{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Gaussian Distribution (Normal or Bell Curve)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Think of a Jupyter Notebook file as a Python script, but with comments given the seriousness they deserve, meaning inserted Youtubes if necessary.  We also adopt a more conversational style with the reader, and with Python, pausing frequently to take stock, because we're telling a story.\n",
    "\n",
    "One might ask, what is the benefit of computer programs if we read through them this slowly?  Isn't the whole point that they run blazingly fast, and nobody needs to read them except those tasked with maintaining them, the programmer cast?\n",
    "\n",
    "First, lets point out the obvious:  even when reading slowly, we're not keeping Python from doing its part as fast as it can, and what it does would have taken a single human ages to do, and would have occupied a team of secretaries for ages.  Were you planning to pay them?  Python effectively puts a huge staff at your disposal, ready to do your bidding.  But that doesn't let you off the hook.  They need to be managed, told what to do.\n",
    "\n",
    "Here's what you'll find at the top of your average script.  A litany of players, a congress of agents, need to be assembled and made ready for the job at hand.  But don't worry, as you remember to include necessary assets, add them at will as you need them.  We rehearse the script over and over while building it.  Nobody groans, except maybe you, when the director says \"take it from the top\" once again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.stats as st\n",
    "import matplotlib.pyplot as plt\n",
    "import math"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You'll be glad to have np.linspace as a friend, as so often you know exactly what the upper and lower bounds, of a domain, might be.  You'll be computing a range.  Do you remember these terms from high school?  A domain is like a pile of cannon balls that we feed to our cannon, which them fires them, testing our knowledge of ballistics.  It traces a parabola.  We plot that in our tables.  A lot of mathematics traces to developing tables for battle field use.  Leonardo da Vinci, a great artist, was also an architect of defensive fortifications.\n",
    "\n",
    "Anyway, np.linspace lets to give exactly the number of points you would like of this linear one dimensional array space, as a closed set, meaning -5 and 5 are included, the minimum and maximum you specify.  Ask for a healthy number of points, as points are cheap.  All they require is memory.  But then it's up to you not to overdo things.  Why waste CPU cycles on way too many points?\n",
    "\n",
    "I bring up this niggling detail about points as a way of introducing what they're calling \"hyperparameters\" in Machine Learning, meaning settings or values that come from outside the data, so also \"metadata\" in some ways.  You'll see in other notebooks how we might pick a few hyperparameters and ask scikit-learn to try all combinations of same.\n",
    "\n",
    "Here's what you'll be saying then:\n",
    "\n",
    "from sklearn.model_selection import GridSearchCV  #CV = cross-validation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "domain = np.linspace(-5, 5, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I know mu sounds like \"mew\", the sound a kitten makes, and that's sometimes insisted upon by sticklers, for when we have a continuous function, versus one that's discrete.  Statisticians make a big deal about the difference between digital and analog, where the former is seen as a \"sampling\" of the latter.  Complete data may be an impossibility.  We're always stuck with something digital trying to approximate something analog, or so it seems.  Turn that around in your head sometimes:  we smooth it over as an approximation, because a discrete treatment would require too high a level of precision.\n",
    "\n",
    "The sticklers say \"mu\" for continuous, but \"x-bar\" (an x with a bar over it) for plain old \"average\" of discrete sets.  I don't see this conventions holding water necessarily, for one thing because it's inconvenient to always reach for the most fancy typography.  Python does have full access to Unicode, and to LaTex, but do we have to bother?  Lets leave that question for another day and move on to...\n",
    "\n",
    "## The Guassian (Binomial if Discrete)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "mu = 0  # might be x-bar if discrete\n",
    "sigma = 1  # standard deviation, more below"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What we have here (below) is a typical Python numeric function, although it does get its pi from numpy instead of math.  That won't matter.  The sigma and mu in this function are globals and set above.  Some LaTex would be in order here, I realize.  Let me scavange the internet for something appropriate...\n",
    "\n",
    "$pdf(x,\\mu,\\sigma) = \\frac{1}{ \\sigma \\sqrt{2 \\pi}} e^{\\left(-\\frac{{\\left(\\mu - x\\right)}^{2}}{2 \\, \\sigma^{2}}\\right)}$\n",
    "\n",
    "Use of dollar signs is key.\n",
    "\n",
    "Here's another way, in a code cell instead of a Markdown cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$ pdf(x,\\mu,\\sigma) = \\frac{1}{ \\sigma\\sqrt{2 \\pi}} e^{\\left(-\\frac{{\\left(\\mu - x\\right)}^{2}}{2 \\, \\sigma^{2}}\\right)} $"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from IPython.display import display, Latex\n",
    "\n",
    "ltx = '$ pdf(x,\\\\mu,\\\\sigma) = \\\\frac{1}{ \\\\sigma' + \\\n",
    " '\\\\sqrt{2 \\\\pi}} e^{\\\\left(-\\\\frac{{\\\\left(\\\\mu - ' + \\\n",
    " 'x\\\\right)}^{2}}{2 \\\\, \\\\sigma^{2}}\\\\right)} $'\n",
    "display(Latex(ltx))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I'm really tempted to try out [PrettyPy](https://github.com/charliekawczynski/prettyPy)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "def g(x):\n",
    "    return (1/(sigma * math.sqrt(2 * np.pi))) * math.exp(-0.5 * ((mu - x)/sigma)**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What I do below is semi-mysterious, and something I'd like to get to in numpy in more detail.  The whole idea behind numpy is every function, or at least the unary ones, are vectorized, meaning the work element-wise through every cell, with no need for any for loops.\n",
    "\n",
    "My Gaussian formula above won't natively understand how to have relations with a numpy array, unless we store it in vectorized form.  I'm not claiming this will make it run any faster than under the control of for loops, we can test that.  Even without a speedup, here we have a recipe for shortening our code.\n",
    "\n",
    "As many have proclaimed around numpy: one of its primary benefits is it allows one to \"lose the loops\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The slowest run took 5.55 times longer than the fastest. This could mean that an intermediate result is being cached.\n",
      "100000 loops, best of 3: 4.1 µs per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit vg = np.vectorize(g)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At any rate, this way, with a list comprehension, is orders of magnitude slower:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1000 loops, best of 3: 263 µs per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit vg2 = np.array([g(x) for x in domain])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "vg = np.vectorize(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The slowest run took 89.97 times longer than the fastest. This could mean that an intermediate result is being cached.\n",
      "1 loop, best of 3: 2.49 ms per loop\n"
     ]
    },
    {
     "data": {
      "image/png": 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dHdc3QWelb8JWybIlAIy64lbHSYKvbnB/ulTByjf+6jqKiTBW+iZsdd6xjz3dYciYKa6j\nBN2gqdcBsOttO5lrgstK34Slw2WlpBc1UJKR6DpKmxhz9hWUJkOH3ELXUUyEsdI3YWnZwidIqIPY\nkSNcR2kTsR4PezISSCv0Uldb4zqOiSABlb6ITBWRLSKSKyJ3N/P690Rko4isE5F3RKSf32v1IrLG\n91jUdFtjTsTBFe/SAIz76hzXUdpM/ZBBJB2BZS8/5TqKiSAtlr6IxAJzgWnAcGCWiAxvMuwzIFtV\nRwEvAg/5vVatqmN8j+kYEwTJu0opToX+w7JdR2kzQy/7BgB7PnzNcRITSQLZ058A5KpqnqrWAvOB\nGf4DVPU9Va3yLa4A0oMb05j/c+hAMWlFDRzI7Ow6SpsadeallHSFTtuLXUcxESSQ0k8DCvyWC33r\njuVm4HW/5Q4ikiMiK0Tk8hPIaMx/+Gj+o8TXQ/zo01xHaXP7MjqQtrue6spy11FMhAik9KWZddrs\nQJHrgGzgYb/VmaqaDVwDPC4iR90ZS0Rm+/4w5JSUlAQQyUSzwznLaRCYcNXtrqO0OR02lE61jT9E\nMyYYAin9QiDDbzkdKGo6SEQuAH4CTFfVLy83UNUi3795wPvA2Kbbquo8Vc1W1ezU1NRWfQETfbrm\nH2R3TyEtq+mppcgzcsZ/A7B/+ZuOk5hIEUjprwIGi0iWiMQDM4H/uApHRMYCT9NY+Pv81qeISILv\neQ9gErAxWOFN9CndW0BasXKwX2Qfz//C0HHnsbcbJO7Y6zqKiRAtlr6qeoE5wBJgE7BAVTeIyP0i\n8sXVOA8DScDCJpdmDgNyRGQt8B7wa1W10jcnbNn8R/A0QIfRE1xHaTclGR1J391AZUWZ6ygmAgQ0\n1ZCqLgYWN1l3r9/zC46x3XJg5MkENMZf5aef4I2BM2Z+13WUdiOnjqDD2lV8tPAJLr7p3pY3MOY4\n7Be5Jqyk5JdR1EvomRYZM2UFYqzvhnKlK95xnMREAit9Ezb25G8jbY9ysF8X11Ha1cCRZ1LcA5J3\n7ncdxUQAK30TNj5Z+DixComnneE6Srvbn5FIWlED5Qf3tTzYmOOw0jdh48iaHLwxMOnqO11HaXex\nI0aR4G280ZwxJ8NK34SNbvkVFPYRuvXKaHlwhBl/5RwagLKVH7iOYsKclb4JC7t3bKTvPqWsX4rr\nKE5kDjmN4lRI3lnqOooJc1b6JiysXPAEMQqdsye7juLMgcwk0oobOHTAbsBmTpyVvgkLdes+pdYD\nk66Onuvzm4ofPY74+sYbzhlzoqz0TVjonn+Ywr4xdO3ex3UUZyZe/V3qpfGGc8acKCt9E/LyNq6k\nbwmU9+/hOopTffoNZXcvoeuug66jmDBmpW9C3mcv/gGAbhPPd5zEvYP9upC2R9m3e7vrKCZMWemb\nkNewYT3V8TD5qjtcR3Eu8bTT8TTAx/Mfdx3FhCkrfRPyUguq2Z0WS2Ln6Lr9QnMmzfw+3hg4smal\n6ygmTFnpm5C2adXb9CqFygG9XEcJCd16ZTT+QC2/wnUUE6as9E1IW//vZwDoddaljpOEjrL+3em7\nT8nfttZ1FBOGrPRNSJPNW6nsAJO/epvrKCEjefxZxCjkvPh711FMGAqo9EVkqohsEZFcEbm7mde/\nJyIbRWSdiLwjIv38XrtBRLb5HjcEM7yJfL0KjlCY7iEuPsF1lJAx+ervUuMB7+e2p29ar8XSF5FY\nYC4wDRgOzBKRpjNSfwZkq+oo4EXgId+23YD7gNOBCcB9IhKdN08xrfbZ0pfpUQZHBqW5jhJSklN6\nUtg3hh75h11HMWEokD39CUCuquapai0wH5jhP0BV31PVKt/iCiDd9/xi4C1VLVXVg8BbwNTgRDeR\nbutrfwUg87yvOU4SeioGpNJnP2xZY3fdNK0TSOmnAQV+y4W+dcdyM/D6CW5rzJfit+RxKBFOv+Qb\nrqOEnJ5nXQLAupeecpzEhJtASl+aWafNDhS5DsgGHm7NtiIyW0RyRCSnpKQkgEgm0tXV1tC3oI7i\nzHhiPR7XcULO5K/OoTIBZONm11FMmAmk9AsB/1kr0oGipoNE5ALgJ8B0Va1pzbaqOk9Vs1U1OzU1\nNdDsJoJ9vOgZkqvAO3SQ6yghKaFjJwozPPTJP0K91+s6jgkjgZT+KmCwiGSJSDwwE1jkP0BExgJP\n01j4/pN4LgEuEpEU3wnci3zrjDmu3e//C4DhM77pOEnoOnJKJt3K4dN3F7qOYsJIi6Wvql5gDo1l\nvQlYoKobROR+EZnuG/YwkAQsFJE1IrLIt20p8ACNfzhWAff71hlzXEm5xeztBiMmTnMdJWQNmDoL\ngO1v/M1xEhNOAjpYqqqLgcVN1t3r9/yC42z7LPDsiQY00edwWSkZuxvYemon11FC2rjzZrKi8y/p\nsDXfdRQTRuwXuSbkfPiPR0ioA8+oMa6jhLRYj4fizATSCr3UVFe1vIExWOmbEFS24n0aBMZffafr\nKCGvYfhQko7AspefdB3FhAkrfRNyuu48yO5eQsbAEa6jhLzRX70VgL0fvOY4iQkXVvompBTv2kx6\nsVLa3+6dH4ghY8+huAck5+1rebAxWOmbELNi/mPEKiRlT3IdJWzs75dEelEDB0t2u45iwoCVvgkp\ndZ/lUBMHU669y3WUsBE/djzxXvjw+YdcRzFhwErfhJTeO6soSI8lOaWn6yhhY/K1P6QuFqpXfew6\nigkDVvomZKx+ZyGph6BysN2TrzV69MmiIC2GnjttCkXTMit9EzK2vfYXAPpPu8ZtkDBUMagXvQ/A\n+hWvtzzYRDUrfRMyOm7ZxYFkGH/hta6jhJ2+530VgI0vz3OcxIQ6K30TEiorysgoqKe4X0e7lfIJ\nmHT5rRxKhLhN211HMSHOSt+EhKXP/4aOtRAzZpTrKGEp1uOhqF8CGfl1dksGc1xW+iYklK94j3qB\nidfYpZonqmHEMBKPwNJ/POo6iglhVvomJHTfUUZhXyEta7jrKGFr3Mw7aQBKP7IpK8yxWekb57Z/\nvpy+e5VDA3q4jhLWBgyfwO5eQte8A66jmBBmpW+cW73gd8QAPaZc4jpK2CsdkEJGsZK/ba3rKCZE\nBVT6IjJVRLaISK6I3N3M61NE5FMR8YrIlU1eq/fNpvXljFrG+PN8vonyTnDWlbe7jhL2UiZfSIzC\nSrslgzmGFktfRGKBucA0YDgwS0SaHnjNB24EXmjmLapVdYzvMb2Z100Uq64sJ3NnLYX9E0joaDNl\nnaxzrrmLio4Qs3a96ygmRAWypz8ByFXVPFWtBeYDM/wHqOpOVV0HNLRBRhPBlr7wMIlHALtUMygS\nOnaioH886Ttr7dJN06xASj8NKPBbLvStC1QHEckRkRUicnlzA0Rktm9MTklJSSve2oS7sg/fwhsD\nE68/6qihOUE6egSdq+H9F+wQjzlaIKUvzazTVnxGpqpmA9cAj4vIwKPeTHWeqmaranZqamor3tqE\nu57by8hPi7FLNYPo9OvvoV7g4Ad26aY5WiClXwhk+C2nA0WBfoCqFvn+zQPeB8a2Ip+JYJ8tfZne\nB6BiSF/XUSJKxsAR5KfFkJp3yHUUE4ICKf1VwGARyRKReGAmENBVOCKSIiIJvuc9gEnAxhMNayLL\nFt/NwbIuu95xkshTfkpv+pbAumWvuI5iQkyLpa+qXmAOsATYBCxQ1Q0icr+ITAcQkfEiUghcBTwt\nIht8mw8DckRkLfAe8GtVtdI3ACRuzmdfCpw+9b9cR4k4mZdeB8DGf/7RcRITagK6naGqLgYWN1l3\nr9/zVTQe9mm63XJg5ElmNBFof/EOMgsa2DwmyXWUiHT6xdez7IGHSNy0y3UUE2LsF7nGiQ//+ivi\n6yHxjLNdR4lIsR4PewYkkVlQT+negpY3MFHDSt84Ub9qJVXxcM71P3YdJWJ1nHgW8V54/9mfu45i\nQoiVvml31ZXlZObVsGtAPEldurmOE7HO+8Z9VHYAXZXjOooJIVb6pt2985cH6FwNOm6M6ygRLbFz\nF3YNSKDf9hoqK8pcxzEhwkrftLuqD9+j1gNnf/NnrqNEvJgJE0isgXefvc91FBMirPRNu6qrrSE9\nt5Kd/Tz06JPlOk7EO+fm+zgSB0eWf+A6igkRVvqmXb33wsOkHIaaMUNdR4kKKalp7OwfR8a2arsB\nmwGs9E07O/juK403WLvxp66jRA3vaSPoUgXvP/8b11FMCLDSN+2m3uulz9ZydmXEkDl4tOs4UWPy\nTfdSFwuH3nvddRQTAqz0TbtZvuhpUg9B5Qg7lt+e+vQbyq7MWNK2VVDv9bqOYxyz0jftZvdr82kA\nxlz3A9dRok7VyEF0L4elC59wHcU4ZqVv2kW910vvTfvJTxOGjD3HdZyoc/o3foo3BkoWL3AdxThm\npW/axdKFT9CrFMrGHjWHjmkH/Ydls6NfLBmby6mrrXEdxzhkpW/aRclr/2i8audm+5GQKzXjR5JS\nAW/ZvXiimpW+aXN1tTVkbK5gR/9Y+g/Ldh0nap37rQepiYPKd99wHcU4ZKVv2tybf/pZ4w+yxo9y\nHSWq9eiTRd7AePpvreZwWanrOMaRgEpfRKaKyBYRyRWRu5t5fYqIfCoiXhG5sslrN4jINt/jhmAF\nN+Gj6t0lHImDc2/9pesoUS9m0hkkHYG359ktraNVi6UvIrHAXGAaMByYJSLDmwzLB24EXmiybTfg\nPuB0YAJwn4iknHxsEy4Ol5WSta2aHQPj7V47IeCC2Q9yuAM0LPvYdRTjSCB7+hOAXFXNU9VaYD4w\nw3+Aqu5U1XVAQ5NtLwbeUtVSVT0IvAVMDUJuEybe+uM9JB6BmMmTXEcxQFKXbuw8pSMDtteyv3iH\n6zjGgUBKPw3wn2+t0LcuEAFtKyKzRSRHRHJKSkoCfGsTFj76mIqOcOEtv3KdxPgknjeVhDp49w8/\nch3FOBBI6Usz6zTA9w9oW1Wdp6rZqpqdmpoa4FubUJe3cSWDttexc3gSiZ27uI5jfC686T5KukLS\nJxtcRzEOBFL6hUCG33I6UBTg+5/MtibMrXzyJ3gaoM+VN7mOYvzExSdQNLoXWYUNrHzzb67jmHYW\nSOmvAgaLSJaIxAMzgUUBvv8S4CIRSfGdwL3It85EuHqvl16fFpLfRzjrim+5jmOaGDP7XuoFdvzt\nD66jmHbWYumrqheYQ2NZbwIWqOoGEblfRKYDiMh4ESkErgKeFpENvm1LgQdo/MOxCrjft85EuLf/\n+gt6l8Kh8ae4jmKaMXTceWwfEEu/9WU2f26UCeg6fVVdrKqnqOpAVf2lb929qrrI93yVqqaraqKq\ndlfVU/22fVZVB/kef26br2FCTcXif1EdD+fe8YjrKOYYGs6eTJcqWPL777mOYtqR/SLXBF3xrs0M\n2lpD3ikd6JlmN1gLVVNve4hDSeD58BPXUUw7stI3Qbf09z8koQ6SL/ua6yjmODomJrNrZAoDd9az\nfoXNqhUtrPRNUNV7vfT4JJfiHnD+dUfdscOEmMHX3QkK65/+hesopp1Y6ZugemPeT0grgZKJg4n1\neFzHMS0Yd/5VbB8QS9baUg4dKHYdx7QDK30TVN5XX6OiI1z0o6dcRzEBirl0GslVsOQ3dmltNLDS\nN0Gz6s2/Myivnrwx3UhJDfROHca1abf8iqJU6Llsi02cHgWs9E3Q7Hj2ERpiYMycB1xHMa0Q6/Fw\nYPJweh+AxXNt0vpIZ6VvgqJg+3oGb6hk69AEho47z3Uc00pT73qKsk7A62+5jmLamJW+CYplj9xO\nhzrofvWNrqOYE5Cc0pOdp/Vg0M4GPn71T67jmDZkpW9O2sGS3fRbWcyOjBjOufq7ruOYEzT+jt9Q\n64GiZ3/vOoppQ1b65qQtuf8bdD0MesVlrqOYkzBw5JlsHp3E0E01dvfNCGalb07KwZLd9F9ewI70\nGC799m9cxzEnacwPH6EuFgqeeth1FNNGrPTNSVny8xvpUglcOaPFsSb0DRkzhS1jkhm6uZYVr//V\ndRzTBqz0zQnbX7yDrOWF5GXGcMmtD7qOY4LktB8+Sp0Hip5+1HUU0was9M0Je/uBb5JcBZ6vX+k6\nigmiwaMnsWVsMkM217J80TOu45ggC6j0RWSqiGwRkVwROeouWiKSICL/8L3+iYj0963vLyLVIrLG\n9/hjcOMbV/K3fMqAj4vYnhnDxd/8ues4JsjG3/17auKh5I9PuI5igqzF0heRWGAuMA0YDswSkeFN\nht0MHFTVQcBjgP8Zve2qOsZXF68XAAAMZUlEQVT3uDVIuY1jK+6bTeIRSJ5t92uJRAOGT2DLmb04\nJa+eRY/f4TqOCaJA9vQnALmqmqeqtcB8oOlZuxnAF2d9XgTOFxEJXkwTSj58aS6nrq1kw+hOTLly\njus4po1c+qsFlHSF5AVv2pSKESSQ0k8DCvyWC33rmh3jm1O3DOjuey1LRD4TkaUictZJ5jWO1Xu9\nVDz5JFUJMPHndrw3kiWn9OTAV8+mVym8co+dt4kUgZR+c3vsGuCYYiBTVccC3wNeEJHkoz5AZLaI\n5IhITklJSQCRjCsvP3gTWYUNbD9vAJlDTnMdx7Sx6d/7A7n9Yxj8QSFb1nzgOo4JgkBKvxDI8FtO\nB4qONUZEPEAXoFRVa1T1AICqrga2A6c0/QBVnaeq2aqanZqa2vpvYdrF7h0b6fPKKopS4YpfLnQd\nx7SDWI+H1DvuIt4LG37+HddxTBAEUvqrgMEikiUi8cBMYFGTMYuAG3zPrwTeVVUVkVTfiWBEZAAw\nGMgLTnTT3lb84Bq6Hob6m68loWMn13FMO5k47QY2TOzGsE21/POXN7qOY05Si6XvO0Y/B1gCbAIW\nqOoGEblfRKb7hv0J6C4iuTQexvniss4pwDoRWUvjCd5bVbU02F/CtL2XHryZ4Rtq+HxiNy668aeu\n45h29pUnFlPYEzJe/MQO84Q5UW16eN6t7OxszcnJcR3D+Nm2dhml3/gmFUkw8ZVlJHXp5jqScWD5\nomdIvOdRdvWL5dJFa2wO5BAjIqtVNbulcfaLXHNc9V4vG398Cx1qoMP377TCj2JnTv9vNp2bzuC8\nel78sV3NE66s9M1xLbztAk7ZXs/Gc9OYPGO26zjGsa89+ip5GTEMXbyFd55/yHUccwKs9M0xvfyb\nWxi5dC+bhsTx9cffcB3HhIC4+ASGPvoMFZ2g42N/ZvPqd11HMq1kpW+atXzRM/R7/gOKesI5816z\n47fmSwNHnon3h9+mUw3s+sEcyg/ucx3JtIKVvjnKljUf4H3wUWo90OvXv6Vbr4yWNzJR5ZyrvsP2\nq8aTWay8fdOF1NXWuI5kAmSlb/7D9s+XU3TbLSRXQtmcWYw681LXkUyI+tq9z7FmSirDNtXy72sn\nWvGHCSt986W8jSvZ9a2bSSmD3d+azsU33es6kglxX3/yXdaekcKpnx/hX9efQb3X6zqSaYGVvgFg\ny2fvk3fLDXQ/BAWzp9l8tyYgsR4PVz3zAesmJDNibTUvXTuBmuoq17HMcVjpG95f+Hv2z/4WPQ7B\nrpsv5Cu32zR5JnCxHg9XPrvsy+J/86sTKN612XUscwxW+lHupQdvpsv9TyIKpT/+JtPv/J3rSCYM\nxXo8XP3cJ6y9ZABZO+vZfM0VrH7HbsoXiqz0o9TBkt3849rxDHluOSXdhW5Pz+W8Wd93HcuEuZmP\nvsbO2y6lcyXo9+9lwf+bacf5Q4yVfhR68y+/YO2MCxi1+jAbR3di/MK3GDruPNexTIS4bM5v0Yfu\n5UCKMHLhWl65fCzbP1/uOpbxsdKPIptXv8vCq04j7TfPE1cHud+exlX/WE1KatOJ0Iw5OeMvmsU5\ni3NYc05vBuzwcuC/bmb+dy7icJndZNc1u8tmFMjf8inLf/0dhuaUElsPm0YnMunB50jLajq/vTHB\n9+HLT1H+5B8YUNDA/i5QfNFovnL3PDomHjWJnjkJgd5l00o/gn3073kU/+2PDN5UTYIXNg+JI+P2\nH5N9/kzX0UyUqfd6efWJO0n619v0LYHSzpA/rjdn3PEw/Ye12FMmAFb6UWrd8tfY8MJjdFtfROYe\npdYD24Z2oOfMm5ly5RzX8UyUq6mu4rXHbyfh3Y8ZUNBAbSzsyPJQN24kZ89+gJ5pA11HDFtBLX0R\nmQo8AcQC/6Oqv27yegLwHDAOOABcrao7fa/dA9wM1AO3q+qS432WlX7r7N6xkZyX5nLks1V031VB\nmm9e+cKecGBEHybM+TUDhk9wG9KYZnz40lyKF/6ZjK2VdK2EWg8U9o2hfEAqqWdexMQZt9r8Da0Q\ntNL3zXG7FbiQxgnQVwGzVHWj35hvA6NU9VYRmQlcoapXi8hw4O/ABKAv8DZwiqrWH+vzrPSbd7Bk\nN+s/WsS+9SuoLdhJh70H6b63jtRDja/Xxf7f/2EGfO2bTLjoOreBjQlQXW0Nbz37cw5/+BY9dh2m\nz37f+ljY20M41LMD9b170LH/EDJPO4ehEy608wHNCGbpnwH8TFUv9i3fA6Cqv/Ibs8Q35mMR8QB7\ngFR8c+V+MdZ/3LE+LxJLv662hsqKUqoPl3H40AGqD5dSXXGQqkP7OVJ+gNqKcuoqSvEeLofKSuRw\nJZ6qGuKrvHQ8XE/yYehc/Z/vWZoMJT09VPftRtKIcUye9X27CsdEhC1rPuDzf82jfvs2kvYcpte+\nBhL97uXWIFCWCBWdherEWOo6xlGf1BESOyFJSXg6dyE+qSvxnbvRMbkbHZO70zG5K0ldUumUnEKn\npBQ6JiZH3O3CAy39QL51GlDgt1wInH6sMarqFZEyoLtv/Yom27ZJM+Vv+ZTt37i2xXES4PuJHv3c\n/18BUN9z3yPG95AGiPX962loXOcvzvc41r5KbSwc7gRVHaEqKZaDfePxdumMp3dfug/LZsSUyxlm\nxz5NhBoyZgpDxkz5crne6yV3/XK2r3iDiu0b0P37iSuvpENFHcmHvCQWeUmqriZGj385aLXvccC3\n7I2BhhiojwGVxj8mDb7nXz5o/Jfj/Nv0+bEEcvb0UGocV/x7XQAjT1wgpd/c12ma/1hjAtkWEZkN\nzAbIzMwMINLR4jp04lD3uMAGH+9/oCavfRlW5MvXtelzAUUgBlRiIEbQGIGYGDQ2BmJjUU8sEhsL\ncfFIfAIx8QnEdkoiLrEz8UldSezRl5Te/UlNH0j3Xv0ibi/EmBMV6/Ec9YegqbraGvYUbGN/4TbK\n9uyiuqyE2opyvJVl1NdUozU1NNTVInW1qLceqa8Hbz2iDVCvSEMDqCKqSIP62l6/3Ln7j+c0LkMz\nVXKsZg/wepmaromBDTwJgTRLIeA/i0Y6UHSMMYW+wztdgNIAt0VV5wHzoPHwTqDh/fXpN5QrXmnb\nv5DGmNAUF59AxsARZAwc4TpKyAvkF7mrgMEikiUi8cBMYFGTMYuAG3zPrwTe1caTBYuAmSKSICJZ\nwGBgZXCiG2OMaa0W9/R9x+jnAEtovGTzWVXdICL3Azmqugj4E/C/IpJL4x7+TN+2G0RkAbAR8AK3\nHe/KHWOMMW3LfpxljDERINCrd+yGa8YYE0Ws9I0xJopY6RtjTBSx0jfGmChipW+MMVEk5K7eEZES\nYJfrHCegB7DfdYh2Zt85Oth3Dg/9VDW1pUEhV/rhSkRyArlcKpLYd44O9p0jix3eMcaYKGKlb4wx\nUcRKP3jmuQ7ggH3n6GDfOYLYMX1jjIkitqdvjDFRxEq/DYjID0RERaSH6yxtTUQeFpHNIrJORF4W\nka6uM7UFEZkqIltEJFdE7nadp62JSIaIvCcim0Rkg4jc4TpTexGRWBH5TERedZ2lLVjpB5mIZNA4\niXy+6yzt5C1ghKqOArYC9zjOE3QiEgvMBaYBw4FZIjLcbao25wW+r6rDgInAbVHwnb9wB7DJdYi2\nYqUffI8BdxHwBGnhTVXfVFWvb3EFjbOjRZoJQK6q5qlqLTAfmOE4U5tS1WJV/dT3vILGEmyT+a1D\niYikA5cC/+M6S1ux0g8iEZkO7FbVta6zOHIT8LrrEG0gDSjwWy4kCgrwCyLSHxgLfOI2Sbt4nMad\ntgbXQdqKzb7dSiLyNtC7mZd+AvwYuKh9E7W9431nVf23b8xPaDwk8Hx7ZmsnR81/TZT8l5yIJAH/\nBL6rquWu87QlEbkM2Keqq0XkHNd52oqVfiup6gXNrReRkUAWsFZEoPEwx6ciMkFV97RjxKA71nf+\ngojcAFwGnK+ReQ1wIZDht5wOFDnK0m5EJI7Gwn9eVV9ynacdTAKmi8glQAcgWUT+pqrXOc4VVHad\nfhsRkZ1AtqqG202bWkVEpgKPAmeraonrPG1BRDw0nqQ+H9gNrAKuUdUNToO1IWncc/krUKqq33Wd\np7359vR/oKqXuc4SbHZM35ysPwCdgbdEZI2I/NF1oGDznaieAyyh8YTmgkgufJ9JwPXAeb7/Xdf4\n9oBNmLM9fWOMiSK2p2+MMVHESt8YY6KIlb4xxkQRK31jjIkiVvrGGBNFrPSNMSaKWOkbY0wUsdI3\nxpgo8v8BnX5Uh+g5oJYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a11ab6b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "%timeit plt.plot(domain, vg(domain))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I bravely built my own version of the Gaussian distribution, a continuous function (any real number input is OK, from negative infinity to infinity, but not those (keep it in between). The thing about a Gaussian is you can shrink it and grow it while keeping the curve itself, self similar.  Remember \"hyperparamters\"?  They control the shape.  We should be sure to play around with those parameters.\n",
    "\n",
    "Of course the stats.norm section of scipy comes pre-equipped with the same PDF (probability distribution function).  You'll see this curve called many things in the literature."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The slowest run took 28.99 times longer than the fastest. This could mean that an intermediate result is being cached.\n",
      "100 loops, best of 3: 2.7 ms per loop\n"
     ]
    },
    {
     "data": {
      "image/png": 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dxQRY/8JbjMuqpq0+jgdKlrqOY/pZMKW/BSgQkTwRSQRmARuCfP+XgC+LSLb/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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a192c22b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%timeit plt.plot(domain, st.norm.pdf(domain))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Gaussian Distributions')"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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d9uhwh1IpdePqcmbKmVXvEllcDNdei6dtO/5S8EDYF8U+nJQUuDH/cbRJE7j2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\nhSy5G2NMFPp/NZOJNL0Ec/8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a19872940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mu = 0\n",
    "sigma = math.sqrt(0.2)\n",
    "plt.plot(domain, vg(domain), color = 'blue')\n",
    "sigma = math.sqrt(1)\n",
    "plt.plot(domain, vg(domain), color = 'red')\n",
    "sigma = math.sqrt(5)\n",
    "plt.plot(domain, vg(domain), color = 'orange')\n",
    "mu = -2\n",
    "sigma = math.sqrt(.5)\n",
    "plt.plot(domain, vg(domain), color = 'green')\n",
    "plt.title(\"Gaussian Distributions\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[see Wikipedia figure](https://en.wikipedia.org/wiki/Gaussian_function#Properties)\n",
    "\n",
    "These are Gaussian PDFs or Probability Density Functions.\n",
    "\n",
    "68.26% of values happen within -1 and 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/jpeg": 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      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"400\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/xgQhefFOXrM\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x10f4026d8>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo(\"xgQhefFOXrM\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.68268949213708585"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = st.norm.cdf(-1)  #  Cumulative distribution function\n",
    "b = st.norm.cdf(1)\n",
    "b - a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.95449973610364158"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = st.norm.cdf(-2)\n",
    "b = st.norm.cdf(2)\n",
    "b - a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.99730020393673979"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 99.73% is more correct than 99.72% \n",
    "a = st.norm.cdf(-3)\n",
    "b = st.norm.cdf(3)\n",
    "b - a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.95000420970355903"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 95%\n",
    "a = st.norm.cdf(-1.96)\n",
    "b = st.norm.cdf(1.96)\n",
    "b - a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9901199684844586"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 99% \n",
    "a = st.norm.cdf(-2.58)\n",
    "b = st.norm.cdf(2.58)\n",
    "b - a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\n",
       "        <iframe\n",
       "            width=\"400\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/zZWd56VlN7w\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x1a193bb828>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo(\"zZWd56VlN7w\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What are the chances a value is less than -1.32?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.093417508993471787"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "st.norm.cdf(-1.32)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What are the chances a value is between -0.21 and 0.85?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.41683383651755768"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - st.norm.sf(-0.21) # filling in from the right (survival function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.80233745687730762"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = st.norm.cdf(0.85) # filling in from the left\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.41683383651755768"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = st.norm.cdf(-0.21) # from the left\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.38550362035974994"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a-b # getting the difference (per the Youtube)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets plot the integral of the Bell Curve.  This curve somewhat describes the temporal pattern whereby a new technology is adopted, first by early adopters, then comes the bandwagon effect, then come the stragglers.  Not the every technology gets adopted in this way.  Only some do."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a1a20d320>]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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WPq2LhEkIqNxFujxaVcPwQbl8dIZWW5LMp3IXAfY0R3j6rR18fNYkCnK12pJkPpW7CPDb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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a19856588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(domain, st.norm.cdf(domain))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Standard Deviation](https://en.wikipedia.org/wiki/Standard_deviation)\n",
    "\n",
    "Above is the Bell Curve integral.\n",
    "\n",
    "Remember the derivative is obtain from small differences: (f(x+h) - f(x))/x\n",
    "\n",
    "Given x is our entire domain and operations are vectorized, it's easy enough to plot said derivative."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a1a2bfd68>]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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GHvJvMzFu49tnuNTpYe38iU6XYvqxYEomxTmjeWq3DfHEm4RgGqnqRmBjn20P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\nMXHEQt8YY+KIhb4xxsSR/w9PoGlpRYLV7wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a198724a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = st.norm.cdf(domain)\n",
    "diff = st.norm.cdf(domain + 0.01)\n",
    "plt.plot(domain, (diff-x)/0.01)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a1a3854e0>]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Oo9Q2d3Hr4gqvS5EIUfiLhNmiyhFsONgU1Zu8rVhTzaicDK6aNtLrUiRCFP4i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\n7QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a1a2b3518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = st.norm.pdf(domain)\n",
    "diff = st.norm.pdf(domain + 0.01)\n",
    "plt.plot(domain, (diff-x)/0.01)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a1a3042e8>]"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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HccopmvDPOksf5cod97+fydvs2TqLcdYsvYkyYfDLLzrFePlyHSvIatz89JP2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\nyd8YYxLQ/we5gZcU2ywAIgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a1a3040f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = st.norm.pdf(domain)\n",
    "plt.plot(domain, x, color = \"red\")\n",
    "\n",
    "x = st.norm.pdf(domain)\n",
    "diff = st.norm.pdf(domain + 0.01)\n",
    "plt.plot(domain, (diff-x)/0.01, color = \"blue\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Integrating the Gaussian\n",
    "\n",
    "Apparently there's no closed form, however sympy is able to do an integration somehow."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$ cdf(a,b,\\mu,\\sigma) = - \\frac{1}{2} \\operatorname{erf}{\\left (\\frac{\\sqrt{2} \\left(a - \\mu\\right)}{2 \\sigma} \\right )} + \\frac{1}{2} \\operatorname{erf}{\\left (\\frac{\\sqrt{2} \\left(b - \\mu\\right)}{2 \\sigma} \\right )}$"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a197a4978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sympy import var, Lambda, integrate, sqrt, pi, exp, latex\n",
    "\n",
    "fig = plt.gcf()\n",
    "fig.set_size_inches(8,5)\n",
    "var('a b x sigma mu')\n",
    "pdf = Lambda((x,mu,sigma),\n",
    "  (1/(sigma * sqrt(2*pi)) * exp(-(mu-x)**2 / (2*sigma**2)))\n",
    ")\n",
    "cdf = Lambda((a,b,mu,sigma),\n",
    "  integrate(\n",
    "    pdf(x,mu,sigma),(x,a,b)\n",
    "  )\n",
    ")\n",
    "display(Latex('$ cdf(a,b,\\mu,\\sigma) = ' + latex(cdf(a,b,mu,sigma)) + '$'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets stop right here and note the pdf and cdf have been defined, using sympy's Lambda and integrate, and the cdf will be fed a lot of data, one hundred points, along with mu and sigma.  Then it's simply a matter of plotting.\n",
    "\n",
    "What's amazing is our ability to get something from sympy that works to give a cdf, independently of scipy.stats.norm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'function'>\n"
     ]
    },
    {
     "data": {
      "image/png": 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fpqa2hsk3TyY1KzXaIUkvoqQgEkfcnfnz57Nv3z4m3DCBfgV6OI6El5KCSBxZtWoVGzdu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\niLRSUhARkVZKCiIi0kpJQUREWv1fzy/vH/H/epsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a1af1b080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(50,159,100)\n",
    "y = np.array([cdf(-1e99,v,100,15) for v in x],dtype='float')\n",
    "plt.grid(True)\n",
    "plt.title('Cumulative Distribution Function')\n",
    "plt.xlabel('IQ')\n",
    "print(type(plt.xlabel))\n",
    "plt.ylabel('Y')\n",
    "plt.text(65,.75,'$\\mu = 100$',fontsize=16)\n",
    "plt.text(65,.65,'$\\sigma = 15$',fontsize=16)\n",
    "plt.plot(x,y,color='gray')\n",
    "plt.fill_between(x,y,0,color='#c0f0c0')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above is truly a testament to Python's power, or the Python ecosystem's power.  We've brought in sympy, able to do symbolic integration, and talk LaTeX at the same time.  That's impressive.  Here's [the high IQ source](https://arachnoid.com/IPython/normal_distribution.html) for the original version of the above code.\n",
    "\n",
    "There's no indefinite integral of the Gaussian, but there's a definite one.  sympy comes with its own generic sympy.stats.cdf function which produces Lambdas (symbolic expressions) when used to integrate different types of probability spaces, such as Normal (a continuous PDF).  It also accepts discrete PMFs as well.\n",
    "\n",
    "<pre>\n",
    "Examples\n",
    "========\n",
    "    \n",
    ">>> from sympy.stats import density, Die, Normal, cdf\n",
    ">>> from sympy import Symbol\n",
    "    \n",
    ">>> D = Die('D', 6)\n",
    ">>> X = Normal('X', 0, 1)\n",
    "    \n",
    ">>> density(D).dict\n",
    "{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}\n",
    ">>> cdf(D)\n",
    "{1: 1/6, 2: 1/3, 3: 1/2, 4: 2/3, 5: 5/6, 6: 1}\n",
    ">>> cdf(3*D, D > 2)\n",
    "{9: 1/4, 12: 1/2, 15: 3/4, 18: 1}\n",
    "    \n",
    ">>> cdf(X)\n",
    "Lambda(_z, -erfc(sqrt(2)*_z/2)/2 + 1)\n",
    "</pre>   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## LAB:  convert the Normal Distribution Below to IQ Curve...\n",
    "\n",
    "\n",
    "That means domain is 0-200, standard deviation 15, mean = 100."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a1b342a90>]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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4K5Q2ME9zwImZlQEbgNOARmAFcIG7/zWpzTeB0e7+T2Y2EzjX3f+fmY0E7gWq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\nM/cWuSMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a1b385710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "domain = np.linspace(0, 200, 3000)\n",
    "IQ = st.norm.pdf(domain, 100, 15)\n",
    "plt.plot(domain, IQ, color = \"red\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a1af06e48>]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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GD4Lyh+RzovEEG/Zt8LsUSQAKBIl7m/Ztor6hPqbGD4I0sCyRpECQuBcLv4HQljFnjaFn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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a1b347b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "domain = np.linspace(0, 200, 3000)\n",
    "mu = 100\n",
    "sigma = 15\n",
    "IQ = vg(domain)\n",
    "plt.plot(domain, IQ, color = \"green\")"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}