Assignment2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Randomly sampled data from normal, exponential and bernoulli"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Submitted By Anu Priya"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Necessary Imports\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import binom,norm, expon, skew"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Normal Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mu, sigma= 0,0.1\n",
    "normal= np.random.normal(mu, sigma, 1000)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.00540314752093 0.0967169010555\n"
     ]
    },
    {
     "data": {
      "image/png": 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JNLVia5FwXgJaoDbn3fyG18VdRGKAT4D7VPVQyd2lfMkpk29VdaKq9lDVHnFx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\nQ1i7dq3bkWocb4r7CqCNiJw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      "text/plain": [
       "<matplotlib.figure.Figure at 0x2d3db6aa6a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mu, std = norm.fit(normal)\n",
    "print(mu,std)\n",
    "plt.hist(normal, bins=25, normed=True, alpha=0.6, color='g')\n",
    "xmin, xmax = plt.xlim()\n",
    "x = np.linspace(xmin, xmax, 100)\n",
    "p = norm.pdf(x, mu, std)\n",
    "plt.plot(x, p, 'k', linewidth=2)\n",
    "title = \"Fit results: mu = %.2f,  std = %.2f\" % (mu, std)\n",
    "plt.title(title)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Here we have generated a dataset with 1000 random points which are normally distributed. After generating the data points, we fit the data to the model. In the above histogram we can see the green area represents the original generated data and the black line represents the fitted data points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Normality test using chi-square\n",
      " NormaltestResult(statistic=0.0021424250810238824, pvalue=0.9989293610028277)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-0.0026366384842279864"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# normality test\n",
    "# normal chi-square test\n",
    "test=stats.normaltest(normal)\n",
    "print(\"Normality test using chi-square\\n\", test)\n",
    "# Testing the skewness of the data\n",
    "skew(normal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exponential Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x2d3db591e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "exponential= np.random.exponential(scale=0.1,size=1000)\n",
    "loc, scale = expon.fit(exponential, floc=0)\n",
    "plt.hist(exponential, bins=25, normed=True, alpha=0.6, color='b')\n",
    "xmin, xmax = plt.xlim()\n",
    "x = np.linspace(xmin, xmax, 100)\n",
    "p = expon.pdf(x, loc, scale)\n",
    "plt.plot(x, p, 'k', linewidth=2)\n",
    "title = \"Fit results: mu = %.2f,  std = %.2f\" % (loc, scale)\n",
    "plt.title(title)\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Normality test using chi-square\n",
      " NormaltestResult(statistic=441.1532273953743, pvalue=1.6024844877605892e-96)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "2.0281824922586966"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# normality test\n",
    "# normal chi-square test\n",
    "test=stats.normaltest(exponential)\n",
    "print(\"Normality test using chi-square\\n\", test)\n",
    "# checking the skewness of the data\n",
    "skew(exponential)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Here we have generated a dataset with 1000 random points which are exponentially distributed. After generating the data points, we fit the data to the model. In the above histogram we can see the purple area represents the original generated data and the black line represents the fitted data points."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bernouli Distribution- As a special case in binomial distribution when p=1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x2d3db65eb38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "binom_sim=data=binom.rvs(n=1,p=0.3,size=1000)\n",
    "plt.hist(binom_sim,bins=10,normed=True)\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8990385001692826"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# checking the skewness of the data\n",
    "skew(binom_sim)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Yes the fit vary as a function of true parameter. The original curve and the fitted curve doesnot overlap completely, so there is a error associated with the fitted distributions."
   ]
  },
  {
   "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": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}