{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hughes and Hase Problem 8.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (a)\n",
    "\n",
    "The reduced chi-squared statistic is defined as\n",
    "\n",
    "$$ \\chi^2_\\nu \\equiv \\frac{\\chi^2_{\\rm min}}{\\nu}, $$\n",
    "\n",
    "where $\\nu$ is the number of degrees of freedom.  So in this part we have\n",
    "\n",
    "$$ \\chi^2_\\nu = \\frac{15.9}{10} = 1.59 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (b)\n",
    "We can use the CDF of the chi-squared distribution to calculate the probability of getting a value of the \n",
    "reduced chi-squared statistic greater than of equal to 1.59:\n",
    "\n",
    "$$ P(\\chi^2_{\\rm min},\\nu)  = 1 - C_{DF} $$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10253102145831061"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - stats.chi2.cdf(15.9, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We find about a 10% chance of getting a value of $\\chi^2_\\nu$ this large."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (c)\n",
    "In this part we have a a larger value of $\\chi^2{\\rm min}$, and more degrees of freedom, \n",
    "giving\n",
    "\n",
    "$$ \\chi^2_\\nu = \\frac{15.9}{100} = 1.59, $$\n",
    "which the same value found in part (a)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (d)\n",
    "Using the CDF again, we the probability of getting this same value of $\\chi^2_\\nu$ now:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0001599798564519972"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - stats.chi2.cdf(159, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The probability is 0.016%. These is a very different probability than than that found in part (b), even though the fits have the same value of $\\chi^2_\\nu$, which illustrates the \n",
    "hazards of relying on that one simple diagnostic to make judgments. With the larger number of degrees of freedom, a value as large as $\\chi^2_\\nu = 1.59$ becomes much\n",
    "more improbable.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Version information\n",
    "`version_information` is from J.R. Johansson (jrjohansson at gmail.com); see <a href='http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-0-Scientific-Computing-with-Python.ipynb'>Introduction to scientific computing with Python</a> for more information and instructions for package installation.\n",
    "\n",
    "`version_information` is installed on the linux network at Bucknell"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "%load_ext version_information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "3.7.7 64bit [GCC 7.3.0]"
        },
        {
         "module": "IPython",
         "version": "7.16.1"
        },
        {
         "module": "OS",
         "version": "Linux 3.10.0 1062.9.1.el7.x86_64 x86_64 with centos 7.7.1908 Core"
        },
        {
         "module": "numpy",
         "version": "1.18.5"
        },
        {
         "module": "scipy",
         "version": "1.5.2"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.7.7 64bit [GCC 7.3.0]</td></tr><tr><td>IPython</td><td>7.16.1</td></tr><tr><td>OS</td><td>Linux 3.10.0 1062.9.1.el7.x86_64 x86_64 with centos 7.7.1908 Core</td></tr><tr><td>numpy</td><td>1.18.5</td></tr><tr><td>scipy</td><td>1.5.2</td></tr><tr><td colspan='2'>Thu Aug 06 14:39:15 2020 EDT</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 3.7.7 64bit [GCC 7.3.0] \\\\ \\hline\n",
       "IPython & 7.16.1 \\\\ \\hline\n",
       "OS & Linux 3.10.0 1062.9.1.el7.x86\\_64 x86\\_64 with centos 7.7.1908 Core \\\\ \\hline\n",
       "numpy & 1.18.5 \\\\ \\hline\n",
       "scipy & 1.5.2 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Thu Aug 06 14:39:15 2020 EDT} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 3.7.7 64bit [GCC 7.3.0]\n",
       "IPython 7.16.1\n",
       "OS Linux 3.10.0 1062.9.1.el7.x86_64 x86_64 with centos 7.7.1908 Core\n",
       "numpy 1.18.5\n",
       "scipy 1.5.2\n",
       "Thu Aug 06 14:39:15 2020 EDT"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version_information numpy, scipy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}