solutions/ch08q02.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generate X and Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Y = np.random.normal(0,1,50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X = np.exp(Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Check distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x10cabd4a8>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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CPLp11uzxe57PtQ3InISiEcuZIsrpi/0AgNlTk2ROQuHukwvDLvU4OWMYhRzL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\nRERECvP/AxmvXmzToechAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cacb710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.distplot(Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x10c8d83c8>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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wuP2oaeoVXQ4R0RVjCBMURUF1Yy/UKgm56fGiyxmRwtwkyGoJR+u6EAiGRJdDRHRFGMKE\nlk4XnAN+ZKfFQadRiy5nRAw6GTOzrBjwBlHVyLNhIpoYGMKEk2cv75kIlyVdTkF2IrSyCpWnuuDz\nB0WXQ0Q0LIZwjHMO+NHU4USCWYukOJ3ocq6JTqPGrOxE+PwhHK/vEV0OEdGwGMIxbt/xdoQUIDc9\nfsINyLqY/Cwr9Fo1jtd3w8HFHYgowjGEY9zuylZIALIjeJ7oq6GRVZiTk4RAUMGHextFl0NEdFkM\n4RjW3OnC6VYH0pJNMOqj55LxvMx4GPUyth1sQo/DK7ocIqJLYgjHsN1HWwEAOenRcRZ8jlqlwtxp\nSfAHQnh/d73ocoiILokhHKOCIQV7jrXBoJORmWIWXc6oy02LR4rVgLLDLejsGxBdDhHRRTGEY9Th\nGjt6nT5cPzMFanX0PQ1UKgl335SNYEjBX3bViy6HiOiiou/Vl67ItgPhQUs3zpksuJKxc31BKiYn\nGbHraBvae9yiyyEi+gKGcAxyewLYe7QVqVYDctOi6/Pg86lUEu6+ORshRcF7O+tFl0NE9AUM4Rh0\noLoDvkAIN86ZHBXXBl/OgvwUZNhM2Hu8Da1dLtHlEBFdgCEcg/ZUtgEAbpiVKriSsaeSJNx9cw4U\nBfjzztOiyyEiugBDOMb0OLyoOdOLWTlJSI43iC5nXBTnJSMr1YLPT3SgqcMpuhwiokEM4Riz/0Q7\nFACL58XOgveSJOHrJdlQALzLs2EiiiAM4Riz73g7VJKEmwrTRJcyrgpzk5CTFoeDNXY0tDlEl0NE\nBIAhHFPau92ob3OgINuKePPEXjHpakmShHtKcgAA7+w4JbgaIqKwYScMDoVCeOaZZ1BdXQ2tVotn\nn30WWVlZg9tfeeUVfPDBBwCAJUuW4Hvf+97YVUvXZN+JdgDA9TOjf0DWxRRMtSIvIx5H6rpQ19yH\n3PR40SURUYwb9kx469at8Pl82Lx5M9asWYMNGzYMbjtz5gzee+89vPHGG9iyZQt27tyJqqqqMS2Y\nRkZRFOw73g6NrEJxnk10OUJIkoR7FofPht/l2TARRYBhQ7i8vBwlJSUAgKKiIlRWVg5umzRpEl5+\n+WWo1WpIkoRAIACdLra6OSeKMx1OtHa5UZibBIMuelZMulozMq0omGrFsfoe1JzpFV0OEcW4YV+N\nnU4nzOahCf7VajUCgQBkWYZGo0FiYiIURcG//uu/oqCgANnZ2Zfdn9VqhCyrr73yEbLZLMKOLdIH\n+8LTVN62aOpgG5z712LWC6trLF3qd/3Nr83Gk7/Zgff3NmD9vIzL3jcWsS3C2A5D2BZDRrsthg1h\ns9kMl2topqFQKARZHnqY1+vF008/DZPJhJ/97GfDHrBH4By+NpsFdnvsjYwNKQq2l5+BQafGVJsR\ndrvjgrZwOD2CKxwbl/pdJxk1KMxNwpG6LpQdaMSS67Ji8nlxMbH6N/L32A5D2BZDRtoWlwvuYbuj\ni4uLUVZWBgCoqKhAXl7e4DZFUfBP//RPmDFjBtatWwe1WtwZLl1aXXMfuvq9KJ5ug0ZgL0Qk+XpJ\nuMfmnbJTUBRFcDVEFKuGPRNevnw5du3ahdLSUiiKgvXr12Pjxo3IzMxEKBTC/v374fP5sGPHDgDA\nj3/8Y8ybN2/MC6crt+/42VHRBbE5Kvpipk6Kw/w8G8pr7Pj8eDuyU0yiSyKiGDRsCKtUKqxbt+6C\n23Jzcwe/P3r06OhXRaMmFFJwoNoOs0GDmVOtosuJKF9fnIODNXa8+uEJ/PTh+VBF+WIWRBR5OFlH\nlKtu7EG/y4cF+SlQq/jrPl96sgk3zJ6E+tZ+fH6iQ3Q5RBSD+Koc5fZXhcNlYX6K4Eoi0903Z0NW\nS3hnxykEgiHR5RBRjGEIR7FAMITyajviTVrkTUkQXU5EsiUYsPz6LHT0DGD32SUeiYjGC0M4ilU1\n9MA54MeCGSlQqfh556WsvDUPGlmFP+88DX8gKLocIoohDOEotv/s55zXzWRX9OUkxRuwbH4Gehxe\nfHqoRXQ5RBRDGMJRKhAM4WCNHVaLDtMyuFDBcO5alAWDTo33d9djwBsQXQ4RxQiGcJSqPN0NtzeA\n6/JTeOnNFTAbNLjj+iw4B/z48OwUn0REY40hHKU+P7tsIbuir9xtC6Yg3qzFx583otfpFV0OEcUA\nhnAU8geCOHSyE8nxeuRMjhNdzoSh06px983Z8PlD+PPO06LLIaIYwBCOQkfquuHxBXFdfgokdkVf\nlZLCyZiUaMSOw61o7XIN/wAiomvAEI5Cn1eFu6IXzuRc0VdLrVLhviW5CCkK3vrslOhyiCjKMYSj\njNcXREVtJ1KsBmSmmod/AH1BcV4ypqXH42CNHbVNfaLLIaIoxhCOMofrOuHzh7BwZiq7okdIkiTc\nf0t4kZIt22u51CERjRmGcJQ5N0HHQo6KvibTMxIwb3oyapv6cKDaLrocIopSDOEo4vYEcKSuC+nJ\nJmTY2BV9rR64ZRrUKglvflrL6SyJaEwwhKPIoZN2BIIhngWPktREI25dkIHOPg8+/vyM6HKIKAox\nhKPIUFc0R0WPlq/eOBVmgwbv72lAHyfwIKJRxhCOEs4BP47XdyNrkgWpiUbR5UQNo16DexbnwOsL\n4u0yXrJERKOLIRwlyqs7EAwp7IoeA4vnTkZ6sgk7j7Sioc0huhwiiiIM4SgxuGxhPkN4tKlVKpQu\nmw4FwBvbTvKSJSIaNQzhKNDr9KKqoQfT0uORHG8QXU5UmpWdiLm5Sag+04t9ZxfHICK6VgzhKHCg\nqgMKeG3wWHtweR40sgqbP6nlmsNENCpk0QXQtdtf1QFJYlf039te0XxF97OY9XA4PVd034KpVhyu\n7cJ/vH30gmUilxalj6hGIoptPBOe4Oy9A6ht6kN+phXxZp3ocqLe7OxEWIwaVDX2oMdxZcFNRHQp\nDOEJbu/x8OeTiwp4bfB4UKtVWDgzFYoC7D3WwUFaRHRNGMITmKIo2HusDbJahfkz2BU9XtJtJmSm\nmmHvHUBdc7/ocohoAmMIT2CN7U60drlRNC0JRj0/3h9P1+WnQFZLOFhjh8fHeaWJaGQYwhPYnmNt\nAIAbZk0SXEnsMRk0KJyWDI8viANVHaLLIaIJiiE8QYVCCvadaIdJL2NObpLocmJSQZYVSXE6nGrp\nx5G6TtHlENEExBCeoE409qDP6cOC/BTIav4aRVCpJNw4ZxIkCfjDR9W8dpiIrhpfvSeovZXsio4E\nVosec3KS0OPw4s3tdaLLIaIJhiE8AXn9QZTX2JEUp8O0jHjR5cS8OblJSE82YfuhZlQ19Iguh4gm\nEIbwBHS4thMeXxCLZk2CSpJElxPz1CoJ37xrJiQJeOXDKng5WpqIrhBDeALac7YrmhN0RI6ctDjc\nfl0mOnoHsPmTk6LLIaIJgiE8wfS7fKg83Y0pKWak28yiy6Hz3LM4Gxk2M7ZXtOBQjV10OUQ0ATCE\nJ5g9x9oQDCm4ec5k0aXQ39HIanznawXQyCps/LAKvU6v6JKIKMIxhCcQRVGw40gr1CoJi2axKzoS\npdvMeOCWaXAO+PH7D04gxLmliegyGMITyKnWfrR0ujAvzwaLUSu6HLqELxWnozA3CcdOd2PrgSbR\n5RBRBGMITyA7DrcCAEoK2RUdySQpPFo6zqjBn7bXoqHNIbokIopQDOEJwusLYv+JdlgtOsyamii6\nHBpGvEmLb325AIGggt++cxTOAb/okogoAjGEJ4gD1R3w+IK4ac5kqFS8NngiKMxNwtdumorOPg9+\n994xhEL8fJiILsQQniB2HAl3Rd/MrugJ5Ws3Z6MwNwmVp7vx7s7TosshoggzbAiHQiGsXbsWK1eu\nxOrVq9HQ0PCF+3R3d+P222+H18tLMsZCe7cbNWd6kZ+ZgJQEg+hy6CqoJAnf/moBbAl6vL+7ntcP\nE9EFhg3hrVu3wufzYfPmzVizZg02bNhwwfYdO3bgW9/6Fux2vriMlZ1Hzw7ImpsmuBIaCZNeg8fv\nmQOtrMLLHxxHa5dLdElEFCGGDeHy8nKUlJQAAIqKilBZWXnhDlQqbNy4EQkJCWNTYYwLhkLYebQV\nBp2M+Xk20eXQCGWmWvDInfkY8Abx71sOo8/lE10SEUUAebg7OJ1OmM1D0yOq1WoEAgHIcvihN910\n01Ud0Go1QpbVV1nm6LHZLMKOPRK7j7Sgz+nDnTdORXra6L7ROdcWFrN+VPc7EV1rG1zJ8+prSy1w\neoPY9HE1fvtuJZ577CbodcP+CY67ifY3MlbYDkPYFkNGuy2GfQUwm81wuYa6z0Kh0GAAj0RPj3vE\nj71WNpsFdvvEumbznU/DiwHcODNlVGs/vy0cTs+o7Xcispj119wGV/q7uXVeGs609mPn0Vb8/Pd7\n8f375kCtipzxkRPxb2QssB2GsC2GjLQtLhfcw/71FxcXo6ysDABQUVGBvLy8qy6ARqapw4mqxl7M\nzLJysYYoIUkSHr5jBmZnJ+JIXRde+7gGCqe2JIpZw4bw8uXLodVqUVpaiueeew7//M//jI0bN2Lb\ntm3jUV9M23YwPOXhsvkZgiuh0SSrVXjs67ORmWLGZxUteGfHKQYxUYwatl9ZpVJh3bp1F9yWm5v7\nhft98skno1cVweXxY8+xNiTF6VE0LVl0OTTKDDoZP3xgLja8dhDv7w5f9ndPSQ4kiROxEMWSyPkw\nii6w43ArfP4QvlSczhmyolSCWYenVs1DSoIB7+9uwNtlPCMmijUM4QgUCin45GATNLKK1wZHucQ4\nPX7yv4qRYjXggz0NeOszBjFRLGEIR6AjdV3o7PNgUUEqzAaN6HJojFktOvxkVTFSrQb8dW8DNm09\nyXmmiWIEQzgCbSs/A4ADsmKJ1aLDU6uKkZZswtbyJvzH20fh8QVEl0VEY4whHGGaO104Vt+DvIx4\nZKbyAvlYYrXo8PRDxZiZZUVFbSc2/PEgehycj50omjGEI8xf99QDAG5bmCm0DhLDqNfgRw/MxeK5\nk9HY7sSz/3MADW2cKIEoWjGEI0hHjxt7j7cj3WZC0XRelhSrZLUKj9yRj/tvyUWvw4tfvHoAH+9v\nRIgDtoiiDkM4gvx1bwMUBfjqjVOh4vWiMU2SJNx5fRaeuH8ujDoZb3xSi19vOYw+J7uniaIJQzhC\ndPV5sOtoG1ITjVgwI0V0ORQhCnOT8H/+4XrMzklE5elurP3v/ThYY+dlTERRIvKWcIlRH+1rRDCk\n4Cs3ZHFyjgloe0XzmO5/3vRkGLQyyqvt+I+3j2J2diIevHU6JieZxvS4wNj/3863tCh93I5FFAl4\nJhwB+pxefHa4BcnxelxfkCq6HIpAkiRh5lQrvnJTFiYnGcNnxb/fjze2nYTbw0uZiCYqnglHgL/t\nP4NAMIS7FmVBVvN9EV1aglmHWxdkIMGkw6ZtJ/Hx52ew62grls5Lx60LpiDepBVdIhFdBYawYA63\nD58eaobVosNNcyaLLocmAEmSMC/Phtk5ifj48zP42/4z+GBPA/62/wxunD0Jt103BWnJY99NTUTX\njiEs2F921cPrD+LeJTnQyDwLpiunkdX48g1TceuCKdh9tBUf7W9E2eEWlB1uQVaqBdcXpGLhzBQk\nxulFl0pEl8AQFqi1y4VPDzUjJcHAASk0YjqNGrcUZ2BJUToO1tix82grjp3uRsOnDmz5tBa56XGY\nmWXFjEwrpqXFQ6dViy6ZiM5iCAu05ZNaBEMKHvjSNJ4F0zVTqSQsyE/BgvwUONw+lNfYsf94O6rP\n9KKuuR/v726AWiVh6iQLMlLMSE82Id1mRlqyCXFGDdcyJhKAISzIsfpuHK7rQn5mAuZxdiwaZRaj\nFkuL0rG0KB1uTwC1zb2oauxFdWMPTrc6UNfSf8H9NbIKVosOk5JMsOhlWExaWIwaWAxaNHU4odOq\nodOEv7QaFQObaJQwhAUIhkJ4Y9tJSABKl03nCxqNKaNeRmFuMgpzw2/2/IEQ2rrdaLY70dzpQmuX\nG139HvT0e3CktnPY/UkAdFo19Fo1DDoZeq0aRr0GZsO5LxlmowZqFXt3iIbDEBZgx+FWNNtduLlw\nMldKonGnkVWYkmLGlBTzF7bFJxhx8nQnHG7/2S8fDtd1wusPwusLwesPwuMLwusLwOUJoNfpu+gx\nJAmIM2o5AWHmAAAPmElEQVSRYNHBatEhMU6HlAQDtBp+Hk10PobwOHN7AnhnxynoNGrcuzhHdDlE\nF9Bq1EixGpFiHboteJkpMoPBEAZ8Qbg9fjgHAnAO+OEc8KPP6UWv04c+l++CVaCsFh1SrAakJhqR\nlmRkKFPMYwiPs7fK6uBw+3HP4hwkmHWiyyG6Jmq1CmaDCmaD5oLgBgBFUcJnyw4v7H0edPS40dnr\nQY/Di+rGXkgSkJpoRIbNhCkpZliMnGiEYg9DeBwdO92NTw82Iz3ZhDu4XjBFOUmSBj8nzjjb9R0M\nKeju86Cly4WmDifautxo63LjQJUdyfF6VDX2YOokC/TasX9pGu/LAjkHN10MQ3icuD1+/PdfT0Ct\nkvDoVwp4SRLFJLVKgs1qgM1qwNxpyXB7AmiyO9HQ5kBblxudfR58fqID6ckm5E1JQJrNxGU9Kaox\nhMfJpq0n0ePw4us3ZyNrEgdjEQHhkdt5UxKQNyUBbk8A9a39qGvpR5PdhSa7C6az26dlxMOg48sV\nRR8+q8fBoRo7dlW2IWuSBXfdkCW6HKKIZNTLKMhOREF2Irr6PKg504vTrf04dLITh2s7kT05DjOn\nWjkNJ0UVhvAY63f78IePqiCrVXj0KwVcJYnoCiTF63FD/CTMn2HDqZZ+VDX2oq4lfJY8OcmIgqlW\npCWbeI09TXgM4TEUCIbwu/eOod/txwO3TEM6V7YhuipajRr5WVbMyExAs92F4/U9aO1yo7XLjQSz\nFrOyEzF1chzUKoYxTUwM4TGiKAo2bT2J4/U9KJqWjNuumyK6JKIJS5IkZKSYkZFiRne/B8dOd6O+\nzYFdR9tw6GQnCqZaMT0jgQMeacJhCI+RTw4249NDzciwmfGPXyuAiu/UiUZFYpweJXPTMC/PjxP1\nPTjZ1IsDVXYcqe3CjMwE5GdZOYiLJgw+U8dA5akuvL61BnFGDZ5YUTgu1zwSxRqzQYPrZqZgTm4S\nahp7cKKhF0dPdeNYfQ9y0+JQMNWKeE6IQxGO6TDKmjtdePHPlVCrVPj+fYVIiudITqKxpNeqUTgt\nGQXZiahr7sfx+m6cbOrDyaY+pCWbMDPLirRkIwdxUURiCI+iMx1O/OqNQxjwBvGPXy1Abnq86JKI\nYoasVmFGZgKmT4nHmXYnTjT0oKXThZZOF+JNWszITEBOWhznq6aIwhAeJXXNffj3LYfh9gbw0G15\nWDRrkuiSiGKSSpKQNcmCrEkWdPV5cKKhB/Wt/dh/ogPl1XZkTbJgekY8FEXh2TEJxxAeBSfqu/Gb\nt47CHwjh0a/MxI2zJ4suiaLcWM1DbDHr4XB6xmTfIiTF63Fz4WTMn2FDXXO4i/pUSz9OtfTjUE0n\nrpuZguvyUzAlxcxAJiEYwtfoQFUHfveX4wAUPPb12Zg/wya6JCL6OwadjNk5SZiVnYj27gGcbOpF\nS6cbH+xpwAd7GpCaaERxXjJmTU3E9IzI/hhJURT4gyEEAiEEggoUBVAQ/lcCoFZL6HN6oZHV0GlV\nUKt42VYkYwiPkD8QxJZP6rDtYBO0GhW+f+9czMpOFF0WEV2GJEmYlGTEpCQjbpg1CUfruvB5VQcO\n13Xiw72N+HBvIzSyCrNzkpAzOdylnZlqQdwYL7MYDCkY8ATg9oa/zn3v8Qbg8QXh8QUw4AvC5w8i\nELz0+s7nvFN2evB7k15GnEkLi1GLOJMWyXF62BL0SE4wwJZggC1Bz6AWiCE8Ai2dLrz050o02V1I\nSzbhu1+bNbhUGxFNDDqNGgvyU7AgPwVeXxDVZ3pxvL4bx+q7cajGjkM19sH7Wi06TEkxIylOj6R4\nPRLjdLCadTDoZOi1aui1MrQaFRQlHKghRUEopGDgXKh6w4tTeHxBDHjDgeo5u83tCQft5ahVEvRa\nNeJMWmjUKmhkFWRZBVmlgiTh7Jd09vghJFr08PmD8PiCcAz40e/yoa3LjYvFt6yWMCnRiLRkE9Jt\nZmSmmJE1ycL1zscJQ/gqBIIhfFbRgjc/rYUvEMLSojSsXDYdOo62JJrQdFo1CnOTUJibBABQ6zTY\nf6QFDe0ONLY70NDuwJG6rlE/rlolwaiXEW/WwqiTYdRrzv4rw6CTYdCFA/5qZwK72HrCoZCCfrcP\nnb0e2PsG0Nk7gI6eAbR0udDS6UaT3QWc6Bi8f7xZi6xUC2bnJiM1QY/syXEwGzTX/H+mCzGEr4Ci\nKKg42Yk3t9ehrdsNo07Go18pwIL8FNGlEdEYSIzTY/4M2wVjPNweP7r6vejq86Cr34NepxceXxDe\ns93FvkAIKkmCJAEqlXT27FWG8WyYNnW6YNCqodfJMGjDt2lk1bgNCFOpJCSYdUgw6zDt7z73DikK\nuvs8aLK70NjuQH3b0BuP8998pCYakZsWh9z0eOSmxSHDZuZsgNeIIXwZiqLgZFMf3v6sDjVNfVBJ\nEpbOS8fdN03lTDxEMcao18Co12DKCD96GqsR7aNBJUlITjAgOcGAounJg7f3uXzodvlxqKoNp1r6\ncbq1H7sr27C7sg1AuAchZ3IcctLikJsWj5z0uDH//DzaMIQvwu0JYM+xNnxW0RzuogFQNC0ZK5bm\nIo0rIRFRjIg3aTFtahKyU8KveyFFQWunC3Ut/aht7kNdcx9ONPTgREPP4GOS4/XISYvD1ElxyD47\nuI1T914aW+YstyeA4/XdqKjtxIHqDvj8IahVEhbkp+DW+RnIm5IgukQiIqFUkoR0mxnpNjMWz00D\nEO6mP9Xaj1PN/aht6UN9qwP7T3Rg/9nPlyUAKYlGZKWaMSXFjMxUC9KTTbBadLw2GzEcwi6PH/Vt\nDpxu6cex0+G5ZkNKeOxgcrweS4rScHNhGuJN7FohIroUo16D2dlJmJ0dHtSmKArsfR7Ut4YnRWls\nd6Cx3XlBMAPhOb/Tkk1ISzIhNdGAVKsRKVYDUqyGmDpzHvZ/GgqF8Mwzz6C6uhparRbPPvsssrKy\nBrdv2bIFb7zxBmRZxmOPPYZbbrllTAu+GsFQCH1OH+y9A+joHYDTG8Tp5j40tjnQ0TsweD8JQHZa\nHApzkjAnNwlZkyxQ8R0aEdFVkyQJKQkGpCQYsHBmKoBwMHf1edDQ7kST3Ynms3N6N7Q5cKql/wv7\nMBs0SIzTISlOj8Q4PRLMWsSbdEgwaweveTYbZGjkiX9lyrAhvHXrVvh8PmzevBkVFRXYsGEDXnzx\nRQCA3W7Hq6++irfeegterxerVq3CTTfdBK12fM4e+90+7DzSCofbB7cnfC2eyxNAv9uHfpcPTrf/\notfFmfQyZk21YurkOGSlWpCXmcDBBEREY0Q6b+DX+SPOA8EQOnrCl0q197jPfu9GV78XbV1uNLY7\nL7tfrUYFs0FzdgT60JdOow5/aVXQadTQyOGR6Bq1CrIsQaMOzyQmq8Oj2NVqFVSSNDiqPTFOB8s4\nZcKwIVxeXo6SkhIAQFFRESorKwe3HTlyBPPmzYNWq4VWq0VmZiaqqqpQWFg4dhWf52C1HX/aXveF\n2w06GfEmLdKSTIg3a5EcH+7iyJuaBK2k8LMIIqIIIKtV4S7piwx4VRQFzgE/uvu96HV60efyoe/s\nv84B/+CXayB86ZjH67roSddImPQy/t8PSsbl8qthQ9jpdMJsHhqSr1arEQgEIMsynE4nLBbL4DaT\nyQSn8/LvXGw2y2W3X437b8vH/bflj9r+Ys2538X9y9mGRH9vNF+rgIn9dzbabXGlUgDkCDnypY12\nWww7DYvZbIbL5Rr8ORQKQZbli25zuVwXhDIRERFd2rAhXFxcjLKyMgBARUUF8vLyBrcVFhaivLwc\nXq8XDocDdXV1F2wnIiKiS5MURblsN/q50dE1NTVQFAXr169HWVkZMjMzsWzZMmzZsgWbN2+Goij4\nzne+g9tvv328aiciIprQhg1hIiIiGhtcRJKIiEgQhjAREZEgMRXCDocD3/3ud/HQQw9h5cqVOHTo\nkOiSxl0oFMLatWuxcuVKrF69Gg0NDaJLEsLv9+PJJ5/EqlWrsGLFCmzbtk10ScJ1dXVhyZIlqKv7\n4rX3seQ///M/sXLlStx777148803RZcjjN/vx5o1a1BaWopVq1bF5PPi8OHDWL16NQCgoaEBDz74\nIFatWoWf/exnCIVCo3KMmArhjRs3YtGiRXjttdfw3HPPYd26daJLGnfnz4C2Zs0abNiwQXRJQrz3\n3ntISEjA66+/jpdffhk///nPRZcklN/vx9q1a6HX60WXItS+fftw6NAhbNq0Ca+++ira2tpElyTM\nZ599hkAggDfeeAOPP/44fv3rX4suaVz913/9F37605/C6/UCAJ577jn88Ic/xOuvvw5FUUbtjXtM\nhfA3vvENlJaWAgCCwSB0uthbE/hyM6DFkjvuuANPPPEEgPDMPGr1xJ+D9lr88pe/RGlpKVJSUkSX\nItTOnTuRl5eHxx9/HN/97nexdOlS0SUJk52djWAwiFAoBKfTOTg/RKzIzMzE888/P/jzsWPHsHDh\nQgDA4sWLsXv37lE5TtS26ptvvok//OEPF9y2fv16FBYWwm6348knn8TTTz8tqDpxLjcDWiwxmcLT\n5DmdTvzgBz/AD3/4Q8EVifP2228jMTERJSUl+N3vfie6HKF6enrQ0tKCl156CU1NTXjsscfw0Ucf\nxeQ0t0ajEc3NzbjzzjvR09ODl156SXRJ4+r2229HU1PT4M+Kogw+D0wmExwOx6gcJ2pfee+//37c\nf//9X7i9uroaP/7xj/HUU08NvquJJZebAS3WtLa24vHHH8eqVavw1a9+VXQ5wrz11luQJAl79uzB\niRMn8JOf/AQvvvgibDbb8A+OMgkJCcjJyYFWq0VOTg50Oh26u7uRlJQkurRx98orr+Dmm2/GmjVr\n0NraikceeQR/+ctfYrIHEQBUqqGOY5fLhbi4uNHZ76jsZYKora3FE088gV/96ldYsmSJ6HKEuNwM\naLGks7MT3/rWt/Dkk09ixYoVossR6o9//CNee+01vPrqq5g5cyZ++ctfxmQAA8D8+fOxY8cOKIqC\n9vZ2DAwMICEhQXRZQsTFxQ1OQxwfH49AIIBgMCi4KnEKCgqwb98+AEBZWRkWLFgwKvuNqVOgX/3q\nV/D5fPjFL34BIHxWeG5ZxlixfPly7Nq1C6WlpYMzoMWil156Cf39/XjhhRfwwgsvAAgPxIj1gUmx\n7pZbbsHnn3+OFStWQFEUrF27NmbHC3zjG9/A008/jVWrVsHv9+NHP/oRjEaj6LKE+clPfoJ/+Zd/\nwb/9278hJydn1GaH5IxZREREgsRUdzQREVEkYQgTEREJwhAmIiIShCFMREQkCEOYiIhIEIYwERGR\nIAxhIiIiQRjCREREgvx/j5MRPDoYwz8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c8e9240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.distplot(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculate skew"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import scipy\n",
    "import scipy.stats as sp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.9899660105233683"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.skew(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Confidence intervals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Normal method\n",
    "\n",
    "Get B estimates of $\\hat{T}$, calculate \n",
    "\n",
    "$ CI = T_n \\pm z_{\\alpha/2} \\cdot \\hat{se}_b $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# number of bootstrap samples\n",
    "B = 1000\n",
    "# sample size\n",
    "s = 1000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "estimates = []\n",
    "\n",
    "for i in range(B):\n",
    "    sample = np.random.choice(X, size=s)\n",
    "    estimates.append(sp.skew(sample))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "mu_hat = np.mean(estimates)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "se_hat = np.std(estimates)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.9829891614960251, 1.8063552332784258, 2.1596230897136244)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(mu_hat, mu_hat-1.96*se_hat, mu_hat+1.96*se_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: The normal method only works if $\\hat{T}$ is normally distributed. We can check:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x10c7d1748>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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KdHV1geQk6W3qvOz5dk4ZU3ZjyRItQOPk+ViW7OxK7VPnZXnxE2U3lizRPAkhcK7DBUuO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\nShKWLBERUZKwZImIiJKEJUtERJQk/z+HD21qAJE/QwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cbedac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.distplot(estimates)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pivot method\n",
    "\n",
    "The pivotal CI is given by:\n",
    "\n",
    "$C_n = (\\hat{a}, \\hat{b})$\n",
    "\n",
    "where:\n",
    "\n",
    "$\\hat{a} = 2 \\hat{\\theta}_n - \\theta^*_{1-\\alpha/2}$\n",
    "\n",
    "$\\hat{b} = 2 \\hat{\\theta}_n - \\theta^*_{\\alpha/2}$\n",
    "\n",
    "where $\\hat{\\theta}_n$ is the sample mean and $\\theta^*_{\\alpha/2}$ is the ${\\alpha/2}^{th}$ quantile."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "theta_hat = np.mean(estimates)\n",
    "theta_star_a = np.percentile(estimates, q=97.5)\n",
    "theta_star_b = np.percentile(estimates, q=2.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.796693978241314, 2.1575170872793867)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2*theta_hat - theta_star_a, 2*theta_hat - theta_star_b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Percentile intervals\n",
    "\n",
    "$C_n = (\\theta^*_{\\alpha/2}, \\theta^*_{1-\\alpha/2})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.8084612357126637, 2.1692843447507362)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(theta_star_b, theta_star_a)"
   ]
  }
 ],
 "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": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}