{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hughes and Hase Problem 3.7\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": [
    "#### Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "58\n"
     ]
    }
   ],
   "source": [
    "counts = np.array([1,3,4,5,6,7,8,9,10,11,12,13])\n",
    "occurences = np.array([1,2,3,6,9,11,8,8,6,2,1,1])\n",
    "\n",
    "n_expts = np.sum(occurences)  # check \n",
    "print(n_expts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (i) Total number of counts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "total # of counts = 423\n"
     ]
    }
   ],
   "source": [
    "total = occurences@counts   # Dot product \n",
    "print(\"total # of counts =\", total)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (ii) Mean number of counts\n",
    "\n",
    "This is the best estimate of the mean of the parent Poisson distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean counts/second = 7.293103448275862\n"
     ]
    }
   ],
   "source": [
    "mean = total/n_expts\n",
    "print('mean counts/second =', mean)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (i) Expected # of trials with 5 or fewer counts¶\n",
    "\n",
    "Use the cumulative distribution function for the Poisson distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "probability of getting 5 or fewer counts = 0.2648489299548199\n",
      "expected number of trials with 5 or fewer counts = 15.361237937379554\n"
     ]
    }
   ],
   "source": [
    "probability1 = stats.poisson.cdf(5, mean)\n",
    "print(\"probability of getting 5 or fewer counts =\", probability1)\n",
    "n_expected1 = n_expts*probability1\n",
    "print(\"expected number of trials with 5 or fewer counts =\", n_expected1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (ii) Expected # of trials with 20 or more counts¶\n",
    "Use the cumulative distribution function to find the probability of 19 or fewer counts. \n",
    "The probability of 20 or more is (1 - the probability of 19 or fewer)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "probability of getting 20 or more counts = 7.674243593669416e-05\n",
      "expected number of trials with 20 or more counts = 0.0044510612843282615\n"
     ]
    }
   ],
   "source": [
    "probability2 = 1 - stats.poisson.cdf(19, mean)\n",
    "print(\"probability of getting 20 or more counts =\", probability2)\n",
    "n_expected2 = n_expts*probability2\n",
    "print(\"expected number of trials with 20 or more counts =\", n_expected2)"
   ]
  },
  {
   "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": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "%load_ext version_information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": {
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         "version": "3.11.5 64bit [MSC v.1916 64 bit (AMD64)]"
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        {
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         "version": "8.15.0"
        },
        {
         "module": "OS",
         "version": "Windows 10 10.0.26100 SP0"
        },
        {
         "module": "scipy",
         "version": "1.11.1"
        },
        {
         "module": "matplotlib",
         "version": "3.7.2"
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       "<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>scipy</td><td>1.11.1</td></tr><tr><td>matplotlib</td><td>3.7.2</td></tr><tr><td colspan='2'>Sat Feb 08 13:57:21 2025 Eastern Standard Time</td></tr></table>"
      ],
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       "\\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",
       "scipy & 1.11.1 \\\\ \\hline\n",
       "matplotlib & 3.7.2 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Sat Feb 08 13:57:21 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",
       "scipy 1.11.1\n",
       "matplotlib 3.7.2\n",
       "Sat Feb 08 13:57:21 2025 Eastern Standard Time"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version_information scipy, matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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