{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hughes and Hase Problem 3.8\n",
    "\n",
    "NOTE: In this notebook I use the `stats` sub-module of `scipy` for all statistics functions, including generation of random numbers. There are other modules with some overlapping functionality, e.g., the regular python random module, and the `scipy.random` module, but I do not use them here. The `stats` sub-module includes tools for a large number of distributions, it includes a large and growing set of statistical functions, and there is a unified class structure. (And namespace issues are minimized.) See https://docs.scipy.org/doc/scipy/reference/stats.html."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (i) Mean count rate\n",
    "\n",
    "If we use the minute as our unit of time, the determination of the mean count rate is trivial: it's $R = 270\\, \\mbox{min$^{−1}$}$. If we choose seconds, it's only slightly less trivial: $R=270/60 = 4.5\\, \\mbox{s$^{−1}$}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (ii) Error in the mean count rate\n",
    "We are sampling from a Poisson distribution, so the error given, by the standard deviation, is $\\sigma = \\sqrt{\\bar \\mu}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sigma (min^-1) = 16.431676725154983 ; sigma (s^-1) =  0.27386127875258304\n"
     ]
    }
   ],
   "source": [
    "sigma = np.sqrt(270)\n",
    "print('sigma (min^-1) =', sigma, '; sigma (s^-1) = ', sigma/60)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (iii) Fractional error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "fractional error = 0.06085806194501846\n"
     ]
    }
   ],
   "source": [
    "print('fractional error =', sigma/270)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Expected counts in 15 minutes: $N = 15\\, \\mbox{min} \\times R$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "expected in 15 minutes = 4050\n"
     ]
    }
   ],
   "source": [
    "n_expected = 15*270\n",
    "print('expected in 15 minutes =', n_expected)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The probability of actually getting this number of counts is given by the probability mass function of the Poisson distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "probability of getting exactly 4050  counts = 0.006268644164779241\n"
     ]
    }
   ],
   "source": [
    "probability = stats.poisson.pmf(n_expected,n_expected)\n",
    "print('probability of getting exactly', n_expected,' counts =', probability)"
   ]
  },
  {
   "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> . If not already installed on your machine, run `pip install --upgrade version_information` from the terminal\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "%load_ext version_information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "3.11.5 64bit [MSC v.1916 64 bit (AMD64)]"
        },
        {
         "module": "IPython",
         "version": "8.15.0"
        },
        {
         "module": "OS",
         "version": "Windows 10 10.0.26100 SP0"
        },
        {
         "module": "numpy",
         "version": "1.23.2"
        },
        {
         "module": "scipy",
         "version": "1.11.1"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.11.5 64bit [MSC v.1916 64 bit (AMD64)]</td></tr><tr><td>IPython</td><td>8.15.0</td></tr><tr><td>OS</td><td>Windows 10 10.0.26100 SP0</td></tr><tr><td>numpy</td><td>1.23.2</td></tr><tr><td>scipy</td><td>1.11.1</td></tr><tr><td colspan='2'>Sat Feb 08 13:58:14 2025 Eastern Standard Time</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 3.11.5 64bit [MSC v.1916 64 bit (AMD64)] \\\\ \\hline\n",
       "IPython & 8.15.0 \\\\ \\hline\n",
       "OS & Windows 10 10.0.26100 SP0 \\\\ \\hline\n",
       "numpy & 1.23.2 \\\\ \\hline\n",
       "scipy & 1.11.1 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Sat Feb 08 13:58:14 2025 Eastern Standard Time} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 3.11.5 64bit [MSC v.1916 64 bit (AMD64)]\n",
       "IPython 8.15.0\n",
       "OS Windows 10 10.0.26100 SP0\n",
       "numpy 1.23.2\n",
       "scipy 1.11.1\n",
       "Sat Feb 08 13:58:14 2025 Eastern Standard Time"
      ]
     },
     "execution_count": 7,
     "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 (ipykernel)",
   "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": "3.11.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}