scipy-stats.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Getting started: import useful modules\n",
    "\n",
    "The python functions must be *imported* from modules before called in a program.\n",
    "\n",
    "A command starts with '```%```' is called a magic command, according to documentatiton:\n",
    "\n",
    ">Magics are specific to and provided by the IPython kernel. Whether magics are available on a kernel is a decision that is made by the kernel developer on a per-kernel basis. \n",
    "\n",
    "In our case, the \n",
    "\n",
    "> ```%matplotlib inline```\n",
    "\n",
    "command will enable Jupyter notebook to show plots generated by matplotlib directly in the notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import scipy.stats as stats\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Read the friendly manual\n",
    "\n",
    "When in trouble, always start with reading the manual/documentation -- you may be be surprised how informative they are.\n",
    "\n",
    "When functions are created in Python, there is an option to include a \"docstring\", and good libraries/modules always include these for their functions.\n",
    "\n",
    "In general in Python, you can access the docstring for a function using\n",
    "> ```help(functionName)``` \n",
    "where functionName is the name of the function\n",
    "\n",
    "\n",
    "However, Jupyter has a better way to read the docstrings in a separate window overlay at the bottom of the browser window. To access it, type\n",
    "> ```?functionName``` \n",
    "\n",
    "or \n",
    "\n",
    "> ```?moduleName``` \n",
    "for help on a module\n",
    "\n",
    "Let's start by reading the docstring for the scipy.stats.norm module:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "?stats.norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The docstring for stats.norm lists all of the functions that the stats.norm module provides, and for each one lists the possible parameters\n",
    "\n",
    "It also provides notes about the distribution in question. For instance, stats.norm contains functions for the Normal distribution (which we call the Gaussian distribution). \n",
    "\n",
    "We see that the pdf (probability density function) for the Normal distribution with no parameters is given by\n",
    "$$\n",
    "\\frac{1}{2\\pi} \\operatorname{exp}\\left( - \\frac{x^2}{2} \\right)\n",
    "$$\n",
    "\n",
    "We also see that when the ``loc`` and ``scale`` parameters are given, then ``norm.pdf(x, loc, scale)`` is identically\n",
    "equivalent to ``norm.pdf(y) / scale`` with\n",
    "``y = (x - loc) / scale`` \n",
    "\n",
    "Later, we will see that this means that ``loc`` is the mean of the distribution, and ``scale`` is the standard deviation (or ``scale``^2 is the variance) \n",
    "\n",
    "Note that the call signature for the cdf function is:\n",
    "``cdf(x, loc=0, scale=1)``\n",
    "    Cumulative density function.\n",
    "    \n",
    "If we inspect the parameters, we will see that there are 2 different types of parameters. The first is a positional parameter ``x``, and the second type are the keyword parameters indicated by ``loc=0`` and ``scale=1``. Positional parameters are required, but keyword parameters are optional and have the default values shown.  In otherwords, if you type\n",
    "\n",
    "> stats.norm.cdf(0.5)\n",
    "\n",
    "it will calculate the same value of the cdf as if you typed\n",
    "\n",
    "> stats.norm.cdf(0.5,loc=0,scale=1)\n",
    "\n",
    "Let's try it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.69146246127401312"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.norm.cdf(0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.69146246127401312"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.norm.cdf(0.5,loc=0,scale=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A search engine is your friend\n",
    "\n",
    "Searching for \"theCommandYouWantToLearn\" plus \"documentation\" can usuallly bring you to the right page"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Typical methods for continuous distributions in ```scipy.stats```\n",
    "\n",
    "|method|discription|\n",
    "|---|---|\n",
    "|rvs|Random variates (return random values from the distribution).|\n",
    "|pdf|Probability density function.|\n",
    "|cdf|Cumulative distribution function.|\n",
    "|mean|Mean of the distribution.|\n",
    "|var|Variance of the distribution.|\n",
    "|std|Standard deviation of the distribution.|\n",
    "|median|Median of the distribution.|\n",
    "|sf |Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).|\n",
    "|ppf|Percent point function (inverse of cdf — percentiles).|\n",
    "|fit|Parameter estimates for generic data.|\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example for discrete random variables\n",
    "\n",
    "> ``` ?stats.poisson```\n",
    "\n",
    "or look at:\n",
    "\n",
    "http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.poisson.html#scipy.stats.poisson\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "?stats.poisson"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that although the Poisson random variable has only one parameter, ``mu`` (the average number of events -- its mean), the scipy.stats.poisson functions also allow another parameter ``loc``. Here ``loc`` can be used to shift the distribution left or right; this is typical of scipy.stats functions. In general, you can omit ``loc`` for distributions that have only one parameter:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3292869816564159"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.poisson.pmf(1, mu=0.6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Typical methods for discrete distributions in ```scipy.stats```\n",
    "\n",
    "|method|discription|\n",
    "|---|---|\n",
    "|rvs|Random variates (return random values from the distribution).|\n",
    "|pmf|Probability mass function.|\n",
    "|cdf|Cumulative distribution function.|\n",
    "|mean|Mean of the distribution.|\n",
    "|var|Variance of the distribution.|\n",
    "|std|Standard deviation of the distribution.|\n",
    "|median|Median of the distribution.|\n",
    "|sf |Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).|\n",
    "|ppf|Percent point function (inverse of cdf — percentiles).|\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The complete list of random distributions in ```scipy.stats```\n",
    "\n",
    "http://docs.scipy.org/doc/scipy/reference/stats.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting CDFs, PMFs, and PDFs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the CDFs for the Gaussian and Poisson distributions. First generate data linearly over a range where the random variable has most of its probability. For instance, the Gaussian is centered at 0 and with standard deviation 1, will have most probability between -4 and 4. To generate 100 equally spaced values from -4 to 4, we can use numpy's ``linspace`` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = np.linspace(-4, 4, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can evaluate the cdf at each of these points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cdfvals = stats.norm.cdf(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can plot the cdf values vs. x using matplotlib:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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O778Po0bBM8+ETiJSvnLXQWwGfAKcZWaPAXpIYglimZdUznTV1dUxciT07g1bbRU6TdNi\nGs+8iyFjOcq9gpgHjHf3W81sY0C7yYg04ZNPYOhQeOih0ElEWqfcHkR74Nvu/nzh9tc+7n5Zm6Vr\nOY96EJJbo0bBfffBhAmhk4h8XirrIMxsbWADd3+vhH/gZu6+oLyYa0YFQvJqxYpkUdyNN8Kee4ZO\nI/J5qayDcPePgT3M7EgzW7eJf1AnMxtA8oxpKSKWeUnlTM/48bDeenVRFIcYxhPiyBlDxnKUsg7i\nPjP7OvBLM/sasE7hz63eauMNYIy7/6dNk4pEwh2GDIGjjgqdRGTN6HkQIil78EE491yYMSMfT4wT\naazNngdhZoc0Om5f7meIVLIhQ2DQIBUHiV9rngexpZl1BTCz9YFL0gxUiWKZl1TONTd1KixYAIcf\nnu+cDSlnemLIWI7WPA9iLNDXzM4DOgCzU00kErErroCzz4a19KQVqQBl9yDM7FngDuARoJ27B9tE\nQD0IyZOZM5PnTc+bB+sWvedPJB/SfB7E2cCbwLvuPtHMtnb3uYWvdQYGuvuFaYQulwqE5Mkxx8C3\nvpX0H0TyLM0m9bfd/VZ3nwiwujgUfv8u8HDrY1aHWOYllbP1/vGPZEuNU0757L085ixGOdMTQ8Zy\nlDJTOgWSBXFAb2AJ8Iy7/xvA3Z9ou3gicbjqKhgwADbaKHQSkfSUMsV0oruPaXB8LfAnd3+prcO1\nRFNMkgeLFkH37jBrFnTpEjqNSMvS7EG8DTwFTAdmAFu5+x8KX+upJrVUu/POS577MGxY6CQipUmz\nB3E5cCXJ1NJhwC/MbL6Z/QU4d81iVodY5iWVs3xLlsDo0cmtrY3lKWdzlDM9MWQsRyl7MV1d+O3f\nVr9XeBbEd4HT2iiXSBSGD4cDD4QttwydRCR9a7QXk5nt7u5/a/nMtqEpJglp6dKkMEyeDDvsEDqN\nSOnabC+mhkIWB5HQxoxJnvWg4iCVao0KhJQmlnlJ5SzdJ58kt7aed17T5+QhZymUMz0xZCyHCoRI\nK9x8M2y/Pey2W+gkIm1Hz4MQKdPKlUlxGD0aamtDpxEpXyY9iNYysz5mNsvMXjWzJm+VNbPdzKze\nzA7OMp9Ic8aPh86d4Qc/CJ1EpG1lXiDMrB1wDcm2HTsCR5pZ9ybOG0IF7PUUy7ykcrZs1Sq4/HK4\n8MKWHwik8UxXDDljyFiOEFcQPYE57v66u9cD44CDipx3OjAeeDvLcCLNmTABOnSAAw4InUSk7WXe\ngyg8srS3uw8oHB8D9HT3Mxqc803gFnff28xuACa4+51FPks9CMmMO/Tsmdy5dLAmPSViue5BlGAo\nn9/GQ0/3leAmTkwWx/XrFzqJSDZCPBhxIbB5g+NNC+819B1gnJkZ0Bk4wMzq3f3exh/Wv39/unbt\nCkCnTp2oqamhtnBryer5wNDHq9/LS56mjocOHZrL8cvLeJ5zTh0/+Qm0a1fa+RrPdI9jGM/p06cz\ncODA3ORZfVxXV8fYsWMBPv1+WRJ3z/QFtAfmAlsAHUl2id2+mfNvAA5u4mseg8mTJ4eOUBLlbNqU\nKe7bbOO+YkXpf0bjma4YcsaQ0d298L2zxe/XQdZBmFkf4E8kU1zXufsQMzupEHp0o3OvB+5z9SAk\noH33TR4pevzxoZOIrLnUngeRZyoQkoUnn4TjjoPZs5M7mERiF3uTuqI0nOvNM+Us7pJL4Pzzyy8O\nGs90xZAzhozlUIEQacbUqTBnTnIFIVJtNMUk0ow+fZI1DwMGhE4ikh5NMYmsoWnT4JVXoH//0ElE\nwlCByEAs85LK+XmXXAKDBkHHjq378xrPdMWQM4aM5QixUE4k955+Gl56Ce66K3QSkXDUgxAp4oc/\nhMMOU+9BKpN6ECKt9Pjj8NprWhQnogKRgVjmJZUz2bH1N7+Biy5a80VxGs90xZAzhozlUIEQaWDy\nZFi4EI49NnQSkfDUgxApcIdeveCUU5J9l0QqlXoQImV66CFYvBiOPDJ0EpF8UIHIQCzzktWcc9Wq\nZL+lyy+H9u3T+cxqHs+2EEPOGDKWQwVCBLj9dlhrLT1KVKQh9SCk6tXXw447wvDhsN9+odOItD31\nIERKdMMNsNlmKg4ijalAZCCWeclqzLlsGVx6KVxxRWof+alqHM+2FEPOGDKWQwVCqtqwYdCzZ/IS\nkc9TD0Kq1uLFsN12ydYa228fOo1IdvRMapEWnHVWMsU0YkToJCLZUpM6R2KZl6ymnPPmwY03wsUX\nr/FHNamaxjMLMeSMIWM5VCCkKp1/Ppx5JnTpEjqJSH5pikmqzrRpyYK4V1+F9dcPnUYke5piEinC\nHc4+Gy67TMVBpCUqEBmIZV6yGnKOHw/vvw8/+1l6eZpSDeOZpRhyxpCxHHomtVSNZcvgnHNg7Nj0\nNuQTqWTqQUjVuPRSmDkz2ZhPpJppHYRIAwsWQE0NPPccdO0aOo1IWGpS50gs85KVnPPcc+HUU7Mt\nDpU8niHEkDOGjOVQD0Iq3pNPwhNPwLXXhk4iEhdNMUlFq6+HHj3gwgvhiCNCpxHJB00xiQBXXw3f\n+AYcfnjoJCLxUYHIQCzzkpWWc8GC5DkPw4aBtfizUvoqbTxDiyFnDBnLoQIhFWvgQDjtNNhmm9BJ\nROKkHoRUpAcegDPOgJdegnXWCZ1GJF9K7UHoLiapOB9+mFw5jBih4iCyJoJMMZlZHzObZWavmtm5\nRb5+lJnNKLyeNLOdQuRMSyzzkpWS84ILYK+9oHfvbPI0pVLGMy9iyBlDxnJkfgVhZu2Aa4B9gTeB\nZ83sHnef1eC0ecD33f0/ZtYHuBbYPeusEp+pU5OtNGbODJ1EJH6Z9yDMbHdgsLsfUDgeBLi7X9nE\n+Z2Ame6+WZGvqQchn1q+HHbZJdnK+9BDQ6cRya88r4PYBFjQ4PiNwntNORF4sE0TSUW47DLYYQcV\nB5G05LpJbWZ7A8cDvZo6p3///nQtbLDTqVMnampqqK2tBT6bDwx9vPq9vORp6njo0KG5HL9SxvO5\n52DYsDquvx4gH3ljHs885YtpPKdPn87AgQNzk2f1cV1dHWPHjgX49PtlSdw90xdJL+GhBseDgHOL\nnLczMAfo1sxneQwmT54cOkJJYs350Ufu223n/te/hsnTlFjHM69iyBlDRnf3wvfOFr9fh+hBtAdm\nkzSp/wU8Axzp7q80OGdz4FHgWHf/WzOf5Vnnl/w5/XRYvBhuuSV0EpE45HYdhLuvNLPTgIkkPZDr\n3P0VMzsp+bKPBi4CvgIMNzMD6t29Z9ZZJf8efhjuuQdmzAidRKTyBFkH4e4Puft27r6Nuw8pvDeq\nUBxw95+7+8bu3sPdd4m9ODSc682z2HK+9x6ccELyCNEvfzlopKJiG8+8iyFnDBnLob2YJEruSXE4\n4gjYZ5/QaUQqk/Zikij98Y/w178mDwPq2DF0GpG46JnUUrGmTYMf/Sj5dcstQ6cRiU+eF8pVnVjm\nJWPIuXgxHHRQHaNH5784xDCeoJxpiiFjOVQgJBqrVkH//tCrF/TrFzqNSOXTFJNEY/BgePRReOwx\n9R1E1kRu10GItMYddyS3sz7zjIqDSFY0xZSBWOYl85pzxgw4+WS46y7o0iW/ORtTznTFkDOGjOVQ\ngZBce+edpN9wzTXQo0foNCLVRT0Iya2PPkoWwe2/f7KVt4ikQ+sgJGorViRXDl/9Klx/PViLf5VF\npFRaB5EjscxL5iWnO5xyCqxcCaNHf7E45CVnS5QzXTHkjCFjOXQXk+TO4MHwwgtQVwcdOoROI1K9\nNMUkufK738FNNyXFoUuX0GlEKpPWQUh0rroqWeswZYqKg0geqAeRgVjmJUPmHDoURo1KVkl/4xvN\nn6vxTJdypieGjOXQFYQE5Q5XXAFjxiTTSptuGjqRiKymHoQEs2oVnHMOTJyYPDr0m98MnUikOqgH\nIbm2YgWceCLMmQOPP57PR4aKVDv1IDIQy7xkVjmXLEke+PP228nVQ7nFQeOZLuVMTwwZy6ECIZma\nMwd23x223RbuvRfWXz90IhFpinoQkplHHoGjj072VRowIHQakeqlHoTkxsqVcPnlMHIk3HYb1NaG\nTiQipdAUUwZimZdsi5xvvgn77ZcsfnvuuXSKQzWPZ1tQzvTEkLEcKhDSZu64A3bdNdmye9Ik3cYq\nEhv1ICR1ixbBqafCSy8lW3V/73uhE4lIQ9ruWzK3alWyl9LOO8M228D06SoOIjFTgchALPOSa5Lz\n6aeT21dHjoQHHki2z1hnnfSyNVQN45kl5UxPDBnLoQIha2Tu3OTW1cMOg9NPh6lTk76DiMRPPQhp\nlXnzkvUMEyYkheFXv4INNgidSkRKoXUQ0ib+/vdka+6HHkoa0XPnQqdOoVOJSFvQFFMGYpmXbCrn\nxx/D//wP9OoFhxwCNTVJYbjkkjDFIfbxzBvlTE8MGcuhKwgpyh2efx5uuAHGjUvuTBo4EPr1g7X0\nt0akKqgHIZ9yT6aQxo9PFrmtWgU/+1ny6to1dDoRSYt6EFKSd95JNtF7+OFk6+0NN4RDD02mlHbZ\nBazFv0IiUqmC9CDMrI+ZzTKzV83s3CbOudrM5pjZdDOryTpjmvIyL7lyJbzyCtx4Y7Kb6g47wNZb\nJ1NIPXvCVVfVMXs2/Pa30KNHfotDXsazJcqZrhhyxpCxHJkXCDNrB1wD9AZ2BI40s+6NzjkA6Obu\n2wAnASOzzpmm6dOnZ/rPc0+2u6irg+HDk7uNevVKGso/+hHcfz/stBPccgssXgz33AO/+AW88062\nOVsr6/FsLeVMVww5Y8hYjhBTTD2BOe7+OoCZjQMOAmY1OOcg4EYAd59mZhuZWRd3X5R52hQsWbIk\n1c9bvjwpAG+9BW+8AQsWJK/58+G115I1CmutlVwhrH4demgyZdTcXUdp52wrypku5UxPDBnLEaJA\nbAIsaHD8BknRaO6chYX3oiwQq1bBsmXwySfJN/fVr2XL4KOPYOnS5NcPP0xeH3wA//lP8mjOJUuS\nn/Lfey95vftucv7Xvw5dusCmm8JmmyW/9uwJ3bolLz3jWUTWVPRN6r59k18b38zU8Hj17xv+2vj3\n7sk38obHq99r/Fq5svhrxYrk1/r6z79WrpzPVVdBhw6w7rrJHkVrrw3rrff514YbJquRN9gg+Ul/\nu+1go42Sb/adO8PGGye/fvnLbdMfmD9/fvof2gaUM13KmZ4YMpYj89tczWx34GJ371M4HgS4u1/Z\n4JyRwGR3v61wPAv4QeMpJjPTPa4iIq2Q19tcnwW2NrMtgH8BPwWObHTOvcCpwG2FgrKkWP+hlH9B\nERFpncwLhLuvNLPTgIkkd1Fd5+6vmNlJyZd9tLs/YGZ9zWwu8BFwfNY5RUSqXdQrqUVEpO1UzGZ9\nZvYrM1tlZl8JnaUYM7vUzGaY2Qtm9pCZfT10pmLM7L/N7JXCAsU7zOxLoTMVY2aHmtlLZrbSzHqE\nztNQKQtB88DMrjOzRWb2YugsTTGzTc3sMTP7XzObaWZnhM5UjJmtbWbTCv9/zzSzwaEzNcfM2pnZ\n82Z2b3PnVUSBMLNNgR8Cr4fO0oz/dvdvu/suwP1AXv8CTQR2dPcaYA5wXuA8TZkJ/ASYEjpIQ6Us\nBM2RG0hy5tkK4Cx33xHYAzg1j+Pp7h8Dexf+/64BDjCzxrfv58mZwMstnVQRBQL4I3BO6BDNcfcP\nGxyuD6wKlaU57v6Iu6/O9jdg05B5muLus919DpC3GxU+XQjq7vXA6oWguePuTwL/Dp2jOe7+lrtP\nL/z+Q+AVkjVRuePuSwu/XZukv5vL+fvCD9R9gTEtnRt9gTCzHwML3H1m6CwtMbPLzeyfwFHAb0Ln\nKcH/Ax4MHSIyxRaC5vIbWmzMrCvJT+fTwiYprjBt8wLwFjDJ3Z8NnakJq3+gbrGARbFQzswmAV0a\nvkXyL3chcD7J9FLDrwXRTM4L3H2Cu18IXFiYlz4duDj7lC3nLJxzAVDv7rcGiEghQ4s5pTqY2QbA\neODMRlfjuVG48t6l0Le728x2cPcWp3GyZGYHAovcfbqZ1dLC98soCoS7/7DY+2b2LaArMMPMjGQ6\n5Dkz6+nub2cYEWg6ZxG3Ag8QqEC0lNPM+pNcgu6TSaAmlDGeebIQ2LzB8aaF96SVzGwtkuJwk7vf\nEzpPS9zvwnsoAAAB4ElEQVT9fTObDPShhHn+jO0J/NjM+gLrAhua2Y3uflyxk6OeYnL3l9z96+6+\nlbtvSXI5v0uI4tASM9u6wWE/krnU3DGzPiSXnz8uNN5ikKc+xKcLQc2sI8lC0GbvFAnMyNf4FXM9\n8LK7/yl0kKaYWWcz26jw+3VJZjVmNf+nsufu57v75u6+FcnfzceaKg4QeYEowsnvX/YhZvaimU0H\n9iO5iyCP/gxsAEwq3AY3PHSgYsysn5ktAHYH7jOzXPRK3H0lsHoh6P8C49w9rz8M3ApMBbY1s3+a\nWe4WpJrZnsDRwD6FW0ifL/wQkzffACYX/v+eBjzs7g8EzrTGtFBORESKqrQrCBERSYkKhIiIFKUC\nISIiRalAiIhIUSoQIiJSlAqEiIgUpQIhIiJFqUCIiEhRKhAiIlJUFJv1icTCzNoDRwBbkWz73RP4\n/+7+j6DBRFpBVxAi6fo2yc6j80j2Bbsd+FfQRCKtpAIhkiJ3f97dPyF5POYUd69z9+Whc4m0hgqE\nSIrMbDcz25jkud7/MLO9QmcSaS31IETS1YfkkZNTzawf8G7gPCKtpu2+RUSkKE0xiYhIUSoQIiJS\nlAqEiIgUpQIhIiJFqUCIiEhRKhAiIlKUCoSIiBSlAiEiIkX9H5xH/qGmmGm9AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b55ae48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, cdfvals)\n",
    "plt.ylabel('$F_X(x)$')\n",
    "plt.xlabel('$x$')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Poisson random variable is a non-negative variable, so its support will start at 0. Let's look at the cdf for the Poisson random variable with mu=5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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N4hKpyHvvQVdX9sfdsCGuYiFSLRWtDWVmZ7j7s+X3rA31LKQvn/wkPPJImJVfP/95uPPO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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110f5b0f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=np.linspace(0,10,100)\n",
    "cdfvals=stats.poisson.cdf(x,mu=5)\n",
    "\n",
    "plt.plot(x, cdfvals)\n",
    "plt.ylabel('$F_X(x)$')\n",
    "plt.xlabel('$x$')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As expected, the cdf has a staircase distribution. However, it doesn't look like a nice staircase because matplotlib is using linear interpolation between the points. We can make a nicer graph using matplotlib's ``step`` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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xnsfaUk+Tx1u3buWcc85pTT1NHm/atGmsvz9s2bIFYO/3y6Vqagz1wczc0D2edwwVES8F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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b7bc0f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=np.linspace(0,10,100)\n",
    "cdfvals=stats.poisson.cdf(x,mu=5)\n",
    "\n",
    "plt.step(x, cdfvals)\n",
    "plt.ylabel('$F_X(x)$')\n",
    "plt.xlabel('$x$')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's plot the PMF for this Poisson. We usually plot PMFs for discrete random variables as stem plots.\n",
    "\n",
    "Since the Poisson only takes values at nonnegative integers, we should evaluate the PMF at those points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x=range(11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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IeFTSzZJepH6rwTEASb8EjADPSfoGEMBdEfGVTv8eZmarxfwtwpdXTHxl+4pm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\n1gZ3AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e669908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmfvals=stats.poisson.pmf(x,mu=5)\n",
    "plt.stem(x, pmfvals)\n",
    "plt.ylabel('$P_X(x)$')\n",
    "plt.xlabel('$x$')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's plot the pdf for the Gaussian random variable with the default parameters:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Vxj/9qe8kxodBg+Chh3ynMJlgxcOzqPSDZirnQw+5F5BszLKab22ZbdnI2auX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\nHsYYY9JmxcMYY0zarHgYY4xJ2/8HTVox6SzzXDUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111831a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-4, 4, 100)\n",
    "pdfvals = stats.norm.pdf(x)\n",
    "\n",
    "plt.plot(x, pdfvals)\n",
    "plt.ylabel('$f_X(x)$')\n",
    "plt.xlabel('$x$')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generating random variables and estimating PMFs and pdfs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's begin by generating 100,000 Poisson random variables with mu=5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "p=stats.poisson.rvs(mu=5,size=100000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check the average value of these variables:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.0007000000000001"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(It is pretty close to 5, which we said was the average number of occurrences)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's calculate the relative occurence of the values from 0 to 10. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can calculate the number of occurences in each bin in several ways. For instance, if we just wanted to count how many times 0 occured, we could compare all the values of p to 0 and then use numpy's ``count_nonzero`` function to count how many times that was true:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "658"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.count_nonzero(p==0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could use a loop to do this over all the bins, but numpy provides an eaier way.\n",
    "\n",
    "First create a set of bin edges that go from 0 to 11:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mybins=range(12) #Note to get 11 included in mybins, we need to use 11+1=12 as the limit in the range function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now call numpy's ``histogram`` function on the data using the specified bins:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(array([  658,  3407,  8439, 13994, 17513, 17530, 14550, 10470,  6675,\n",
      "        3617,  2641]), array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11]))\n"
     ]
    }
   ],
   "source": [
    "phist=np.histogram(p,bins=mybins)\n",
    "print(phist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot it and compare to the PMF:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 3 artists>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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5bg5mZpbj5mBmZjluDmZmluPmYGZmOW4OZmaWU+/9HKwgAwODdHb2MjQ0QkvLDHp6OnzH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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11172cc50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.stem(phist[1][:-1],phist[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It does have the same shape, but it would be easier to compare if we changed these to relative frequencies. We can do that ourselves by dividing the numbers in the bins by the total number across all bins:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 3 artists>"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1118e59b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "phist2=phist[0]/np.sum(phist[0])\n",
    "\n",
    "plt.stem(phist[1][:-1],phist2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can overlay this with the true histogram to see if they match up:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 3 artists>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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8Zja2byS1KcXQhiEHvpldMhWN9CUtl7Rf0gFJ90zS535JByXtlXR9SfvnJP1S0s8lbZN0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\nDn0zswRx6JuZJcj/B1GhHWKZ5sZpAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11190a358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.stem(phist[1][:-1],phist2,markerfmt='bo')\n",
    "#plt.hold(True)\n",
    "x=range(11)\n",
    "pmfvals=stats.poisson.pmf(x,mu=5)\n",
    "\n",
    "# I am slightly offsetting the true PMF values, so that\n",
    "# both sets of results are visible:\n",
    "plt.stem(np.array(x)+0.2,pmfvals,'g',markerfmt='gx')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that we didn't have to do the normalization ourselves. Numpy can do it for us:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 3 artists>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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8Zja2byS1KcXQhiEHvpldMhWN9CUtl7Rf0gFJ90zS535JByXtlXR9SfvnJP1S0s8lbZN0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\nDn0zswRx6JuZJcj/B1GhHWKZ5sZpAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1113bfa90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "phist=np.histogram(p,bins=mybins,normed=True)\n",
    "\n",
    "plt.stem(phist[1][:-1],phist[0],markerfmt='bo')\n",
    "#plt.hold(True)\n",
    "x=range(11)\n",
    "pmfvals=stats.poisson.pmf(x,mu=5)\n",
    "\n",
    "# I am slightly offsetting the true PMF values, so that\n",
    "# both sets of results are visible:\n",
    "plt.stem(np.array(x)+0.2,pmfvals,'g',markerfmt='gx')\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can get a plot of the histogram directly from matplotlib:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 0.00661346,  0.03424327,  0.08481919,  0.1406517 ,  0.17602066,\n",
       "         0.17619153,  0.14623997,  0.10523248,  0.06708947,  0.03635395,\n",
       "         0.02654431]),\n",
       " array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11]),\n",
       " <a list of 11 Patch objects>)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX0AAAEACAYAAABfxaZOAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAFYxJREFUeJzt3W+MXfWd3/H3xzjOkgBO1dXaxQ52+BNYLCUIqY432Si3\nIhKGKOs8aWVnVzTsE0vln4IUmdIHnn1Sha6yCYiVqBUHLRWpo7BIa0UkdRG5qip2wfyxQsCOjVCM\nDfFUJKEEKnZt+PbBvbFuZsczdzz3j+ee90sa+Z5zfud8f0ee+7m/Off8SVUhSWqGZePugCRpdAx9\nSWoQQ1+SGsTQl6QGMfQlqUEMfUlqkL5CP8nmJIeSHE6yY5blVyZ5Msm7Se6cseyrSX6a5CdJHk6y\nYlCdlyQtzLyhn2QZcD9wPbAB2JbkqhnNfgncBvzljHUv7s6/tqo+ASwHtg6g35Kks9DPSH8jcKSq\njlbVSWAPsKW3QVW9UVXPAqdmWf884MNJlgMfAl5fZJ8lSWepn9BfAxzrmT7enTevqnod+AbwKvAa\n8GZVPb7QTkqSBmOoX+Qm+QidvwrWARcDFyT58jBrSpLObHkfbV4DLumZXtud14/PA69U1a8AkjwK\nfBr47syGSbwJkCQtUFVlIe37GenvBy5Psq575s1WYO8c7Xs78CqwKcnvJQlwHXDwTCtW1UT+7Ny5\nc+x9cP/cP/dv8n7Oxrwj/ap6L8mtwD46HxK7q+pgku2dxbUrySrgGeBC4P0kdwBXV9XTSR4BngdO\ndv/ddVY9lSQtWj+Hd6iqHwFXzpj3X3teTwMfPcO6fwH8xSL6KEkaEK/IHYFWqzXuLgzVqPdv9er1\nJBnZz333PTDS/Rs1fz+bJWd7XGjQktS50hctzOrV65mePjriqqP8XclZHz+VhikJtcAvcg19LVrn\nO/rRhrChL51d6Ht4R5IaxNCXpAYx9CWpQQx9SWoQQ1+SGsTQl6QGMfQlqUEMfUlqEENfkhrE0Jek\nBjH0JalBDH1JahBDX5IapK/QT7I5yaEkh5PsmGX5lUmeTPJukjtnLFuZ5PtJDiZ5McmnBtV5SdLC\nzPvkrCTLgPvpPN/2dWB/kr+rqkM9zX4J3AZ8aZZN3As8VlX/Nsly4EOL77Yk6Wz0M9LfCBypqqNV\ndRLYA2zpbVBVb1TVs8Cp3vlJLgI+W1UPdtudqqq3BtN1SdJC9RP6a4BjPdPHu/P68THgjSQPJnku\nya4k5y+0k5KkwejrweiL3P61wC1V9UySbwF3ATtnazw1NXX6davV8tmWktSj3W7TbrcXtY15H5eY\nZBMwVVWbu9N3AVVV98zSdifwm6r6q+70KuDvq+rS7vQfAzuq6ouzrOvjEpcoH5cojcewHpe4H7g8\nybokK4CtwN65+vHbF1U1DRxL8vHurOuAlxbSQUnS4PT1YPQkm+mchbMM2F1VX0+ync6If1d3RP8M\ncCHwPvA2cHVVvZ3kk8C3gQ8ArwA3V9X/naWGI/0lypG+NB5nM9LvK/RHwdBfugx9aTyGdXhHkjQh\nDH1JahBDX5IaxNCXpAYx9CWpQQx9SWoQQ1+SGsTQl6QGMfQlqUEMfUlqEENfkhrE0JekBhn2Q1Sk\nCfDB7k3lRmPVqnWcOPHzkdVTs3iXTS1aE+6y6V09dS7yLpuSpDkZ+pLUIH2FfpLNSQ4lOZxkxyzL\nr0zyZJJ3k9w5y/JlSZ5LMtdjFiVJQzZv6CdZBtwPXA9sALYluWpGs18CtwF/eYbN3IHPxpWksetn\npL8ROFJVR6vqJLAH2NLboKreqKpngVMzV06yFriRznNyJUlj1E/orwGO9Uwf787r1zeBrzHa0x8a\nbfXq9SQZ2Y+kpWOo5+kn+QIwXVUHkrTonPt2RlNTU6dft1otWq3WMLs3saanjzL6UxolDVu73abd\nbi9qG/Oep59kEzBVVZu703cBVVX3zNJ2J/Cbqvqr7vR/Bv6MzmGf84ELgUer6qZZ1vU8/QHxvPml\nX8/3gvoxrPP09wOXJ1mXZAWwFZjrLJzTHaiqu6vqkqq6tLveE7MFviRpNOY9vFNV7yW5FdhH50Ni\nd1UdTLK9s7h2JVkFPENnJP9+kjuAq6vq7WF2XpK0MN6GYQJ5eGfp1/O9oH54GwZJ0pwMfUlqEENf\nkhrE0JekBjH0JalBDH1JahBDX5IaxNCXpAYx9CWpQQx9SWoQQ1+SGsTQl6QGMfQlqUEMfUlqEENf\nkhrE0JekBukr9JNsTnIoyeEkO2ZZfmWSJ5O8m+TOnvlrkzyR5MUkLyS5fZCdlyQtTD8PRl8GHAau\nA16n88zcrVV1qKfN7wPrgC8Bv+55MPpqYHVVHUhyAfAssKV33Z5t+OSsAfHJWUu/nu8F9WNYT87a\nCBypqqNVdRLYA2zpbVBVb1TVs8CpGfNPVNWB7uu3gYPAmoV0UJI0OP2E/hrgWM/0cc4iuJOsB64B\nnlroupKkwVg+iiLdQzuPAHd0R/yzmpqaOv261WrRarWG3jdJWira7TbtdntR2+jnmP4mYKqqNnen\n7wKqqu6Zpe1O4De/Pabfnbcc+AHww6q6d446HtMfEI/pL/16vhfUj2Ed098PXJ5kXZIVwFZg71z9\nmDH9HeCluQJfkjQa8470oXPKJnAvnQ+J3VX19STb6Yz4dyVZBTwDXAi8D7wNXA18EvhfwAt0hkoF\n3F1VP5qlhiP9AXGkv/Tr+V5QP85mpN9X6I+CoT84hv7Sr+d7Qf0Y1uEdSdKEMPQlqUEMfUlqEENf\nkhrE0JekBjH0JalBDH1JahBDX5IaxNCXpAYx9CWpQQx9SWoQQ1+SGsTQl6QGGcmTsyQtxAe7d0od\njVWr1nHixM9HVk/j5a2VJ5C3VrbeQuv53luahnZr5SSbkxxKcjjJjlmWX5nkySTvJrlzIetKkkan\nn2fkLgMOA9cBr9N5fOLWqjrU0+b3gXXAl4Bf//YZuf2s27MNR/oD4kjfegut53tvaRrWSH8jcKSq\njlbVSWAPsKW3QVW9UVXPAqcWuq4kaXT6Cf01wLGe6ePdef1YzLqSpAHzlE1JapB+Ttl8DbikZ3pt\nd14/FrTu1NTU6detVotWq9VnGUmafO12m3a7vaht9PNF7nnAz+h8GfsL4GlgW1UdnKXtTuDtqvrG\nWazrF7kD4he51ltoPd97S9PZfJE770i/qt5Lciuwj87hoN1VdTDJ9s7i2pVkFfAMcCHwfpI7gKur\n6u3Z1l3gfkmSBsSLsyaQI33rLbSe772laWgXZ0mSJoOhL0kNYuhLUoMY+pLUIIa+JDWIoS9JDWLo\nS1KDGPqS1CCGviQ1iKEvSQ1i6EtSgxj6ktQghr4kNYihL0kNYuhLUoP087hELdLq1euZnj467m5I\nUn8j/SSbkxxKcjjJjjO0uS/JkSQHklzTM/+rSX6a5CdJHk6yYlCdXyo6gV8j/JGk2c0b+kmWAfcD\n1wMbgG1JrprR5gbgsqq6AtgOPNCdfzFwG3BtVX2Czl8WWwe6B5KkvvUz0t8IHKmqo1V1EtgDbJnR\nZgvwEEBVPQWs7D43F+A84MNJlgMfAl4fSM8lSQvWT+ivAY71TB/vzpurzWvAmqp6HfgG8Gp33ptV\n9fjZd1eStBhDPXsnyUfo/BWwDrgYuCDJl4dZU5J0Zv2cvfMacEnP9NruvJltPjpLm88Dr1TVrwCS\nPAp8GvjubIWmpqZOv261WrRarT66J0nN0G63abfbi9pGquY+2yPJecDPgOuAXwBPA9uq6mBPmxuB\nW6rqC0k2Ad+qqk1JNgK7gX8N/CPwILC/qv56ljo1X1+WqiSM9qwa61lvYfUm9b036ZJQVVnIOvOO\n9KvqvSS3AvvoHA7aXVUHk2zvLK5dVfVYkhuTvAy8A9zcXffpJI8AzwMnu//uWthuSZIGZd6R/qg4\n0h9oRetZb0H1JvW9N+nOZqTvbRgkqUEMfUlqEENfkhrE0JekBjH0JalBDH1JahBDX5IaxNCXpAYx\n9CWpQQx9SWoQQ1+SGsTQl6QG6ed++pIm2ge7NwUcjVWr1nHixM9HVk+/y7tsjoB32bSe9X633qS+\n10fNu2xKkuZk6EtSg/QV+kk2JzmU5HCSHWdoc1+SI0kOJLmmZ/7KJN9PcjDJi0k+NajOS5IWZt7Q\nT7IMuB+4HtgAbEty1Yw2NwCXVdUVwHbggZ7F9wKPVdUfAp8EDiJJGot+RvobgSNVdbSqTgJ7gC0z\n2mwBHgKoqqeAlUlWJbkI+GxVPdhddqqq3hpc9yVJC9FP6K8BjvVMH+/Om6vNa915HwPeSPJgkueS\n7Epy/mI6LEk6e8M+T385cC1wS1U9k+RbwF3AztkaT01NnX7darVotVpD7p4kLR3tdpt2u72obcx7\nnn6STcBUVW3uTt8FVFXd09PmAeDHVfW97vQh4HPdxX9fVZd25/8xsKOqvjhLHc/TH1xF61nvnK43\nqe/1URvWefr7gcuTrEuyAtgK7J3RZi9wU7cTm4A3q2q6qqaBY0k+3m13HfDSQjooSRqceQ/vVNV7\nSW4F9tH5kNhdVQeTbO8srl1V9ViSG5O8DLwD3NyziduBh5N8AHhlxjJJ0gh5G4YR8PCO9az3u/Um\n9b0+at6GQZI0J0NfkhrE0JekBjH0JalBDH1JahBDX5IaxNCXpAYx9CWpQQx9SWoQQ1+SGsTQl6QG\nMfQlqUEMfUlqEENfkhrE0JekBjH0JalB+gr9JJuTHEpyOMmOM7S5L8mRJAeSXDNj2bIkzyWZ+ZhF\nSdIIzRv6SZYB9wPXAxuAbUmumtHmBuCyqroC2A48MGMzd+CzcSVp7PoZ6W8EjlTV0ao6CewBtsxo\nswV4CKCqngJWJlkFkGQtcCPw7YH1WpJ0VvoJ/TXAsZ7p4915c7V5rafNN4GvMdqHcEqSZrF8mBtP\n8gVguqoOJGnReQLzGU1NTZ1+3Wq1aLVaw+yeJC0p7Xabdru9qG1kvqfSJ9kETFXV5u70XUBV1T09\nbR4AflxV3+tOHwI+R+dY/p8Bp4DzgQuBR6vqplnq1Hx9WaqSMNo/dKxnvXO73qS+10ctCVU152B6\npn4O7+wHLk+yLskKYCsw8yycvcBN3U5sAt6squmquruqLqmqS7vrPTFb4EuSRmPewztV9V6SW4F9\ndD4kdlfVwSTbO4trV1U9luTGJC8D7wA3D7fbkqSzMe/hnVHx8M5AK1rPeud0vUl9r4/a2RzeGeoX\nuZL0z32wOxAajVWr1nHixM9HVu9c50h/BBzpW8964603ydniSL8Pq1evZ3r66Li7IUkj18iRviNv\n61mvWfXOlZwbtGGdsilJmhCGviQ1iKEvSQ1i6EtSgxj6ktQghr4kNYihL0kNYuhLUoMY+pImXOde\nP6P4Wb16/bh3dl5ekTuaitaznvUaUW+0V/96Ra4kaU6GviQ1SF+hn2RzkkNJDifZcYY29yU5kuRA\nkmu689YmeSLJi0leSHL7IDsvSVqYeUM/yTLgfuB6YAOwLclVM9rcAFxWVVcA24EHuotOAXdW1Qbg\nj4BbZq4rSRqdfkb6G4EjVXW0qk4Ce4AtM9psAR4CqKqngJVJVlXViao60J3/NnAQWDOw3kuSFqSf\n0F8DHOuZPs4/D+6ZbV6b2SbJeuAa4KmFdlKSNBgjeXJWkguAR4A7uiP+WU1NTZ1+3Wq1aLVaQ++b\nJC0V7Xabdru9qG3Me55+kk3AVFVt7k7fBVRV3dPT5gHgx1X1ve70IeBzVTWdZDnwA+CHVXXvHHU8\nT9961rPeEq83Gefp7wcuT7IuyQpgK7B3Rpu9wE3dTmwC3qyq6e6y7wAvzRX4kqTRmPfwTlW9l+RW\nYB+dD4ndVXUwyfbO4tpVVY8luTHJy8A7wFcAknwG+FPghSTP0/m4vbuqfjSk/ZEkzcHbMIymovWs\nZ71G1JuMwzuSpAlh6EtSgxj6ktQghr4kNYihL0kNYuhLUoMY+pLUIIa+JDWIoS9JDWLoS1KDGPqS\n1CAjuZ9+v1au/Ffj7oIkTbRz6oZr8PoIKv0TsJ7JveGT9axnvfHVO/dvuHZOjfRhFCP9fxpBDUk6\nN3lMX5IapK/QT7I5yaEkh5PsOEOb+5IcSXIgyTULWVeSNBrzhn6SZcD9wPXABmBbkqtmtLkBuKyq\nrgC2Aw/0u64kaXT6GelvBI5U1dGqOgnsAbbMaLMFeAigqp4CViZZ1ee6kqQR6Sf01wDHeqaPd+f1\n06afdSVJIzKss3cWdArRb1100RcH3Y9ZvM9bb42gjCSdg/oJ/deAS3qm13bnzWzz0VnarOhj3dPe\neusHfXRnUM7qc8l61rOe9eaulFHv28L0E/r7gcuTrAN+AWwFts1osxe4Bfhekk3Am1U1neSNPtYF\nWPAFBpKkhZs39KvqvSS3AvvofAewu6oOJtneWVy7quqxJDcmeRl4B7h5rnWHtjeSpDmdM7dhkCQN\n39ivyJ3ki7eSrE3yRJIXk7yQ5PZx92nQkixL8lySvePuyzAkWZnk+0kOdv8fPzXuPg1Kkq8m+WmS\nnyR5OMmKcfdpMZLsTjKd5Cc98/5Fkn1JfpbkfyRZOc4+LsYZ9u+/dH83DyT52yQXzbedsYZ+Ay7e\nOgXcWVUbgD8Cbpmw/QO4A3hp3J0YonuBx6rqD4FPAhNxeDLJxcBtwLVV9Qk6h3q3jrdXi/YgnSzp\ndRfweFVdCTwB/MeR92pwZtu/fcCGqroGOEIf+zfukf5EX7xVVSeq6kD39dt0AmNirlNIsha4Efj2\nuPsyDN1R02er6kGAqjpVVZN0wu95wIeTLAc+xGhuczs0VfW/gV/PmL0F+Jvu678BvjTSTg3QbPtX\nVY9X1fvdyX+gc4bknMYd+o25eCvJeuAa4Knx9mSgvgl8jdHeJ3eUPga8keTB7iGsXUnOH3enBqGq\nXge+AbxK5zTqN6vq8fH2aij+oKqmoTMIA/5gzP0Zpj8Hfjhfo3GHfiMkuQB4BLijO+Jf8pJ8AZju\n/iUTRn/i9SgsB64F/rqqrgX+H53DBUteko/QGQWvAy4GLkjy5fH2aiQmcoCS5D8BJ6vqu/O1HXfo\n93Ph15LW/dP5EeC/VdXfjbs/A/QZ4E+SvAL8d+DfJHlozH0atOPAsap6pjv9CJ0PgUnweeCVqvpV\nVb0HPAp8esx9Gobp7n3ASLIa+D9j7s/AJfkKncOsfX1ojzv0T1/41T1zYCudC70myXeAl6rq3nF3\nZJCq6u6quqSqLqXz//ZEVd007n4NUvewwLEkH+/Ouo7J+dL6VWBTkt9L5xLS65iML6ln/tW5F/hK\n9/W/B5b6wOt39i/JZjqHWP+kqv6xnw2M9clZk37xVpLPAH8KvJDkeTp/Wt5dVT8ab8+0ALcDDyf5\nAPAK3QsPl7qqejrJI8DzwMnuv7vG26vFSfJdoAX8yySvAjuBrwPfT/LnwFHg342vh4tzhv27m87t\nbv5n9/YP/1BV/2HO7XhxliQ1x7gP70iSRsjQl6QGMfQlqUEMfUlqEENfkhrE0JekBjH0JalBDH1J\napD/D/xJ0zQxfDBJAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111901cf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(p,bins=mybins,normed=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, the matplotlib version is not a nice stem plot!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's generate some Gaussian random variables and compare them to their pdf:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "g = stats.norm.rvs(size=100000) # generate 100000 random numbers with normal distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gaussian random variables can take on any real value, so it does not make sense to count the values. Instead, we need to count the values that occur in intervals. Let's make 50 intervals from -4 to 4:20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mybins=np.linspace(-4,4,20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can use the numpy.histogram function to count the values in the bins:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(array([   17,    60,   206,   733,  1868,  4193,  7466, 11960, 15358,\n",
      "       16742, 15332, 11718,  7503,  3924,  1854,   740,   246,    62,    16]), array([-4.        , -3.57894737, -3.15789474, -2.73684211, -2.31578947,\n",
      "       -1.89473684, -1.47368421, -1.05263158, -0.63157895, -0.21052632,\n",
      "        0.21052632,  0.63157895,  1.05263158,  1.47368421,  1.89473684,\n",
      "        2.31578947,  2.73684211,  3.15789474,  3.57894737,  4.        ]))\n"
     ]
    }
   ],
   "source": [
    "ghist=np.histogram(g,bins=mybins)\n",
    "print(ghist)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1113e65f8>]"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x111249550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.step(ghist[1][:-1],ghist[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It looks like a blocky, scaled version of the pdf. If we normalize this in the right way, we can compare it to the pdf. The pdf does not show probabilities. It shows probability densities. So, instead of normalizing so that the values of the bins add to 1, we need to normalize so that the integral over the bins is equal to 1. Since the value over each bin is constant, then the integral over the bin is the value at that bin times the width of the bin. If we denote the bin values by $b_i$ and the bin width by $w$, then we want to find the normalizing constant $c$ such that\n",
    "$$\n",
    "c \\sum_{i} b_i w = 1 ,\n",
    "$$\n",
    "which implies that \n",
    "$$\n",
    "c=\\frac{1}{w \\sum_i b_i}\n",
    "$$\n",
    "Let's test this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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s/hID6wSMIyc/lZnd7Z/PT/UucAwwPM8hmoAGd09RggQpl0svhWefhcceix2J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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110f62dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "binwidth=mybins[1]-mybins[0]\n",
    "totalvals=sum(ghist[0])\n",
    "c=1/binwidth/totalvals\n",
    "\n",
    "plt.subplot(211) \n",
    "plt.step(ghist[1][:-1],ghist[0]*c)\n",
    "\n",
    "plt.subplot(212)\n",
    "x = np.linspace(-4, 4, 100)\n",
    "pdfvals = stats.norm.pdf(x)\n",
    "\n",
    "plt.plot(x, pdfvals)\n",
    "plt.ylabel('$f_X(x)$')\n",
    "plt.xlabel('$x$')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we increase the number of bins to 50, it looks better"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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h1XI+0yL639nq//JMJUYsVLbS78FVzrvvhg7aG28MHbYuPU47Dbp0gQsvTDoS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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111ee90f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mybins=np.linspace(-4,4,50)\n",
    "ghist=np.histogram(g,bins=mybins)\n",
    "\n",
    "\n",
    "binwidth=mybins[1]-mybins[0]\n",
    "totalvals=sum(ghist[0])\n",
    "c=1/binwidth/totalvals\n",
    "\n",
    "plt.subplot(211) \n",
    "plt.step(ghist[1][:-1],ghist[0]*c)\n",
    "\n",
    "plt.subplot(212)\n",
    "x = np.linspace(-4, 4, 100)\n",
    "pdfvals = stats.norm.pdf(x)\n",
    "\n",
    "plt.plot(x, pdfvals)\n",
    "plt.ylabel('$f_X(x)$')\n",
    "plt.xlabel('$x$')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, we have to be careful not to make the bins so small that there is not enough data points in the bins:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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wN9/Yd/Cii+DWW1O/9uuv23wqgIEDbXvllTav6rbb7NzSpTbJsVs3++527gx1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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111e76668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mybins=np.linspace(-4,4,1000)\n",
    "ghist=np.histogram(g,bins=mybins)\n",
    "\n",
    "\n",
    "binwidth=mybins[1]-mybins[0]\n",
    "totalvals=sum(ghist[0])\n",
    "c=1/binwidth/totalvals\n",
    "\n",
    "plt.subplot(211) \n",
    "plt.step(ghist[1][:-1],ghist[0]*c)\n",
    "\n",
    "plt.subplot(212)\n",
    "x = np.linspace(-4, 4, 100)\n",
    "pdfvals = stats.norm.pdf(x)\n",
    "\n",
    "plt.plot(x, pdfvals)\n",
    "plt.ylabel('$f_X(x)$')\n",
    "plt.xlabel('$x$')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As before, we can generate a histogram directly from matplotlib:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x111f02940>]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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PCe3G2o6jgox+Vg9jyw95blU3KnUaPbpOa9i/cqWugR5o+saPpENUV044i9lb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\nKaVCnBZ6pZQKcf8fc7Hn2RDG+/kAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111f02278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(g,bins=50,normed=True,alpha=0.3) # add some transparency\n",
    "plt.plot(x, pdfvals,'g-',lw=2) # bump up the line width"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, the emperical estimate (from the data) matches the pdf well."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that we can also generate a cumulative histogram and compare to the CDF:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1122fad68>]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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s0I8///nf6TNiYNxzHYrS0jW+I+xl2bIShg0r8h1jD8oUu0TMlYiZYhXLCH04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\nIpIiVOgiIini/wNMH5sLRYat+wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1122a0438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(g,bins=50,normed=True,cumulative=True,alpha=0.3) # add some transparency\n",
    "cdfvals = stats.norm.cdf(x)\n",
    "\n",
    "\n",
    "plt.plot(x, cdfvals,'g-',lw=2) # bump up the line width"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": 1
}