Normal Distribution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "#### Let's learn the normal distribution! Note: You should check out the binomial distribution first."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll start with the definition of the PDF, we'll see how to create the distribution in python using scipy and numpy, and discuss some properties of the normal distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The distribution is defined by the probability density function equation:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## $$  f(x,\\mu,\\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}}e^\\frac{-1}{2z^2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where: $$z=\\frac{(X-\\mu)}{\\sigma}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where: μ=mean , σ=standard deviation , π=3.14... , e=2.718... The total area bounded by curve of the probability density function equation and the X axis is 1; thus the area under the curve between two ordinates X=a and X=b, where a<b, represents the probability that X lies between a and b. This probability can be expressed as: $$Pr(a<X<b)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "----------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at the curve. The normal distribution has several characteristics:\n",
    "\n",
    "    1.) It has a lower tail (on the left) and an upper tail (on the right)\n",
    "    2.) The curve is symmetric (for the theoretical distribution)\n",
    "    3.) The peak occurs at the mean.\n",
    "    4.) The standard deviation gives the curve a different shape:\n",
    "        -Narrow and tall for a smaller standard deviation.\n",
    "        -Shallower and fatter for a larger standard deviation.\n",
    "    5.) The area under the curve is equal to 1 (the total probaility space)\n",
    "    6.) The mean=median=mode.\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the normal distribution, we can see what percentage of values lie between +/- a standard deviation. 68% of the values lie within 1 TSD, 95% between 2 STDs, and 99.7% between 3 STDs. The number of standard deviations is also called the z-score, which we saw above in the PDF."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/2/25/The_Normal_Distribution.svg/725px-The_Normal_Distribution.svg.png\"/>"
      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "Image(url='https://upload.wikimedia.org/wikipedia/commons/thumb/2/25/The_Normal_Distribution.svg/725px-The_Normal_Distribution.svg.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's learn how to use scipy to create a normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PolyCollection at 0x239a6f28>"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x239a6ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Import\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "#Import the stats library\n",
    "from scipy import stats\n",
    "\n",
    "# Set the mean\n",
    "mean = 0\n",
    "\n",
    "#Set the standard deviation\n",
    "std = 1\n",
    "\n",
    "\n",
    "# Create a range\n",
    "X = np.arange(-4,4,0.01)\n",
    "\n",
    "#Create the normal distribution for the range\n",
    "Y = stats.norm.pdf(X,mean,std)\n",
    "\n",
    "#\n",
    "plt.plot(X,Y)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's learn how to use numpy to create the normal distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "#Set the mean and the standard deviaiton\n",
    "mu,sigma = 0,0.1\n",
    "\n",
    "# Now grab 1000 random numbers from the normal distribution\n",
    "norm_set = np.random.normal(mu,sigma,1000)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([  1.,   0.,   0.,   1.,   1.,   0.,   0.,   2.,   3.,   1.,   5.,\n",
       "          4.,   7.,  12.,  13.,  12.,  19.,  21.,  24.,  24.,  34.,  34.,\n",
       "         44.,  56.,  52.,  32.,  52.,  62.,  49.,  44.,  58.,  43.,  42.,\n",
       "         44.,  33.,  27.,  28.,  21.,  16.,  18.,  16.,  17.,  11.,   3.,\n",
       "          1.,   2.,   6.,   2.,   1.,   2.]),\n",
       " array([-0.34999642, -0.33728494, -0.32457346, -0.31186198, -0.2991505 ,\n",
       "        -0.28643903, -0.27372755, -0.26101607, -0.24830459, -0.23559312,\n",
       "        -0.22288164, -0.21017016, -0.19745868, -0.1847472 , -0.17203573,\n",
       "        -0.15932425, -0.14661277, -0.13390129, -0.12118982, -0.10847834,\n",
       "        -0.09576686, -0.08305538, -0.0703439 , -0.05763243, -0.04492095,\n",
       "        -0.03220947, -0.01949799, -0.00678652,  0.00592496,  0.01863644,\n",
       "         0.03134792,  0.0440594 ,  0.05677087,  0.06948235,  0.08219383,\n",
       "         0.09490531,  0.10761678,  0.12032826,  0.13303974,  0.14575122,\n",
       "         0.1584627 ,  0.17117417,  0.18388565,  0.19659713,  0.20930861,\n",
       "         0.22202008,  0.23473156,  0.24744304,  0.26015452,  0.272866  ,\n",
       "         0.28557747]),\n",
       " <a list of 50 Patch objects>)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "AABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x20fd7470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Now let's plot it using seaborn\n",
    "\n",
    "import seaborn as sns\n",
    "\n",
    "plt.hist(norm_set,bins=50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "There's a ton on information to go over for the normal distribution, this notebook should just serve as a very mild introduction, for more info check out the following sources:\n",
    "\n",
    "1.) https://en.wikipedia.org/wiki/Normal_distribution\n",
    "\n",
    "2.) https://mathworld.wolfram.com/NormalDistribution.html\n",
    "\n",
    "3.) https://stattrek.com/probability-distributions/normal.aspx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thanks!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}