQRAUgYU4QBFEBSJgTBEFUABLmBEEQFYCEOUEQRAUg\nYU4QBFEBSJgTBEFUABLmBEEQFYCEOUEQRAUgYU4QBFEBSJgTBEFUABLmBEEQFYCEOUEQRAUgYU4Q\nBFEBSJgTBEFUABLmBEEQFSC1MGeMvYcx9kXG2GOMsXsYY7vyLBhBEARhTxbN/CEArwbwowB2Ajie\nS4nGjHPnzg27CIVR5boBVL9xp+r1cyW1MOecP8w53+acbwPoA9iXX7HGhyo3qCrXDaD6jTtVr58r\nednMrwfwezldiyAIgnCkbvqRMfYwgCsUP/0q5/z3vWNuB/A9zvnvFlA+giAIwgLGOU9/MmNvhdDK\nX8c5/wfNMelvQBAEMcFwzpntsUbN3ARj7A0AbgXwEzpB7loYgiAIIh2pNXPG2JcBTAF4wfvqjzjn\n/zGvghEEQRD2ZDKzEARBEKNB4StAGWN3MMa+xRh73Pu8oeh7lgFj7Ce8BVNfYIy9fdjlyRvG2De8\nuj3OGPuTYZcnK4yxDzLGnmOMXQh912SMfcKr5xnGWGOYZcyCpn6V6HuMsf2MsQ3G2JOMsXOer64y\n789QP6f3V7hmzhg7ARHt8r5Cb1QijLFLAHwJwOsB/BWAPwXwJs75F4dasBxhjH0dwD/jnL+QePAY\nwBj7lwD+DsCHOOcL3nfvBrDJOX83Y+w2ANOc83cMs5xp0dSvEn2PMXYFgCs4559njM0C+HMAPwng\nF1CB92eo37+Fw/srKzdL1ZygPwbgK5zzb3DOvw/gowB+dshlKoLKvDfO+WcBfFf6+loAp73/nwbw\nc6UWKkc09QMq8A45589yzj/v/X8TQnn6AVTk/RnqBzi8v7KE+dsZY3/BGPsdxtjuku5ZJD8A4Juh\nv7+F4OFXBQ7gD7zp3fXDLkxBXM45f877/3MALh9mYQqiUn2PMfZKiDQif4wKvr9Q/f7I+8r6/eUi\nzBljDzPGLig+1wL4HwBeAeC1AP4RwHoe9xwyk+A1XuKcXw3gzQB+1ZvGVxYu7I1Ve6+V6nueTfyj\nAG7mnP9d+LcqvD+pfn8Px/eXOs48DOf8iMVh/5cx9t8BPJDHPYfMXwHYH/p7P4R2Xhk453/t/ftF\nxtgZCNPSZ4dbqtx5jjF2Bef8WcbYHIBvD7tAecI59+sz9n2PMfYSAB8H8ADn3E8dUpn3p6qf6/sr\nI5plzvu3DqHlXTCfMRb8GYBXMcauYoxNAfh3AM4OuUy5wRh7GWOs6f1/L4CfRjXem8xZAGve/9cA\nfGKIZcmdqvQ9xhgD8DsAnuSc3xP6qRLvT1c/1/dXRjTLhwBcA2ALwGcA3B2yc40tjLFlAPdAzG4+\nwDn/jSEXKTcYY68AcMb783kAH+Oc//YQi5QZxthHACwD2AOhwb0LwP8C8GEAPwjgqwB+Xp6+jwuh\n+s1C2I9PADiMCvQ9xtiPQ5T/CwhMKccBfA4VeH+a+v0qgDfB4f3RoiGCIIgKQNvGEQRBVAAS5gRB\nEBWAhDlBEEQFIGFOEARRAUiYEwRBVAAS5gRBEBWAhDlBEEQFIGFOEARRAf4/LKScg8JzWvIAAAAA\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1162c9fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "marbles_that_hit_the_wall = filter(lambda x: (x[0] < window_lower_left[0] or x[0] > window_upper_right[0]) or\n",
    "                                            (x[1] < window_lower_left[1] or x[1] > window_upper_right[1]), \n",
    "                                  random_marbles_thrown)\n",
    "\n",
    "plt.scatter(x=map(lambda x: x[0], marbles_that_hit_the_wall), y=map(lambda x: x[1], marbles_that_hit_the_wall))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we find the percentage of marbles that went through the window to find what percentage of the wall the window takes up."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.749"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(marbles_that_hit_the_wall) / (1.0*number_of_marbles)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.251"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - len(marbles_that_hit_the_wall) / (1.0*number_of_marbles)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## FACT CHECK"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's pretty straight forward to do this manually with math."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(10 * 5) / (1.0 * wall_height * wall_width)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulating integration of an arbitrary function\n",
    "\n",
    "The same technique works when trying to evaluate a definite integral. Let's use a normal curve and find out how much area of the curve is between -1 and 1. Or in other words, within one standard deviation. Given what we've learned in stats 101 we would expect the answer to be ~0.682."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def normal(x):\n",
    "    standard_deviation = 1.0\n",
    "    mean = 0.0\n",
    "    return np.exp(-((x-mean)**2)/(2.0 * standard_deviation)) / np.sqrt(2*np.pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.680116799628\n"
     ]
    }
   ],
   "source": [
    "max_value = normal(0)\n",
    "x_values = np.random.random(5000) * 2.0 - 1 # Generate x values between -1 and 1.\n",
    "y_values = np.random.random(5000) * max_value\n",
    "\n",
    "points = zip(x_values, y_values)\n",
    "points_under_curve = filter(lambda x: x[1] <= normal(x[0]) , \n",
    "                                  points)\n",
    "\n",
    "print(2.0 * max_value * len(points_under_curve) / (1.0 * len(points)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "+/- a standard deviation is ~68.2% of the area by the way. Our approximation comes very close."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And hey let's graph this for good measure. This is the region that is under the curve of our function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x116525ed0>"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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VIFqyl2X8XXcl88xWgR6Q+ohFT6rTdgBGbSVUG8wWaeiwAyWVmGVL0EQj9fFi\nGnn3dP9KeV5UVa7DTsCx8iUf3e9seOUzFHgjnNRZNhTOz5/FwsKFooGYVTkt0Ue5rTJpe8/od7yb\nfTZOCOLfRzBxe0bH0kApecZYxMHlsr6Wn9MDRcr/4Ad/ceh5HtC8srAI06q/p2rQ9ZO8PanOat/K\nypWhHaBWHWrGYU3gOsyEPKtVIfqeVYalDvOiqlp1120uTYa6DiW1mWWrAbYnD72D3DLSejYPmdzk\n0CDPBuVJ9BpeMEKdT6CMIP59gqVzZele4G3nF3UPu13WfP9lMGufdnans4xx2vNETxZAXfLS9ecV\njNTdOlyE866RG0uzrXrx1lWFEPitW18ZSgdgIJyEXjXoyKDWcY7asGnZPfjd65VZbTKsTRrWqkRP\nwFafeqsGgUXccmZxrd5LS8MbBmtG9MDOEMS/T5BBw+ECLP9rj8AsgvMGQc55IPCZGDkBP5yBQEjX\ni/woBFFzu7SMcnpl4EmjrHMvgZ+v6ZA1aj7ykuepUw+ZZQL+ebs8wXM7S4ZKy+7BE3apL1snSCbP\nF198qR8nR7ct54zFxQuYmChHAOVntCprVK8dyUuvXHsrjov9MePZpTwBKPT9NoL49wnWwODrlmum\nZSwVspLnZQVg6ZN12RoWCVn2Az14vMmJ0+pTmnT7rLJ0Ot0X1n0LnmuiN/HoPuL0Wucs0jeAvqpH\n56t3zPJE5unGS+Qo4RVa3qNWxXFMJW4rB0+z+nhzczs/8XSyXE+9dlghsy1471N/SysrVzA394R7\n6pcl2MTqwEcQ/z7Dc3HUkrz3IbNumb1WdHAr/eyoOmSrXoCtO7ckdO33zaomi6hYGrVOyKr1TanO\nXnme9Oi5PGqPp1K4B2u1IvdKnl2cTy1Qm0WqCwvPYnMTSAl99YrVTtaV64l7dfVq35jLh620xunh\nMrkfWA1Uc9PkewsLT2Nu7omhdrMKrLR7OTCMzsSfUpoB8JGtvD6ac/4ldf9RAD+x9fMPAHww5/z7\nXcs9yqgRsxcLX8hKk0pKqR9iF9hdFzeWcC0i9vKXv+fnzw5s4OF7Vl1rRy6OQjrWb/Zo4v954vGk\na0kvhC1E5qmKAN/PnH3adRmtbbUmwYmJ3jdQO0/BUlnJNe1gUJrMLMO/ZS8ABgP81dqoT46zDnzh\nMNlxMtdo6ET8KaU3AVgA8C4ArwN4OaX0qZzzK5TsDwHM5JzfSCn9GICPAbCDfd+hGNVAZakLGPr3\n4uLF/uZsG+GaAAAgAElEQVQgNhLOz5/F2tqNvu50JyipRDiN1U5g22OEDdmW1F1yO9R5d5HktBSt\nVwPSRqstmlzYwD2KOq1U/5J6qiUfIUiJLdTyrGdrsZ7RxmrLJsH7Enhympl5EOvrN/r6+hKE1Hsr\nF1+9pM86YISO30dXif/tAK7nnL8IACml5wA8AqBP/Dlnjtr12wA+0LHMI4dWaVDf1y6fnnTFZCjS\nuRB2ztvugi26b329pBLRRFxSb/BzlipFSGNm5kGcP39hiBxaVxo1eHpkKYPj62hpmlcDNQO3lni1\nMblFmmejpldnLx/vIBWGZ1fy0ki99GpBJHPpH2/VlLMdELAkMJTqXFslxwrAR1fivxvAa/T7SwDe\nUUj/jwH8ZscyjxwsF03rlCoNa+B4xCQDwgqDYEm3vHFIQ6e7du065uaGvS1q9bXuLS5eNEmBJXHJ\nm3XC3sBuOagcaDNSe1Jl7VlrMtN9rglT103HvxGjpi6/tLry2uKB6+NNrLpd/F2JYCJ7HVo25ol0\nznaOUepvqTGtENwRkrmMrsTfPKWmlP4WgMcAPOilOXfuXP/v2dlZzM7Odqja4YH+WGUZOz//mKvO\nAYZ9oocH0IWt5XDqh3YQKVPSWGoa9pe2pC6dzoLW/es8rDZ4m6qsti0tXcLHPvbs0CHr3jMlCIGt\nrFzF+fMXcObMjwzpn716AOWVDJOjN3HU6utNCpYjQKu+3brmrRY8FVfN3gEMb/7yYH1fYufQap/S\nJi5LGNL1LE2yRx3Ly8tYXl7unE9X4n8dwD30+x70pP4BpJS+B8B/APBQzvmPvcyY+O8kWANR/JVL\nbm9W+AW+L4Y8lhYFejWgQwZzXHS5psHpvAGnr2lcu3YdOYtb4PCBL7pfuI7e2biMFqOeEMPa2o3+\nbznkXXD69PAxk3yvlncpTYt0Xlst6XfYotqQSakUnVSe0yGkS5M0sB11c2pqYsCDiydUa3LkNnkY\nZQMfMOwKq1cld5LUrwXip556akf5dCX+TwOYTCndC+DLAN4L4H2cIKX0VwD8FwCP5Zxt8XGMoImi\nJMHVDGB33ZX6g2R9/UZVOhN40mgLuXB6S9LXeczNPdavg45OydDeNq0Dt0Wy8yYP/WwpzpHlQtii\nq2+tI5dReoclN0irfTWyFfKsHVyj6ya7cblebNC16qHL9/rOc3nWkIi0XhiIO1Xq3w10Iv6c8zdS\nSnMAnse2O+crKaXHt+4/A+BfAngrgF/Zkjy+nnN+e7dqHz1Y5GhtuhplF6K4x01Onui7ckpawCc8\nT5prHbAaJV1xzQBXy0sHf9NxfDwys3S8NaN1aaIt7aL1JF2rD1rbruEdXajbWtoQZU2ikh+/G63C\n43SSv+4r+Z5rB8NolNLU+m1y8j6zbrVVYmAX/Phzzi8AeJu69gz9fQbAma7lHHVY5OgtzVvipQPb\nB2p74Xfld82oKagRVolkvGdqbdAQFYLOS55tiT7q6Xi7SN6l+Ebe+yzlJ9dLqweeULyQytzWVoN9\nzSOr9ShHKx+OPGvtbG6plzeB6n60Jh8vbWAQsXN3n2AZ0uS3SGKWpKWfZ1gSOV/zJpLa6oKhjX08\nUEs7OUsbezjInN552cvTb7eWYr0QyLVVzij6aHnWC5A3yopIX7fuSd6Lixdw8+bt/p4OC5p0awZ7\nrz7e5F2SonlFxv0uBNziUmr1u/ddSXnerucW4SbQQxD/PqNkkGsZKIyaV0pNVdMyOKzBretbW1VY\nemBg27uJMT39gBnfR+BNRLqejDNnzuKFF14a8vyQOnAZVp9Zrrclwq5NVqUJisubmDiBzU0UXX4Z\n4jklahfLmF4z2ur26BPCSjGhBDs5/lMLPaurV4dUU/y9e/tbBCHplxHEvw/gD1OfkKWlYO95rbMu\nqYVal7ktg6O03JaBaIVBAAYlZGDYh19L+sDggNa6W01mLdI20FMPHT9+rMmrxLrOah5Op99bSywb\ny26h01vlCWo2IC6ndL/0jXBafXqZzsOq506kbdndXRoH8g1oAclra5C/jyD+fQAPdktH7RmneHOK\nhqcW8iaEUYzG+ronya+sXOnHCOLt+SIdW6sRJktLQiv5rOdsewYtLV3CwsKzQ2GEPUOkrpMFdu2U\nNNaxfzxR1Q5k1x40rbYWRkltpJ+Vd2Hd9zxm2O22NSTzKPX3wJK+l8fp09tnWFh1K3k8BQYRxL8P\n4I+4tCllZuahrciKqU+qOWPoZKmcs6sWEtLV0naNMBiWgdjykllbu4GNjdtDu3p1m7V6pja4refk\nt2fglDDCgpmZh3Hz5q2+AXxUMrImaA4yZq02SgS0tHQJH/jAh3H8+LGhUM8eWlQYJVfa0mlurDbh\n9wvAdbst6fs9m06pLRreIUS6Xz03U2vyC9gI4t8ntHyMsiFLdNcyAC2J21MRyXWt3y2lZzDp86Rh\nTxw9FUpNt8rP8n0rTK/V1pIkJ1K0JlNR7wC9YyTl6L/WFY81QfOpWy3t1W2Rc5FbIO+Bd2HrOutD\n2rm/JIa+t9rRaiGus/etllRZnk1Hk7Jug+4jyw5j9Qc/U9qsGLARxH9IoKXZFoLypGfxjNFS204G\nn+RtDSTZnFXKz2oL68SttmjdNx9xqL2CPDuHTAQSpVLPG55UbkmY1sRWSuutZFqJiSdu6/Q0Vt94\nmwFrKg+u9+rqVTfgnOV5ZbWf22Md3Wgd/m7ViY3Tus3We9YTovc9BAYRxH/AsIxullrFkrY8vagM\n0pRSP4SuJTlaA7BmUGRpU9zqahK59gYRsBcK628991PpBw61wK6rXA6fHWzFa5e/+bpuQ4k8S2k9\nY3yL/ll88SXoWc1zy0JJatf1kyBrOnyHuE2KutFyLLC+KWD7W+XVhnYA0O9ep9PfAMeG0mpOnvBq\nR4IGegjiP2BYRjeLIEpRDPWh7VZeLSTvxaqxCFMkuoWFXhkScM4jRE+i5/TSRu8s2+28huuvDcsS\nI8jb/i/Pzs+f7ROSzrNEGKVVjaUa0xONR0qbm7l/+pXUu2X3tAdr4uX6sZAgJMz9L+pGFiIAuHp2\nyVNDrq2uXh06jEXnwxPe4Hsvu9B6E1AQ/zBSixSyH0gp5cNSl71CqwQiEhmAIcnfyodVIS3lW9L3\n0tJ2NMzaWbiM6emHkDNw5syPuLtLBVJPIWivHCuf6emH+xMMYLtCcj8IoWsvHy2J6zpZ58q2ouX9\nlmwEXD8dE2jUciQNT4alcArcd9b3tLh4sb8a0+VK+hb/fZ0P10PXQVYc2sOIYb1nXa87mfi3+rwc\nhc9ASPz7iBYdu6TLuedBImqNktqnFgOHpSYmAl1mKUy0JT0tLV3qS4SlXceCEoExUem0co93pWoC\n1T7lltSZc95SXdi66dXVq2a4Cy9OkIbXR0zgJULnowStVQO3o7SDlesibbG+MQarR/RvaZMVilv+\nbtWvs2Sun7FsIJ5uX1Da9HgnE35XBPHvI1oMUXzwCRNqScqTQezZB7Q6SYfg5XulMNH6utSND31v\nfZbL4jpau4EXFp4FgKJfeYsevFQfS8WjVQyiPvLgTTY598IviKTrEbqObe8RlzfRc3u8TWfW3/y8\nd6CM3LPKknTeao8nUG3Q97zWpI6WAFDbPRy6/TqC+A8AljpDPlYJdwsMfviypOVBLWCJywIPgJqv\ntCdRW5uO9K5bTZgaFjlrLx2ZuDituLnK/ZWVl3Ds2KA/vEXqlrqMdcdejHiPOHhDFJcjZKRPlBLp\neWHhaczNPdEPne2puLx26Lp4JKtJc1RY+XqG05rKigma85eVbFb7UwSlA34ADKh/PNVS6PbrCOI/\nAPAA07rYmZkHh4xp4q8tfumWxCeDpRbDXw9uj6R1GTUJenX1at910tMj601DPJi5fpyW3VwXFy/i\nxRdfwsbGLRw7dqzaTvZI4eus7iqtoAAMHF7vqch4QtAhJ+Q37wGQVZ0cYiJpvXZoeITmSb+1ENU1\nN1aelFpcXj01k7zL6ekH+isgT82pr/GkmjOG9jDodoVLZxl3HXQFxhGiL2WvGyZ4YHAA5Dy4+cci\nA/bM8CA+2/K3twPVK8MD+21rCU/ir3CevDrhyaCn1nl6qB8Ep08/iunpB/DDP/yD5u5XaZOknZt7\nbMBuIQTG+VnPS58sLW0fPSlkIhMyl3Pq1Mm+JK93y66t3cD58xf6bQWGDzGRtNb7895PC6Sfud+t\nmDu9VdSgCobhqXustAD6/WFttpI2TUycwPr6jYG+lHuy25qvSb+eOnUS8/OPDZShVYZdvKDGBSHx\n7xMsnSgwLDXztZp+lpfUQDlappSp9dZe2pZDUwTWakNI1gobwGoDy12R26TrJZI3lyP19/T8THYs\n7XvvhFdh7OVU03MDGArCJ4eFcP56p3Jvw9YFbGzcGuqLneqrLXWiFQcJGAyQZvU7T+y6ftYqztrA\nJX0j0VhlBcQTDveTjIWaQb1lJ3VgGEH8+4QW4+MoYZl5d6deUrNvutbflwzMmkRrniOS3ppweHlu\nTVSjnj0g8KRRwJ+sLF2wTDq63sA2mWh9vfQVE5uerHWMHyYsb7OY5KUjiPI7ttQsJfAKxVrZeYZt\nq/+0kOJNtgsLz2JzE32XW11Ptl/JfV0nvUtY96dnkwqMhiD+fULNcMhpvHR6wFmxfIBe8LSSnpP1\n61rfz0Rm7erkwanrzTps2TVrhVm2pMFSX3DZck9LzDLZeZOYft6qt6cPlzrzKkUmNT1ZeztSJQ9u\nX4u6jSf21hWYtNPTg1t56e+M699qRBYjvPfN8srJ6x+eZNi+49XbgrcRMbCNIP59hqem0NCeCVo6\nTym5ukxRL5QmDX3f+q3dC6X+1vNMqgsLF/pLeu01I/WoDUrPm4X/1kbh6ekHcPnyVSwsXNjqhxMD\n3jYWaXC92eZiuQ5a/WTZCTyytVZbnkpL7DGsJ7dWTjkP7y/gCZaft6T8Wj/rOpVWHNY7tTxs5Jp1\nfnKtHi0SfsvxnOOOIP59hny4pfNRgWG1j166A37scj0AdVmlwePpb/l+ieyWlnqbuuQxrQNuPSvA\nUqN4hMuT4MrKFWxs3O6rTXiiKoWU4LZxPZm4tE3FU31Yu1J1W7k+1nssrYq0ukjvL+CyOchbKS+G\npzKzhBY+fMeaFErhG2p7VFrtTPr50glugR6C+A8I1oHiDE+aFpXGzMyDQ7tQPUj8l1EHQKtqgScl\nIUwrRo6WQEvl1OqqJXW5JpOAF7VS0nkqHQDmxjTLpsL198hbypO2e2EtrDhGTOAM3v17+fJV5Azc\nf/9E3zgr9dErq1Z7ivferbZZNgtuc6lMtklZO9R1v5acFoDhMN8RqdNHEP8+wPpgdUjjUloeVLyR\nikmoJF2OquscVVIqeX2U1CL6umW4s/TBelAzgVy7dn3AhdPa5SkqHS8MtEjJTNKLixeHbCpeP3nG\nYt0/lpSspWp95q2sKiSdTOp6gx0b3Et9X6p3CZ4NRt5Vq9Cwvadhe5MeCwjWHgxdvvzNHkItIUTG\nGUH8ewxPl8vkwdJnzQbg/V1bmpfyLD1XIjfLyHvmzNn+oSdWXBqLzJeWLg15hFiSLtfPkprZJmH5\nq3M9S2kkX/FqEhtBKWppKbyz19eWkZTbze3UrpiW+suCBLcrxQiy6sN1KUnbtfbUYEUj5by8792q\nD7+fUTzkxhFB/HsMGbwaeoAvLFwcOktUE7AOQeC55AksH2tRAbTq+fl5Jg4OKMZkLi57Opa77hP9\nu3fk5HCaWjx67gP2oR9FVaSvS37i1aSlUT2JaY+iVunaSsv9LfdLXiq1VYcY+mWVUAqC59UFKIdB\n6LLXoORyaoFXd96ENsoqZ1wRxL/H8CRl1ot7ekhNwJpIa8tpa+B6qiFPVST10OcEaI8fuceSmyeZ\nM2FKHpZUK/no/uN6cp3Ym6NGRjXDqW6jt+dBnhl1t6hlND59+tEBtVlLu0rt0Cdr6T5mYcDSsfPf\nvMmrpHYbda9BqQ0lzyoeA9rW1apmGmcE8e8jZJAA6LsZCmnUpDmRQLXE6aXX99m7w/Lvtpb52n20\n5FrIuzIlXcm2YEmgJUOgVv1YBlpvtWT1T83eIO3wyJ1VETuVLK0DSSwVR6ldXtmeR4y1mtOurp6K\nUR/4I3m0xH7SsPZk6Pvy/ZXaIV5kVrC9gI8g/n0Af9js/cK/axt+WlBKXxqs+r6VxpKsuMzNzd79\n8+cvDOnpPULXEmgJIiHfdVfqbw6rtffMmbMD6rNR+5OlcmvV0gVMttruYdlAdD1aVjSlc3T1362T\nZk2y1+V5bp5chlZHnj79aP9ZzlPXh1exNSEoMIgg/n3A+fMXtqTbXmCxWohbwNZT6pgqWlXD6WuG\nYSsNnwXgDWi93Jf/U0pYW7uBlOBumGqpk5fOCgHN9bZ04Hojj6d20zYT6Vvpb1ZJtaBEOpZaBxj0\nWNErO08a5s1pVngOhkW0ulzvWUlX2uXNdWNC9tJZE5wXS8iCrBityS2lNBDqIjCMzsSfUpoB8JGt\nvD6ac/4ldf87ASwCeBuAn8k5f7hrmUcNHKgL8H30GbUBqclAp5ffpeP29PJeIlHKb2tiATCw3OfV\njEj6KytXzHJL0NKeVw9rIHs7NUvqMyGHjY3bQ+VqdYsVSMyqm6DU1hrpAv5h4nrlyHWr6cm9ybaV\nGDVRl1YaEi+/NvHV1Falull14Pdn7bcIbKMT8aeU3gRgAcC7ALwO4OWU0qdyzq9Qsv8F4CyAH+xS\n1lGFeC6UBoqkqxGcvs4DUet0rQNXtBGud/D1dSwsXMDk5In+wRhWWvnteehoMrFO17IkQ8uNtTTB\nWWjZp6AnHyEHhheeofbuNHQ7rfcqE6kV88iDSNK1TVm6jt4O75Y8LKKuqfCsZ7k+AssOUVMRlbCT\nCW1c0VXifzuA6znnLwJASuk5AI8A6BN/zvkrAL6SUvr+jmUdSdSIy4r02PrR8zPAcMA1rU7S93PO\nuHnzNgAMqHesZT1HpbQGWGnisshA94219C9dl7xWVq5gff06Jib8A7m5r/SeAgm41mprYLSQi+cG\n6U2Glh5eYEVB9dRVrIIrfYM1tRRvIJP61vrKC8us22vdt76NEkKXvzN0Jf67AbxGv78E4B0d87yj\nUNK5AzYBtA5UTZBizJSDLCwC9VwoPaJmvX9JnTBKDB7ADjLHEqW3GmCIWmFt7Tpu3rw9YBy1CHR1\ndTuAmzeJ6T7aK2Lx9Oienp+f0aspT38P2DtYtRTvCRw84Vs7pT1Y3zTDcx3Wkr42eNfUajEJtKMr\n8e+qw+y5c+f6f8/OzmJ2dnY3sz9QtKgFNOHqDV2cj/WRi67bGwyWFFYbJPoMYK9ttTNgdX0tf3K+\n56l+GCklTE6eGFLZsJTNtg5R77Abrc5fJhNPD8311SRl/T3KDtKSnt9LZwkANVh9q/sEKIfiaG1H\nrXx9jb9vnmxq48cTPu40LC8vY3l5uXM+qXVJZT6c0kkA53LO7976/dMANnPOHzLSPgngz7Jj3E0p\n5S51OSpolUrm5p4AsH3ilE7PMeEtyUl2sY6ym1TXTwe92ok0pXXr1vN61WEZlq2yZfemnhz1dR3D\nRx+oYvWtV1e+L9B56kBsLXrrUhr9TkeJvcRB/Th/XQ63mTd2Wf1r1Xu3pOyW78WCNx7udGy9x+HI\nhxV0lfg/DWAypXQvgC8DeC+A9zlpR67cnQhPnaGxsPA0zpw5i4997FkzdHNJ+uZQA6MMAnYTBHoR\nRDc2blfP8S0N/Nqyn+9pwmGVgwVLKhbVFPeZjuEjpCxRIdklUt6Jt7fCaoscMq/z131QEmz0zmqG\nPDfKzmRuOweck3JK7p9SntW/Nc+wko2i5RqrAnPO1Wd4UuRDg5aWesdZTk7eFweyGOhE/Dnnb6SU\n5gA8j213zldSSo9v3X8mpfRtAF4G8C0ANlNK/xzAd+ec/6xj3Y8k5GPVh1YDw4NAAlh5JGQZPkWX\nzYGvSmBJUy+V5+Yec6Mcan/tWnutrf56QHv5eeRm1emFF17C8ePHBqJzclquDwDkPEjMlsrBW33I\n3zXPnFEm39Lznj2kNAGMogbyhJHSN6L7U9JbR0ZasFSPrLYpqYX4bx163FJrBrbR2Y8/5/wCej76\nfO0Z+nsDwD1dy7mT0KITB7aP8bNC8wJ+nJTaoetWmTKwgUEJ3Nv9ael+PeKw4tJYkHq3EJTVLqkP\nE5QuS7um6rrqturnLdIp1bcmBfMxlS3pdT1bvJFayd8jS/5GSt8t18k6MrJlUuT34wkKVnu095JO\nFxhE7Nw9ALRKsMDgYBM9pnhh6I0y1gActS686cuqCz+nB+b5888i5+FDylmK5zpaqxXPddOrh26L\n9JPVNq3K8spgiVWvmkqThO6PlZUrA+/ImoRefXVQJWftfyipLXZiu/Fw+vTwfhC53jIJ1ci5Bd4K\nsya9W/fDy8dHEP8+gpfMtdjo1kcrf8vgqAWn2gkp8AHro+SXs334O68k2Bdc7ml4UStb66HL0HWU\nA+DZrVO/i5zz1o5eX9XDOnJLz221T+utgd7JWVpVpCe/+fmz2Ni4NbQDXLe7RHISNqS2OhMbiw4F\n0RLOe7dUK+vrN4ZcaoE2O9Fe1OdORBD/PsLSJQP1HY5aktKkWDIUt0g9TLaW+qMFLOV5kp8lSep6\n6hOXWqB10J5Ez7/X1q5jZeUKpqcfGHDfXFrq7Wg+fvwYpqZOuKoevfoR8O7a06d7BuQXX3wJc3NP\nuKoPHVpD6+9L76SkOuP3oCcNT62jhQsm/ZqrZO3ba/kWJcaT1M07m7rmHFE6wyAQxL+v8Ja/lqHO\nksL4Oa1rlo1MksYjBC2x8rmyul6SftRJRddXD3wLmgxrUp62N0geLZvISmoIIR0JX6FRU3tonDp1\nsni+siZavqZdLK3gfDzRLC0Nbj7jvrZURN5EAgy6EfdCe9xwDxTyvg+9a7k0kXvf/srKFdM7Td6T\nF45CH0AfGEQQ/wHC2rhkLbNLhMF/M3lYRMoDjz1oarGELN10aRCX8qo9Y+VdM2rv1B7g/V0jdE1o\nTH5WgLVaftYuZrku9fc8nvQKiYUI3Ta5Xpq0dHmyCvT6kU9i07D6wrPheO/ck9ilX72DerwzrQM9\nBPHvI/Sg0/psoDdYdMRFPYgtIgRQDN6lJwIrnScde5LcKEcBWmVbkHqK7h0YDl9QczHVPvmj2FFa\noL2mLLuIloRLIa9rqyA2kJfeH78rz7BvrS49aVvSAcBdd6WhQ1ikL7zwFtb3qstviddfAn8nnP+o\n73TcEMS/j9Afs5bWvEO9ZRB7knZJpaMHocDaIJNzxuLihX68n5y3d4vqnZ/sXso7SmvSulzzSJmJ\ni/OS06pYRSUbp1i1If24tnYdn/vcF/Diiy+ZZHb+/AVsbNzqk2mNiLw6SvmWkZTrz6EvdF0kT90v\nXE5pVeYJAhbx6tVlaxs3N3P/HdRUlhraHlLz09f5lvT1up/5u44NXD6C+PcRNWmt9pwMQn2IuvV8\nTWKS++ztcfr0o0OrDdGVsmFSu5fygeQt9fAMi7rNFqmwukPHyeflf849z5zjx48N5b2ycgUbG7dw\n/PixIYNry8TF5bXW3yMfbauxyvfca+W+Nmhrqdqrn/W3XgnIfT52U6fT+VgCgW4zHyLjrWQF1nkL\nNQcCecclT6hxRhD/IUJteaongJJfemtemjy9uDxWfjIBeYZQa1Ky1FJyvUSwJeK19gOU2m9J6DWV\ng1c/LaW3qI9K/u5W+d5kpHfSau8gyzhstdlqr/5bq9dKq0xRzXk4fXrbbZiveX3tSfpW2dy+cYvb\nMwqC+PcZo7iZ1fzYRQXQkod1VKOnp9dLcw+SrwcrFr0Qg87bu15rW0vYiJraS8q3IKR5/vyzA9f4\nOSYsb5c116UURdKSuFm/rtUkHGyO7UMteyasa54ELxvRrHScH7t+1s4CSMn+1jyDbUkQ0eUE4ZcR\nxL/P8I4JZGjJqVWa1s+zX7smRx4oLcTYmo7TWOoJkfa0oVD+1lJrqR7Sjtph2602kJLHCZNeiy6+\nZFwthSOwJG7+ZqznORqpR55W+2oqFt3+0uSqQ0+U+gDoSeOW+7Fe9cikq49ybP1mAzaC+PcZokop\nhSXQkhOAoRDJnorHkggBDBnyWoxy2hPFksq1VMdSfo1Y9OpH8lpfv1ElIqlDLY03QXqrEau+ouOW\naxLmWSY23Y6SXr622rAmQ31Nfz8SgZOvecZ7b7Wl4U1uVrqcc39y4u+hJPHXJHa+bk06JcGldj0Q\nxH8gaDFs6gFqSX0ABtIx6WvSbT3YA9gmsvX1G9jYuD1Ur5WVK+6g1uVYqg1Jw/YF3qHZskOUyc9S\nZ3mSr1dP7gfrGW6rdmGUeDuctkV1pdU6TNwl0tKTVklXz9dTSgO7lPm6pcPXE6FlB5E0MulZsYY8\ncFv5N/cP58114T7zUCt/nBHEfwBokUB0oDMmc73rVlwwrRUAS8c1aOlNNsFog1kpX2/1wdctbw8O\nqyv1qLnx6b8tiZ3L5fxKZFxrlyYdbdyWcmtB8ryVktxrtXmw+sdy7eT+mZwcDvPNJM9t0OdBW7aU\n0kRVUp/putX6x+qn2rcdkr6PIP5DitJSXO+6FbSqFjwJTktvtXotLW2HCPCkQUv/btk5uEyt2/Yk\neo9U9L4HeU7nV1O7WGCyEmK2opGW1CMCrY4pGYU5b/2eWeL3JH9ge0Kanz87sGpjkueyrEm7xTbh\nla/VR95GwFYVTc14HPARxH8AGGWzEDD8gcvAl+vWwCnF1/E2z4yyKpCBK7puawAysXF9ah5NrNsu\n1dfS0/NzYhSUydFSR3hlW8dOsrFc1D3aHbYmqc/MPIzNzYzV1U8OkLXeG8HtkI1q3kTq2Qi8OuhV\nm1VvXVbpO9L39eTHqyPrO9bPeO9VnllcvDAQ9E4/H6gjiP8AIB+/PgbPO7zb07VqImRpVEeclL9l\nw9ao4Ra47izplmIKcTu9eC4tHiXertWS3ULUGpYHDVD3SOKdqrouAkuq9/KVCeLmzVv9TWWeTtuL\nv8hYzboAAA4RSURBVOP1w6hk15JHzS5h1cdT3cg1HZOIn9Wxgawypb+tkB0h+Y+GIP4DgHywQqDW\nMt2S4rTUpNUdcvDH1NSEeQLSVs64dev2kGG0JmnpunD5nl865+9t8rIGrCcx6vrW6jmqKocnWFaj\n1Yi15lrI5P3+9/+4Kx1bqg9Ww0j79Yax0uquFq/Iqn8Jnl6djfPeJKKPZ/TKLK1oAFsVWbMnBAYR\nxH8A0Lpa60P1TkLiPLTKgmOZW5LV0tIlTE6eGPBJ95beJfAgqxkgZYXC9dYkqQespXaw6stoJS+P\ndHklYIXGKMFS7+gJqiUYmXdkpm5/6bdeKUrdeG+Ep2tvIUyrfcDwmbe6b3X+ukzLJVWXWTL2swAS\nap86gvgPAFpqsnT+nq6Yl9OLixcwMbF9PGBNCpYlshfFs3Wg8CCrTRa67np1oK9pdQcTcklN46m9\nSnaU0gqrtJKx2qg3pY1qj+A4+npC1e+o9Fu3Q/LjHb3nzz+7dcKYv9GuBksK5wm8NllbIZs9aG8z\nrz61zXOBbQTx7yM0GQm0qkc2Tolkbun7WdfplWNNGC1Epp9ng6TUsxQtUoOX+nqPgfYc0ZNiq1+4\nJ0m3PuNNUN4KQaNGZDV7hKXa43Jbw01717g/ZdVXOnxnFJUft8O6P0o9dR1SSgPnH1vqnKWlwQ1z\nofapI4h/nzCKN01POoQpwXDamZkHsbJyZegD9wxsFpExoVi6/t4GpXUAGAqDXGuvJg89IK066Xx5\nkqzpsJksrPRe3bQKQddL0lshNPSkqMvwYiSVpHbdbp4Y9CTRAq12qknX3kqnNiHs5JvwwO3lScrb\ncV1SDwaGEcS/TxhFSraOyfPy9PS7Gp6EJt4/rGrh372j7ybM82ctWJ4dnr93TYfOLpnePgH2XpK8\nLHIo+Zxfu3Yd165dx/T0Q/3wFJbawtLBe+oHL3hczR7A6QSarK2dtrqdvDKQ8wlqMZCkXAmwpvdo\nWBPfKEEHd2JL4L/1t8neUt7GsoCNIP59gNZT1yTAVjAheIdScx0sSVP/rdUxox6+bknn2sAopGIR\n47CXC4ZWSV6f1ewbHBSMMTf32Jab63A7gOGQzbwCE0lfrz48w738rkmnNXK2Qi9wvfWkXnpGoA3c\nWl1mqavkvAZrwhlF3dWCUlgJIDZ0jYIg/j2GXjq3Gv28vDyDpfaq0PAGRIk8S8ZUgd7oVMqTD3nx\nJEvuK2+fQInEAZsASufGygTKq5WS6kxCTljqBt1mr7zV1atYWLhgpq15xIyqe5f2jRJCQp6pSfRT\nUxNYW7s+EAWWg+y1hMnQbdbXrfOM5VsTdaeuf6CMIP49hlbxeFJQbYCxRG994K27YS14E4qlstHS\nHKs6eEOaTmepK4QsWbLkmDPynBAWh/6VZ0pqklF021Z7vb7T6h3r3Wnyto4B3Ni4bZK/FY/Gq1NJ\nGBhlRQTYE2YtjLhMqNPTD/RXMXzQufe8/pa8dsrmPylLf5v8/bCKrnVX/LgiiH+PUfvwtDRsoWdg\nvV6U6K1nWlVKMnAWFp7F5ibcScpSFzCRcSiAmvTFg7VUJ64HL/UttYtAJPKSgdLbJV2DbO6S54SY\npqZOmHYEWfGNsgJhybZWJ21L8Dy6LPWLt1+A05cECpmcLZuUEHDpyMnaCknqx+rL2gqoZOcJbCOI\n/5BADjNnsOqBd0S27MjUeVlGRYEmVh2HX3D58tW+zt1Cqzpip/c53n2p/SJlliYU7p/S6sF6XvpR\njg+Ud8PB4aR8kT69CJ6tbbfAxs7SROhNxJrogcFNgLocPWlo75+SekzXG7BXqVwXVgdK+Vb7Wr25\nAtvoTPwppRkAH9nK66M5518y0vwcgO8H8FUA/zDn/GrXcu8EeJ4m2jNGS+1AXZ856jIf2B6I09MP\n9ePwaw+WwkmLTaiRayv5ltpfIhRrsiiRpq4TG2dFGtUrJM/9thVaTcSkzN+CEK+2k0h6Dn9t6c6l\nfnKwjJwpoCclb9LwvNRqtqHSu+N3q92ZPScJmYyD/NvRifhTSm8CsADgXQBeB/BySulTOedXKM3D\nAP5azvl7UkrvALAEoJt5/4jBIzNvqWvpLHly8KTFFj/rlo0tHIefSWxh4emixNWiNvGW4DypeSsT\ni4BbIYRSCjHRSlTWyobrpnXl0i+8O7dWVz4xTRtM5ZqQnLaTiDpOgyVjy4he6m8NLy1vuPLUSqVv\nVwdy47J4AvAigwba0FXifzuA6znnLwJASuk5AI8AeIXSvAfArwJAzvl3U0rfmlI6nnO+1bHsIwOP\ncFqW+mwwtZ6prRqsycaK3cJpWw2EIo3KQdwsfUlanbcXipcnNWCY/EsbizRKm6R0oLCaysySXuUa\nS9oMS3VUW6V43i9WvXV9eLLR1y3BgVUo1i5Y63uqYVC3PqwOLKnT5N1aZ0PzM7Ka0fXnsWHtIwkM\noyvx3w3gNfr9JQDvaEjzHQDGhvgtMmtRaXgSGsOSLEsqCwBDceS9QekNUk1mvLPSO4PX8vTxVjw6\nD/ZosshKQ+885jZ4O2I9WHYXeRcrK1f6hlu2P0h7+D1YkqyQ/czMg0PHN7ao6qwDWPRExRM9MKgu\n0pOghOwWY2rLtyl9zOktgaJmg2CvIF2uOB3Mzz82IGRwtFp9prSuY2AQXYm/dWrVmuGxm5JLah4P\nWvrxltyeOsjKT+6VltKlOmqyttqlN4DJ9ZLKpJQH71FoCcCl49mPUn+NWvv0O6qp7xhihNa2glZY\nZXkrA8trTNcv5/JeEK/sUdo8iopNMDFxwkxXyyskfx+pS+eklE4COJdzfvfW758GsJlz/hCl+RUA\nyznn57Z+vwrgnVrVk1LKTz75ZP/37OwsZmdnd1y3cYIM9pBsAoE7G8vLy1heXu7/fuqpp5BzHtnl\noivxvxnANQB/G8CXAfwegPcZxt0ncs4Pb00UH8k5Dxl3U0o5ZuhAIBBox5a6b2Ti76TqyTl/I6U0\nB+B5bLtzvpJSenzr/jM550+klGZSSl8A8H8AnO5SZiAQCAS6oZPEv5sIiT8QCARGw04l/rv2ojKB\nQCAQOLwI4g8EAoExQxB/IBAIjBmC+AOBQGDMEMQfCAQCY4Yg/kAgEBgzBPEHAoHAmCGIPxAIBMYM\nQfyBQCAwZgjiDwQCgTFDEH8gEAiMGYL4A4FAYMwQxB8IBAJjhiD+QCAQGDME8QcCgcCYIYg/EAgE\nxgxB/IFAIDBmCOIPBAKBMUMQfyAQCIwZgvgDgUBgzBDEHwgEAmOGIP5AIBAYMwTxBwKBwJghiD8Q\nCATGDEH8gUAgMGYI4g8EAoExQxB/IBAIjBmC+AOBQGDMsGPiTym9JaX0Gymlz6eUnk8pfbOTbiGl\ndCul9IWdVzMQCAQCu4UuEv+/APBSzvl7AFwF8LNOukUA7+5QzpHH8vLyQVdhTxHtO9qI9o0fuhD/\newD86tbfvwrgB61EOecVAP+7QzlHHnf6hxftO9qI9o0fuhD/8Zzzra2/bwE4vgv1CQQCgcAe482l\nmyml/wbg24xbP8M/cs45pZR3s2KBQCAQ2BuknHfG1ymlVwHM5pw3Ukp/GcDv5Jy/00l7L4Dfyjn/\n1UJ+MXEEAoHAiMg5p1GfKUr8FXwcwI8B+NDW/7/RIa8dVT4QCAQCo6OLjv9fA3ggpfR5AO8A8G8A\nIKX07Sml35ZEKaVfA/ASgKmU0msppdNdKhwIBAKBbtixqicQCAQCRxMHsnM3pfTDKaU/SCn9v5TS\n9xbSfXFrg9hnU0q/t5917IIR2jeTUvrMVhvP7mcdu2CEzXtH6v21vI+U0s9t3b+aUjJtWocRtbal\nlGZTSm9svavPppS8fTmHDi2bRI/qewPq7dvRu8s57/s/AN8JYArA7wD43kK6/wngrQdRx71uH4A3\nAbgO4F4Afw7A5wB810HXvbF9vwDgJ7b+/kkAP3/U31/L+wDwMIBPbP39DgBXD7reu9i2WQAfP+i6\n7rB90wDeBuALzv0j+d5GaN/I7+5AJP6c86s557XG5EfO6NvYvrcDuJ5z/mLO+esAngPwyN7XblfQ\ntHlvC0fl/bW8j367c86/C+BbU0pHYf9K67d2VN7VAHJ9k+hRfW8AmjfBjvTuDnuQtgzgf2wtX/7R\nQVdml3E3gNfo95e2rh0FtG7eO0rvr+V9WGm+Y4/rtRtoaVsG8OCWivITKaXv3rfa7T2O6ntrxcjv\nros7ZxGFzV/vzzn/VmM2fzPnfDOl9F0APpFSenVr9jtw7EL7DrVVfZc27x3a92eg9X1oyepQv8ct\ntNTxMwDuAfB19NyzPw5gYi8rtc84iu+tFSO/uz0j/pzz39mFPG5u/f9KSul59Jash4I4dqF9r6P3\nsgT3oCeJHAqU2rdlaPq2vL1577aTx6F9fwZa3odO8x1b1w47qm3LOf+p/J1SOg/gQymlt+ac/2h/\nqrinOKrvrQk7eXeHQdVj6qZSSn8+pfSWrb//EnoGmqMY2tnTvX0awGRK6d6U0jcBeC96M/VRgGze\nA5zNe0fw/bW8j48D+FEASCmdBPDHpPI6zKi2LaV0PKUk3+oPAPjaHUL6wNF9b03Y0bs7ICv1D6Gn\nc/sagA0An9y6/u0Afnvr7/vQ8z74HID/DuDxg7au72b7tn6/E8Bn0SPEf3bQ9R6hfW9Bj+w/D+B5\nAN98J7w/630AeJzrDuDnt+5fxRHxwmppG4B/CuD3t97XfwTwfQdd5xHa9msAvgzg/26Nu7k75b21\ntG8n7y42cAUCgcCY4TCoegKBQCCwjwjiDwQCgTFDEH8gEAiMGYL4A4FAYMwQxB8IBAJjhiD+QCAQ\nGDME8QcCgcCYIYg/EAgExgz/H9f92LBoE8b6AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1161a0710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x=map(lambda x: x[0], points_under_curve), y=map(lambda y: y[1], points_under_curve), s=0.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The basic idea is to form a rectangular window around the function you want to integrate, randomly place points in the window, then see what percent are below your curve. After this, you can multiply the known area of the rectangle with the percentage of the points that were under your curve to get your answer."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Key Moral: Simulation empowers us to explore statistics and probability instead of consulting an oracle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Statistical tests that make no (read: fewer) assumptions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's say we have two samples of data and we want to know how likely it is that the mean of the population of sample A is greater than the mean of the population of sample B. Let's not T-test. Let's choose an even more flexible test- bootstrapping."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "''"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x116536250>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample_a = np.random.gamma(3.5, 5, 40)\n",
    "sample_b = np.random.gamma(4.5, 5, 20)\n",
    "\n",
    "sample_a_samples = [np.random.choice(sample_a, 40).mean() for i in range(10000)]\n",
    "sample_b_samples = [np.random.choice(sample_b, 20).mean() for i in range(10000)]\n",
    "\n",
    "plt.hist(sample_a_samples, bins=50, alpha=.8)\n",
    "plt.hist(sample_b_samples, bins=50, alpha=.8)\n",
    "''"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Key take aways here: We've come up with a distribution of possible population means by resampling from observed data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "''"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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H/1TV0SS7gUMM/hrY6SG8JE3OWGWcpWIZZ2WzjLMc67KMo/ktWRlHkrS6GPaS1ADDXpIa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\nqWM2NMr6AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x116548150>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "random_pairings = zip(sample_a_samples, sample_b_samples)\n",
    "difference_of_pairings = map(lambda x: x[0] - x[1], random_pairings)\n",
    "\n",
    "plt.hist(difference_of_pairings, bins=20)\n",
    "''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "95% Credible region: -13.1169462263 to -3.10525291921\n"
     ]
    }
   ],
   "source": [
    "difference_of_pairings = sorted(difference_of_pairings)\n",
    "print('95% Credible region: {0} to {1}'.format(difference_of_pairings[25], difference_of_pairings[-25]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is 95% probability that the true value lies in the above range. So there is evidence that the population of sample_a has an average that is lower."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Parametric Tests\n",
    "\n",
    "Now let's say we want to determine if a coin is fair. We can solve this using bootstrapping or by using a parametric distribution. First we'll start with the bootstrapping approach."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "''"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11655e2d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample_coin_flips = [0,0,0,0,1,1,0,0,0,0]\n",
    "sample_coin_flip_averages = [np.random.choice(sample_coin_flips, len(sample_coin_flips)).mean() for i in range(10000)]\n",
    "plt.hist(sample_coin_flip_averages, bins=6)\n",
    "''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "''"
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1168c2290>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "possibilities = [np.random.beta(sum(sample_coin_flips), 10-sum(sample_coin_flips)) for i in range(15000)]\n",
    "plt.hist(possibilities, bins=100)\n",
    "''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "''"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x117c3ff50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "coin_flip_values = [0,1]\n",
    "sample_coin_flips = [8,2]\n",
    "sample_coin_flip_averages = [(np.random.dirichlet(sample_coin_flips) * coin_flip_values).sum() for i in range(15000)]\n",
    "plt.hist(sample_coin_flip_averages, bins=100)\n",
    "''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0312727360328348"
      ]
     },
     "execution_count": 152,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(np.random.dirichlet(sample_coin_flips) * coin_flip_values).sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Split Test Conversion\n",
    "\n",
    "What can we say about the below split test? Also a great opportunity to explore the affects of sample size on test results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "''"
      ]
     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x117ee7350>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "variant_a_orders = 23\n",
    "variant_a_visits = 144\n",
    "\n",
    "variant_b_orders = 21\n",
    "variant_b_visits = 115\n",
    "\n",
    "variant_a_samples = [np.random.beta(variant_a_orders, variant_a_visits) for i in range(10000)]\n",
    "variant_b_samples = [np.random.beta(variant_b_orders, variant_b_visits) for i in range(10000)]\n",
    "\n",
    "plt.hist(variant_a_samples, bins=100, alpha=.8)\n",
    "plt.hist(variant_b_samples, bins=100, alpha=.8)\n",
    "''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "95% chance the true change is between -0.135451127774% points and 0.0934575475419% points.\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x117b836d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "variant_a_improvement = sorted(map(lambda x: x[0] - x[1], zip(variant_a_samples, variant_b_samples)))\n",
    "\n",
    "plt.hist(variant_a_improvement, bins = 100)\n",
    "print('95% chance the true change is between {0}% points and {1}% points.'.format(variant_a_improvement[25], variant_a_improvement[-25]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exploring unintuitive probability puzzles\n",
    "\n",
    "Monty Hall is a classic example. You're probably familiar but I'll repeat. \n",
    "\n",
    "Scenario:\n",
    "\n",
    "You're asked to choose one of three doors in order to win a car that's behind one of them. After you pick a door, the game show host opens one door that is not a winner (it has a goat behind it) and then asks if you want to switch your door and stick with the one you chose.\n",
    "\n",
    "What's the optimal strategy to win the car? Stick or switch? Spoiler Alert! You want to switch. Does that seem crazy?\n",
    "\n",
    "Let's simulate both strategies and see which is better."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Games won with switching strategy: 6713\n",
      "Games won with sticking strategy: 3388\n"
     ]
    }
   ],
   "source": [
    "def simulate_game(switch = True, iteration_count = 1000):\n",
    "  wins = 0\n",
    "  for i in range(iteration_count):\n",
    "    prizes = ['goat','car','goat']\n",
    "    random.shuffle(prizes)\n",
    "    doors = {\n",
    "        1: prizes[0],\n",
    "        2: prizes[1],\n",
    "        3: prizes[2]\n",
    "    }\n",
    "    chosen_door = np.random.choice([1,2,3])\n",
    "    \n",
    "    # Host chooses which door to open\n",
    "    for key, value in doors.iteritems():\n",
    "        if key != chosen_door and doors[key] != 'car':\n",
    "            del doors[key]\n",
    "            break\n",
    "    \n",
    "    # Does the contestant switch or stick??\n",
    "    if switch:\n",
    "        for key, value in doors.iteritems():\n",
    "            if key != chosen_door:\n",
    "                chosen_door = key\n",
    "                break\n",
    "    \n",
    "    if doors[chosen_door] == 'car':\n",
    "        wins += 1\n",
    "  return wins\n",
    "\n",
    "print('Games won with switching strategy: {0}'.format(simulate_game(switch = True, iteration_count=10000)))\n",
    "print('Games won with sticking strategy: {0}'.format(simulate_game(switch = False, iteration_count=10000)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Continuing our statistical exploration\n",
    "\n",
    "How do we calculate sample size? I bet the answer is math isn't it? Nope. Simulation. Here's how."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "''"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Wcsy/T04GS1oJDP++ORks6dLnsI8ktZDhL0kt5LDPEM02H+BcgKTlxvAfqtl+eE6SlhfD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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x117b7ae50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p_values = []\n",
    "\n",
    "for i in range(10000):\n",
    "    control = np.random.normal(1.2,1,80)\n",
    "    variant = np.random.normal(1.0,1,80)\n",
    "    t_stat, p_value, df = statsmodels.stats.weightstats.ttest_ind(control, variant, alternative='two-sided')\n",
    "    p_values.append(p_value)\n",
    "\n",
    "plt.hist(p_values, bins=50)\n",
    "''"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at power of our test as a function of sample size again using simulation.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 249,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x118027490>]"
      ]
     },
     "execution_count": 249,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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STgV+BnwVmJTz0nrAPRFxevHDWxxLiyhRvP469OgB77zjaU3NrPiKfme2pLWB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\nKMzMrFH/D4dV92qx46ZWAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x117cbc110>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "percentage_of_runs_correct = []\n",
    "sample_sizes = list(xrange(30, 1500, 20))\n",
    "for sample_size in xrange(30, 1500, 20):\n",
    "    correct_test = 0\n",
    "    runs = 1000\n",
    "    for i in range(runs):\n",
    "        control = np.random.normal(1.15,1,sample_size)\n",
    "        variant = np.random.normal(1.0,1,sample_size)\n",
    "        t_stat, p_value, df = statsmodels.stats.weightstats.ttest_ind(control, variant, alternative='two-sided')\n",
    "        if p_value <= 0.05:\n",
    "            correct_test += 1\n",
    "    percentage_of_runs_correct.append(correct_test/(1.0*runs))\n",
    "    \n",
    "fig = plt.figure()\n",
    "fig.suptitle('Power Analysis by Sample Size')\n",
    "plt.xlabel('sample size')\n",
    "plt.ylabel('test power')\n",
    "plt.plot(sample_sizes, percentage_of_runs_correct)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now you can play with different factors in your experiment and tangibly see how they affect your experiment design."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How about asking a question like: How does the conversion rate of our population affect our ability to detect improvements?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 279,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "''"
      ]
     },
     "execution_count": 279,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x119b28e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p_values = []\n",
    "success_rates = []\n",
    "population_rates = list(xrange(0,100,5))\n",
    "\n",
    "for population_rate in population_rates:\n",
    "    sims_to_run = 500\n",
    "    number_of_trials = 1000\n",
    "    total_success = 0\n",
    "    successes = np.random.binomial(number_of_trials, 1.04 * population_rate/100., sims_to_run)\n",
    "    for success in successes:\n",
    "        p_value = scipy.stats.binom_test(success, number_of_trials, population_rate/100.)\n",
    "        p_values.append(p_value)\n",
    "        if p_value <= 0.05:\n",
    "            total_success += 1\n",
    "    success_rates.append(total_success/(1.0*sims_to_run))\n",
    "plt.plot(population_rates, success_rates)\n",
    "''"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What if instead of using a percentage lift we used percentage points?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 282,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "''"
      ]
     },
     "execution_count": 282,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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CT38KRxxR1suKSEq98UYYoLF2LXTqFHc0gZYqaIBmtIpIc0ydCqeempzkXqyKTvCa0Soi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\nFAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x119b337d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p_values = []\n",
    "success_rates = []\n",
    "population_rates = list(xrange(0,100,5))\n",
    "\n",
    "for population_rate in population_rates:\n",
    "    sims_to_run = 500\n",
    "    number_of_trials = 1000\n",
    "    total_success = 0\n",
    "    successes = np.random.binomial(number_of_trials, 0.02 + population_rate/100., sims_to_run)\n",
    "    for success in successes:\n",
    "        p_value = scipy.stats.binom_test(success, number_of_trials, population_rate/100.)\n",
    "        p_values.append(p_value)\n",
    "        if p_value <= 0.05:\n",
    "            total_success += 1\n",
    "    success_rates.append(total_success/(1.0*sims_to_run))\n",
    "plt.plot(population_rates, success_rates)\n",
    "''"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Why? The variance is higher in the middle. The left and right edges are more consistent so a deviation from that consistency is easier to detect."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Extra Credit: Touching on Bayesian Statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use some of these ideas along with probability to tackle problems with non-obvious solutions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You're given 5 dice (d6, d8, d12, d20, d22). One of these die are rolled (but you don't know which). You're told it rolled a 7, what's the probability the die is 8-sided?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we call out our hypotheses and make sure they add up to 100%."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 319,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{6: 0.2, 8: 0.2, 12: 0.2, 20: 0.2, 22: 0.2}"
      ]
     },
     "execution_count": 319,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hypotheses = {\n",
    "    6: 1,\n",
    "    8: 1,\n",
    "    12:1,\n",
    "    20:1,\n",
    "    22:1\n",
    "}\n",
    "\n",
    "# Normalize\n",
    "total_possibilities = sum(hypotheses.values())\n",
    "for key, value in hypotheses.iteritems():\n",
    "    hypotheses[key] = hypotheses[key]/(1.0*total_possibilities)\n",
    "\n",
    "hypotheses"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we need to be able to come up with our likelihood function and update our hypotheses using Bayes magic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 320,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def die_rolled(rolled_value, hypotheses):\n",
    "    for die_sides, value in hypotheses.iteritems():\n",
    "        if die_sides < rolled_value:\n",
    "            likelihood_of_event_given_die = 0\n",
    "        else:\n",
    "            likelihood_of_event_given_die = 1/(1.0*die_sides)\n",
    "        hypotheses[die_sides] *= likelihood_of_event_given_die\n",
    "        \n",
    "    # normalize\n",
    "    possibility_total = sum(hypotheses.values())\n",
    "    for key, value in hypotheses.iteritems():\n",
    "        hypotheses[key] = value/(1.0*possibility_total)\n",
    "    return hypotheses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 321,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{6: 0.35426731078904994,\n",
       " 8: 0.26570048309178745,\n",
       " 12: 0.17713365539452497,\n",
       " 20: 0.106280193236715,\n",
       " 22: 0.096618357487922718}"
      ]
     },
     "execution_count": 321,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "die_rolled(5, hypotheses)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 322,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{6: 0.50584755594109587,\n",
       " 8: 0.28453925021686649,\n",
       " 12: 0.12646188898527397,\n",
       " 20: 0.045526280034698649,\n",
       " 22: 0.037625024822064992}"
      ]
     },
     "execution_count": 322,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "die_rolled(6, hypotheses)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 323,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{6: 0.0,\n",
       " 8: 0.0,\n",
       " 12: 0.72553987580240009,\n",
       " 20: 0.15671661317331853,\n",
       " 22: 0.11774351102428135}"
      ]
     },
     "execution_count": 323,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "die_rolled(9, hypotheses)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A simple introduction to PyMC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 331,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " [-----------------100%-----------------] 18000 of 18000 complete in 0.7 sec\n",
      "conversion_rate:\n",
      " \n",
      "\tMean             SD               MC Error        95% HPD interval\n",
      "\t------------------------------------------------------------------\n",
      "\t0.374            0.161            0.003            [ 0.083  0.684]\n",
      "\t\n",
      "\t\n",
      "\tPosterior quantiles:\n",
      "\t\n",
      "\t2.5             25              50              75             97.5\n",
      "\t |---------------|===============|===============|---------------|\n",
      "\t0.101            0.256           0.362          0.485         0.714\n",
      "\t\n"
     ]
    }
   ],
   "source": [
    "import pymc\n",
    "\n",
    "conversion_rate = pymc.Uniform('conversion_rate', lower=0, upper=1.0)\n",
    "observations = pymc.Bernoulli(\"observations\", conversion_rate, value=[0,0,0,1,0,1], observed=True)\n",
    "mcmc = pymc.MCMC([conversion_rate, observations])\n",
    "mcmc.sample(18000, 1000)\n",
    "mcmc.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Extra Extra Credit: Let's Simulate a Product Funnel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We want to understand how much our conversion rates can fluctuate day to day and not be a cause for alarm. Said another way, we want to establish control limits of a sort for our conversion metric. Let's set up a simple model for this:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Visits\n",
    "\n",
    "* New\n",
    "    * SEO\n",
    "        * Have a conversion rate around 2%\n",
    "        * Receive 1,000,000 visits a day\n",
    "    * SEM\n",
    "        * Have a conversion rate around 7%\n",
    "        * Receive 100,000 visits a day\n",
    "    * Other\n",
    "        * Have a conversion rate around 3%\n",
    "        * Receive 650,000 visits a day\n",
    "* Return\n",
    "    * Have a conversion rate around 20%\n",
    "    * Receive 50,000 visits a day\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simple modeling approach"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 343,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "''"
      ]
     },
     "execution_count": 343,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x119b2e3d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Our model inputs\n",
    "seo_conversion_rate = 0.02\n",
    "seo_visits_per_day = 1000000\n",
    "\n",
    "sem_conversion_rate = 0.07\n",
    "sem_visits_per_day = 100000\n",
    "\n",
    "other_conversion_rate = 0.03\n",
    "other_visits_per_day = 650000\n",
    "\n",
    "return_conversion_rate = 0.2\n",
    "return_visits_per_day = 50000\n",
    "\n",
    "orders_per_day = []\n",
    "for random_day in range(5000):\n",
    "    days_orders = 0\n",
    "    \n",
    "    sampled_seo_visits_per_day = np.random.normal(seo_visits_per_day, seo_visits_per_day * 0.1)\n",
    "    seo_conversions = np.random.binomial(sampled_seo_visits_per_day, seo_conversion_rate)\n",
    "    \n",
    "    sampled_sem_visits_per_day = np.random.normal(sem_visits_per_day, sem_visits_per_day * 0.1)\n",
    "    sem_conversions = np.random.binomial(sampled_sem_visits_per_day, sem_conversion_rate)\n",
    "    \n",
    "    sampled_other_visits_per_day = np.random.normal(other_visits_per_day, other_visits_per_day * 0.1)\n",
    "    other_conversions = np.random.binomial(sampled_other_visits_per_day, other_conversion_rate)\n",
    "    \n",
    "    sampled_return_visits_per_day = np.random.normal(return_visits_per_day, return_visits_per_day * 0.1)\n",
    "    return_conversions = np.random.binomial(sampled_return_visits_per_day, return_conversion_rate)\n",
    "    \n",
    "    orders_per_day.append(seo_conversions + sem_conversions + other_conversions + return_conversions)\n",
    "    \n",
    "plt.hist(orders_per_day, bins=50)\n",
    "''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 342,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "56500.0"
      ]
     },
     "execution_count": 342,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "seo_conversion_rate * seo_visits_per_day + \\\n",
    "sem_conversion_rate * sem_visits_per_day + \\\n",
    "other_conversion_rate * other_visits_per_day + \\\n",
    "return_conversion_rate * return_visits_per_day"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
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