Efficient_sampling_from_a_Binomial_distribution.ipynb

{
 "cells": [
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   "source": [
    "# Table of Contents\n",
    " <p><div class=\"lev1 toc-item\"><a href=\"#Bernoulli-and-binomial-distribution\" data-toc-modified-id=\"Bernoulli-and-binomial-distribution-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Bernoulli and binomial distribution</a></div><div class=\"lev2 toc-item\"><a href=\"#Requirements\" data-toc-modified-id=\"Requirements-11\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>Requirements</a></div><div class=\"lev2 toc-item\"><a href=\"#A-naive-generator\" data-toc-modified-id=\"A-naive-generator-12\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>A naive generator</a></div><div class=\"lev2 toc-item\"><a href=\"#The-generator-included-in-numpy.random\" data-toc-modified-id=\"The-generator-included-in-numpy.random-13\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>The generator included in <code>numpy.random</code></a></div><div class=\"lev2 toc-item\"><a href=\"#An-efficient-generator-using-the-inverse-transform-method\" data-toc-modified-id=\"An-efficient-generator-using-the-inverse-transform-method-14\"><span class=\"toc-item-num\">1.4&nbsp;&nbsp;</span>An efficient generator using the inverse transform method</a></div><div class=\"lev3 toc-item\"><a href=\"#Explicit-computation-of-the-probabilities\" data-toc-modified-id=\"Explicit-computation-of-the-probabilities-141\"><span class=\"toc-item-num\">1.4.1&nbsp;&nbsp;</span>Explicit computation of the probabilities</a></div><div class=\"lev3 toc-item\"><a href=\"#First-function-using-the-inversion-method\" data-toc-modified-id=\"First-function-using-the-inversion-method-142\"><span class=\"toc-item-num\">1.4.2&nbsp;&nbsp;</span>First function using the inversion method</a></div><div class=\"lev3 toc-item\"><a href=\"#Simplified-code-of-the-inversion-method\" data-toc-modified-id=\"Simplified-code-of-the-inversion-method-143\"><span class=\"toc-item-num\">1.4.3&nbsp;&nbsp;</span>Simplified code of the inversion method</a></div><div class=\"lev3 toc-item\"><a href=\"#In-Cython\" data-toc-modified-id=\"In-Cython-144\"><span class=\"toc-item-num\">1.4.4&nbsp;&nbsp;</span>In Cython</a></div><div class=\"lev2 toc-item\"><a href=\"#Numerical-experiments-to-check-time-cost-of-the-different-versions\" data-toc-modified-id=\"Numerical-experiments-to-check-time-cost-of-the-different-versions-15\"><span class=\"toc-item-num\">1.5&nbsp;&nbsp;</span>Numerical experiments to check time cost of the different versions</a></div><div class=\"lev2 toc-item\"><a href=\"#Checking-that-sampling-from-$Bin(n,p)$-requires-a-time-$\\Omega(n)$.\" data-toc-modified-id=\"Checking-that-sampling-from-$Bin(n,p)$-requires-a-time-$\\Omega(n)$.-16\"><span class=\"toc-item-num\">1.6&nbsp;&nbsp;</span>Checking that sampling from <span class=\"MathJax_Preview\" style=\"color: inherit;\"></span><span class=\"MathJax\" id=\"MathJax-Element-461-Frame\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>B</mi><mi>i</mi><mi>n</mi><mo stretchy=&quot;false&quot;>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo stretchy=&quot;false&quot;>)</mo></math>\" role=\"presentation\" style=\"position: relative;\"><nobr aria-hidden=\"true\"><span class=\"math\" id=\"MathJax-Span-3306\" style=\"width: 3.761em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.586em; height: 0px; font-size: 104%;\"><span style=\"position: absolute; clip: rect(1.794em, 1003.54em, 2.887em, -999.998em); top: -2.576em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-3307\"><span class=\"mi\" id=\"MathJax-Span-3308\" style=\"font-family: STIXMathJax_Normal-italic;\">𝐵</span><span class=\"mi\" id=\"MathJax-Span-3309\" style=\"font-family: STIXMathJax_Normal-italic;\">𝑖</span><span class=\"mi\" id=\"MathJax-Span-3310\" style=\"font-family: STIXMathJax_Normal-italic;\">𝑛</span><span class=\"mo\" id=\"MathJax-Span-3311\" style=\"font-family: STIXMathJax_Main;\">(</span><span class=\"mi\" id=\"MathJax-Span-3312\" style=\"font-family: STIXMathJax_Normal-italic;\">𝑛</span><span class=\"mo\" id=\"MathJax-Span-3313\" style=\"font-family: STIXMathJax_Main;\">,</span><span class=\"mi\" id=\"MathJax-Span-3314\" style=\"font-family: STIXMathJax_Normal-italic; padding-left: 0.177em;\">𝑝</span><span class=\"mo\" id=\"MathJax-Span-3315\" style=\"font-family: STIXMathJax_Main;\">)</span></span><span style=\"display: inline-block; width: 0px; height: 2.581em;\"></span></span></span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.225em; border-left: 0px solid; width: 0px; height: 1.002em;\"></span></span></nobr><span class=\"MJX_Assistive_MathML\" role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi><mi>i</mi><mi>n</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo stretchy=\"false\">)</mo></math></span></span><script type=\"math/tex\" id=\"MathJax-Element-461\">Bin(n,p)</script> requires a time <span class=\"MathJax_Preview\" style=\"color: inherit;\"></span><span class=\"MathJax\" id=\"MathJax-Element-462-Frame\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi mathvariant=&quot;normal&quot;>&amp;#x03A9;</mi><mo stretchy=&quot;false&quot;>(</mo><mi>n</mi><mo stretchy=&quot;false&quot;>)</mo></math>\" role=\"presentation\" style=\"position: relative;\"><nobr aria-hidden=\"true\"><span class=\"math\" id=\"MathJax-Span-3316\" style=\"width: 1.969em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.882em; height: 0px; font-size: 104%;\"><span style=\"position: absolute; clip: rect(1.794em, 1001.84em, 2.887em, -999.998em); top: -2.576em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-3317\"><span class=\"mi\" id=\"MathJax-Span-3318\" style=\"font-family: STIXMathJax_Main;\">Ω</span><span class=\"mo\" id=\"MathJax-Span-3319\" style=\"font-family: STIXMathJax_Main;\">(</span><span class=\"mi\" id=\"MathJax-Span-3320\" style=\"font-family: STIXMathJax_Normal-italic;\">𝑛</span><span class=\"mo\" id=\"MathJax-Span-3321\" style=\"font-family: STIXMathJax_Main;\">)</span></span><span style=\"display: inline-block; width: 0px; height: 2.581em;\"></span></span></span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.225em; border-left: 0px solid; width: 0px; height: 1.002em;\"></span></span></nobr><span class=\"MJX_Assistive_MathML\" role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span></span><script type=\"math/tex\" id=\"MathJax-Element-462\">\\Omega(n)</script>.</a></div><div class=\"lev2 toc-item\"><a href=\"#Conclusion\" data-toc-modified-id=\"Conclusion-17\"><span class=\"toc-item-num\">1.7&nbsp;&nbsp;</span>Conclusion</a></div>"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bernoulli and binomial distribution\n",
    "- References: [Bernoulli distribution on Wikipedia](https://en.wikipedia.org/wiki/Bernoulli_distribution) and [Binomial distribution on Wikipedia](https://en.wikipedia.org/wiki/Binomial_distribution#Generating_binomial_random_variates).\n",
    "\n",
    "The Bernoulli distribution of mean $p\\in[0,1]$ is defined as the distribution on $\\{0,1\\}$ such that $\\mathbb{P}(X=1) = p$ and $\\mathbb{P}(X=0) = 1-p$.\n",
    "\n",
    "If $X$ follows a Binomial distribution of mean $p\\in[0,1]$ and $n$ samples, $X$ is defined as the sum of $n$ independent and identically distributed (iid) samples from a Bernoulli distribution of mean $p$, that is $X\\in\\{0,\\dots,n\\}$ ($X\\in\\mathbb{N}$) and $\\forall k\\in\\{0,\\dots,n\\}, \\mathbb{P}(X=k) = {n \\choose k} p^k (1-p)^{n-k}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Requirements\n",
    "Let's import the modules required for this notebook."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%load_ext cython"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Lilian Besson (Naereen) 2019-02-28T14:09:13+01:00\n",
      "\n",
      "CPython 3.6.7\n",
      "IPython 7.2.0\n",
      "\n",
      "numpy 1.15.4\n",
      "matplotlib 3.0.2\n",
      "cython 0.29.2\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -a \"Lilian Besson (Naereen)\" -i -v -p numpy,matplotlib,cython"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A naive generator\n",
    "Using the pseudo-random generator of (`float`) random numbers in $[0,1]$ from the `random` or `numpy.random` module, we can easily generate a sample from a Bernoulli distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "\n",
    "def uniform_01() -> float:\n",
    "    return random.random()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.02240955612218365,\n",
       " 0.7770015834088284,\n",
       " 0.8358045673716832,\n",
       " 0.4095374878779069,\n",
       " 0.987872833694527]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ uniform_01() for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's very quick now:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bernoulli(p: float) -> int:\n",
    "    return 1 if uniform_01() <= p else 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 0, 0, 0, 0]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ bernoulli(0) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 0, 0, 0, 0]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ bernoulli(0.12345) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 1, 1, 1]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ bernoulli(1) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we can naively generate samples from a Binomial distribution by summing iid samples generated using this `bernoulli` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def naive_binomial(n: int, p: float) -> int:\n",
    "    result = 0\n",
    "    for k in range(n):  # sum of n iid samples from Bernoulli(p)\n",
    "        result += bernoulli(p)\n",
    "    return result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 2, 0, 1]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ naive_binomial(10, 0.1) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[8, 4, 6, 5, 8]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ naive_binomial(10, 0.5) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[10, 9, 8, 10, 10]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ naive_binomial(10, 0.9) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can quickly illustrate the generated distribution, to check it has the correct \"shape\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([254.,   0., 378.,   0., 248.,   0.,  92.,   0.,  24.,   4.]),\n",
       " array([0. , 0.5, 1. , 1.5, 2. , 2.5, 3. , 3.5, 4. , 4.5, 5. ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.12345.')"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.12345\n",
    "X = [ naive_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([  1.,  10.,  41.,  98., 215., 236., 219., 116.,  51.,  13.]),\n",
       " array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9., 10.]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.5.')"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.5\n",
    "X = [ naive_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([  1.,   0.,   0.,   5.,   0.,   0., 101.,   0.,   0., 893.]),\n",
       " array([ 7. ,  7.3,  7.6,  7.9,  8.2,  8.5,  8.8,  9.1,  9.4,  9.7, 10. ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.98765.')"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcYAAAEICAYAAADFgFTtAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvOIA7rQAAHDxJREFUeJzt3XuYHVWZ7/HvK5G7QoCIEC5BQRQ4MmgUVEQ0jgo4hDPe8EZw8CBn0FFxHsx4QwedAcZHHMfxgqKAF0RRDwiiMnITFZzgIBcBidwCBAgIgXBT4D1/rNWw2Ozu3t1peu9uvp/n6ad3Va2qWqt2Vf2qVlU6kZlIkqTiSf2ugCRJg8RglCSpYTBKktQwGCVJahiMkiQ1DEZJkhoG4xhExDER8ckJXuYaEfGjiFgeEd+byGX3Q0RcGhG7DvJ6IyIjYssey348Ir5ZP28WESsiYpWVqGq77C9FxEfr510j4vqJWG5d3ksj4oqJWt5Erz8i5tTvYcZk1muqiohrIuKV/a7HE8WowRgR746IRRFxf0Qc02X6vIi4PCLuiYgzI2LzZtpqEfG1iLgzIm6KiIN6nfcJ5PXAhsD6mfmGfldmJM3JbEX9uTkivhARTx4qk5nbZuZZk123yVhvZl6XmWtn5oMjlYuIfSPi3B6Wd0BmHjoRdesM+8z8RWZuPRHLHo/O9U+FE3tErBoRJ9a6ZueFVhSHR8Rt9efwiIg+VXfSjeV8HREvjojfRMRdEXFRROzcMf09EXF1zYZF7fSIOK05x6yIiD9HxMUd87+3zn93RFwWEc+q43eNiIc65l8w1rb2csd4I/BJ4GtdGr8B8APgo8B6wCLghKbIx4GtgM2BlwMHR8Rrepz3iWJz4A+Z+UC3iQN6Rb1uZq4N/C/gRcCBfa7PlDNRd52acOcCbwNu6jJtf2AvYHvgucDfAO+avKr1z1jO1xGxHvAj4N+AdYEjgB9FxMw6fUfgMMpNwTrA0cAPh46JzNytXoCuXc8zvwK+1yz/ncB+wB7A2sBrgVubKtzYzp+Zx465wZnZ0w8lHI/pGLc/8KtmeC3gXuDZdfhG4FXN9EOB7/Qyb5f1fxC4AbgLuAKYV8e/EPg1cAewFPg8sGozXwJ/D1xZ5z0UeGbd2HcC3x0qD+wKXA98qG7oa4C3Nss6BvhkM/xa4MK67l8Bzx2tvh1t+gTwZ+AvwArKl70v8EvgSOC2ut2fBHwEuBa4BTgOWKcuY05t4zuAJcDtwAHAC4CLat0+P8L3OuL26yg7tK4ZzbgjgKOa4WuAV9bPH6/b97i6HS4F5jZlnwOcVdd9KbBnx7b+AnBa3Ta/BJ4OfLa28XJgh2HW28s+seUwbdwCOLvW9/Q67ze7tb9+V1fVslcDb61tug94sNb7jqY9XwR+DNwNvJJmf2L0fe8s4J3N8L7AufXzObVed9d1vmloeWPY1v8JnFrbcj7wzGG2z7HAB+rn2XW9B9bhZwJ/ouyvD68f+AbwEOX4XgEc3GzLBcB1tc0fHmE/7bmOE/FTv4tdO8b9Cti/Gd4POG+Y+WcCpwDLKPvrKcAmHd/noZT9+i7gZ8AGzfS3U47324AP0+zfw2ybL1H217so++/mE7w9ej5fU86Ll3aM+wOwX/38JuA3HctKYKNhzjkPAnPq8JMo57nHnE/b42hl27uyzxi3BX43NJCZdwN/BLatVwcbtdPr521Hm7dzJRGxNfBu4AWZ+RTg1ZQdBcpGez+wAeXuZR4lCFuvBp4P7EQ5KI+iXBVuCmwHvLkp+/S6rNmUg/aouv7OOu1AuYt+F7A+8GXg5Np9PFJ9H5aZhwD/ApyQ5crm6DppR8oJd0PgU5ST4L6Uu+5nUK6SPt+xuB0pd+dvooTHhykn322BN0bEyzrXX/Wy/bqKiI1r284bodiewHcoV44nD9W7dr/+iHJCeBrwHuBbHdv6jZQLgg2A+ylh99s6fCLwmYluE/Bt4II676GUfeAxImIt4HPAbvU7fjFwYWZeRrkw+XX9TtdtZnsL5ft8CuXOpFNP+16nzNylfty+rvNRV/I9buu9KRdqM4HFtZ7dnE05+QC8jLKf7tIM/yIzH+qo39sp4fc3tX5HNJN3BramfEcfi4jnjNDUXutIRNwxws/CEdYxkkeds3j0+azTk4CvU3qENqOESOcx+xbKBe3TgFWBf6x134ZyEfV2YGPK+WWTUer2Vsr+ugHlYv1bwxUc57bp+Xw9tJouw9vVz6cBq0TEjvUu8e9qnbvdpe9D2aeuqcOb1J/tImJJ7U79RES0Wfa0+pjn6og4sh6rY7Kywbg2sLxj3HLKgb92M9w5bbR5Oz0IrAZsExFPzsxrMvOPAJl5QWael5kP1I33ZcoB2joiM+/MzEuBS4CfZeZVmbmc8iXt0FH+o5l5f2aeTblCfWOXOu0PfDkzz8/MB7Pcrt9PCd9h69ujGzPzP2qb7qXs9J+pdV4B/BOwd0c366GZeV9m/oxy53B8Zt6SmTcAv+jSRqDn7dfp1oi4g3JHfDclpIZzbmb+OMtzuW9QuqGgbKe1gcMy88+ZeQblqrq9SPlhrd99wA+B+zLzuLqsEya4TUTEZpQ77aHv/xxKoAznIcoBukZmLq3710hOysxfZuZDtU3d9LLvjVWv2/o3Wbr0vwX81TDLOhvYuZ6IdqH0GLykTntZnT4Wn8jMezPzd5QT7/YjlO21jmTmuiP8HDbGOg7pPGctB9bu9pwxM2/LzO9n5j2ZeRclxDv3wa9n5h/qMf7dpj2vB07JzHMy835K9+VDjOzUpvyHgRdFxKbdCo5z24zlfP1rYOOIeHNEPLk+43smsGadfhfwfcrF4f3AIZQ78W5/uHsfyh3xkKELhFdRHuW8nLIf71fHX07ZjhsBr6DcEA13AT2slQ3GFcBTO8Y9ldLwFc1w57TR5n2UzFwMvI/SNXdLRHyn3q0QEc+KiFOivNxzJ+UObIOORdzcfL63y/DazfDt9WpoyLWUq7ZOmwMfaK+2KHegG49U3x4t6RjeuNajrdMMyh3lkLG08WE9br9OG9Q7oTUpXUE/HaFsexV4D7B6DfSNgSUddxfXUu6W+tEmap26ff+PUcu8iXJ3uDQiTo2IZ4+y/M7vtVOv+95Y9bKtO7+nrtu2XuDdTTn5vJQSsDfWu8/xBGNP6x1H2cdD5znrqcCKbif0iFgzIr4cEdfWffAcYN2OZ8vDtWdjmn2l7hO3jVK3tvwKSpf2ROw7Q8Zyvr4NmA8cRDlmXwP8F6V7GkqIvYNyt7kqpffulM5zZH0h5+k8+sL73vr7iMy8o7nw3b2u+6bM/H29+Lya0kP4urE2dmWD8VKaK7x6y/pMSv/y7ZTnO+0V4PZ1nhHn7baizPx2Zu5MCaQEDq+Tvki5StgqM59KeUazMm+Kzey49d6M8qy00xLgUx1XW2tm5vGj1LcXnQfajXU5bZ0e4NFBMV7j3n71SvcYYKf6cH4sbgQ27egC2YxyF7qyxtumpXT//rvKzJ9m5l9Trk4vB74yNGm4WUZZ/0j73t08csUN5YTRq4ne1mdT7mpWrT0SZ1O6fmdSusS6mdT/xqfjrcTOnw+Nc7GPOmfx6PNZpw9Quoh3rPvgUHdzr/vhw3d7EbEmpTt1JG35tSkvyHQ7b41324z1fH12Zr4gM9ejdAk/G/hNnfxXlDviP9QA+0lt84s7FrMA+EEN+iFXUN7LaPenkfatZBw518s/15gREasDq1D6hYeu+KF0b20XEa+rZT4GXJSZl9fpxwEfiYiZ9Wr6//DIbfFo87Z12DoiXhERq1FebLiXR7oWnkJ5iWZFXcf/HetG6OITUV7dfinlQXK3f1/4FeCA2k8eEbFWROwREU8Zpb7jcTzw/ojYou70Q88lu77JOkbj3n61fW+nXPmOdkXb6XzKVfLBtbtlV8pbft8Z43K6GVebMvNaytt2Q9//zrVOjxERG0bE/HqCuJ9yRT30Hd8MbBIRq46j7sPtexcCf1vvRLbkka6jITdTnj93M9Hb+mzKM/Rz6vBZdfjcHP6fsoxUvwmXj34rsfPnX4abL8o7AqvXwVXr+W4ozI4DDoqI2fXu5gM8upuv9RTKcX9HlLc0DxlD9U8EXhsRO9d96J8Z/Vy9e1P+UMpLQV17KMa5bXo+X0N5B6Pua08FPk3psRjqWfpvYI+IeEY9d/418CzKY66h+degPEY4pqPu91Aeoxxcz7WbUB5rnVLne3lEbF6Xuynl7deTRtl2j9FLkn6E8gUvpNzy3lvHkZnLKLepn6K8ebUj5QH5kEMoD2ivpRxM/1avDnqZt7UapYG3Uk7CT6M8Z4PywPotlFv6r7Dy/+TjplqfGynPMQ7o9uVn5iJK0H++ll9MeUFmtPqOx9coz+fOobz9eB/lBYqJMJ7td0dErKCc7F5EecNxTHcEmflnysl5N8p2+gKwz3AH2hitzD7xFsq++CfK/nvcMOWeROkqurGWfRmPBPAZlCvpmyLi1u6zdzXSvnck5Ur5ZsqboZ0vV3wcODZKt/6jnks+Dtv6bMqJfygYz6XczZ4z7Bzwr5SL5Dsi4h/Hud7JcAXlHDeb8ojgXh7prfky5ZnzxZST+Kl1XDefBdagbO/zgJ/0WoH6rPpAyotgSyn7xGh//OHblP31T5Tnam/rdX091mnE83WUP1bxpWaWgyltX0LpUfnfzbTjKBdlZ1EuYD8HvKtjf9yL8gb1mV2q827KheiNlOeZ3+aRf064A+Xt4bvr74uBf2jqeVovPQYxxvPZtFavpL+ZmaO9ASZJAyHKH165PjM/0u+6TBf+SThJkhoGoyRJDbtSJUlqeMcoSVJjEP9A9eNqgw02yDlz5vS7GpI0ZVxwwQW3ZuasftdjsjzhgnHOnDksWrSo39WQpCkjIrr+Bajpyq5USZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaT7i/fCNJ/TRn4al9We81h+3Rl/VORd4xSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUGLhgjIj3R8SlEXFJRBwfEatHxBYRcX5ELI6IEyJi1Vp2tTq8uE6f09/aS5KmuoEKxoiYDfwDMDcztwNWAfYGDgeOzMwtgduB/eos+wG31/FH1nKSJI3bQAVjNQNYIyJmAGsCS4FXACfW6ccCe9XP8+swdfq8iIhJrKskaZoZqGDMzBuATwPXUQJxOXABcEdmPlCLXQ/Mrp9nA0vqvA/U8ut3Ljci9o+IRRGxaNmyZY9vIyRJU9pABWNEzKTcBW4BbAysBbxmZZebmUdl5tzMnDtr1qyVXZwkaRobqGAEXglcnZnLMvMvwA+AlwDr1q5VgE2AG+rnG4BNAer0dYDbJrfKkqTpZNCC8Tpgp4hYsz4rnAf8HjgTeH0tswA4qX4+uQ5Tp5+RmTmJ9ZUkTTMDFYyZeT7lJZrfAhdT6ncU8EHgoIhYTHmGeHSd5Whg/Tr+IGDhpFdakjStzBi9yOTKzEOAQzpGXwW8sEvZ+4A3TEa9JElPDAN1xyhJUr8ZjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVJj4IIxItaNiBMj4vKIuCwiXhQR60XE6RFxZf09s5aNiPhcRCyOiIsi4nn9rr8kaWobuGAE/h34SWY+G9geuAxYCPw8M7cCfl6HAXYDtqo/+wNfnPzqSpKmk4EKxohYB9gFOBogM/+cmXcA84Fja7Fjgb3q5/nAcVmcB6wbERtNcrUlSdPIQAUjsAWwDPh6RPxPRHw1ItYCNszMpbXMTcCG9fNsYEkz//V13KNExP4RsSgiFi1btuxxrL4kaaobtGCcATwP+GJm7gDczSPdpgBkZgI5loVm5lGZOTcz586aNWvCKitJmn4GLRivB67PzPPr8ImUoLx5qIu0/r6lTr8B2LSZf5M6TpKkcRmoYMzMm4AlEbF1HTUP+D1wMrCgjlsAnFQ/nwzsU99O3QlY3nS5SpI0ZjP6XYEu3gN8KyJWBa4C3kEJ8O9GxH7AtcAba9kfA7sDi4F7allJksZt4IIxMy8E5naZNK9L2QQOfNwrJUl6whiorlRJkvrNYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQYuGCNilYj4n4g4pQ5vERHnR8TiiDghIlat41erw4vr9Dn9rLckaXoYuGAE3gtc1gwfDhyZmVsCtwP71fH7AbfX8UfWcpIkrZSBCsaI2ATYA/hqHQ7gFcCJtcixwF718/w6TJ0+r5aXJGncBioYgc8CBwMP1eH1gTsy84E6fD0wu36eDSwBqNOX1/KPERH7R8SiiFi0bNmyx6vukqRpYGCCMSJeC9ySmRdM9LIz86jMnJuZc2fNmjXRi5ckTSMz+l2BxkuAPSNid2B14KnAvwPrRsSMele4CXBDLX8DsClwfUTMANYBbpv8akuSppOBuWPMzH/KzE0ycw6wN3BGZr4VOBN4fS22ADipfj65DlOnn5GZOYlVliRNQwMTjCP4IHBQRCymPEM8uo4/Gli/jj8IWNin+kmSppFB6kp9WGaeBZxVP18FvLBLmfuAN0xqxSRJ095UuGOUJGnSGIySJDUMRkmSGgajJEkNg1GSpIbBKElSw2CUJKlhMEqS1DAYJUlqGIySJDUMRkmSGgajJEkNg1GSpIbBKElSw2CUJKlhMEqS1DAYJUlqGIySJDUMRkmSGgajJEkNg1GSpIbBKElSw2CUJKlhMEqS1DAYJUlqGIySJDUMRkmSGgajJEkNg1GSpIbBKElSw2CUJKlhMEqS1DAYJUlqGIySJDUMRkmSGgajJEkNg1GSpMZABWNEbBoRZ0bE7yPi0oh4bx2/XkScHhFX1t8z6/iIiM9FxOKIuCgintffFkiSprqBCkbgAeADmbkNsBNwYERsAywEfp6ZWwE/r8MAuwFb1Z/9gS9OfpUlSdPJQAVjZi7NzN/Wz3cBlwGzgfnAsbXYscBe9fN84LgszgPWjYiNJrnakqRpZKCCsRURc4AdgPOBDTNzaZ10E7Bh/TwbWNLMdn0d17ms/SNiUUQsWrZs2eNWZ0nS1DeQwRgRawPfB96XmXe20zIzgRzL8jLzqMycm5lzZ82aNYE1lSRNNwMXjBHxZEoofiszf1BH3zzURVp/31LH3wBs2sy+SR0nSdK4DFQwRkQARwOXZeZnmkknAwvq5wXASc34ferbqTsBy5suV0mSxmxGvyvQ4SXA24GLI+LCOu5DwGHAdyNiP+Ba4I112o+B3YHFwD3AOya3upKk6WaggjEzzwVimMnzupRP4MDHtVKSpCeUgepKlSSp3wxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNQxGSZIaBqMkSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNWb0uwKS+m/OwlP7st5rDtujL+uVRuIdoyRJDYNRkqSGwShJUsNglCSpYTBKktQwGCVJahiMkiQ1DEZJkhoGoyRJDYNRkqSGwShJUsNglCSpYTBKktQwGCVJahiMkiQ1DEZJkhoGoyRJDYNRkqSGwShJUsNglCSpMeWDMSJeExFXRMTiiFjY7/pIkqa2KR2MEbEK8J/AbsA2wJsjYpv+1kqSNJXN6HcFVtILgcWZeRVARHwHmA/8vq+10kqbs/DUvq37msP26Nu6JfXfVA/G2cCSZvh6YMfOQhGxP7B/HVwREVeMc30bALeOc95BM13aMuHtiMMncmljMl2+E+ixLX3c1mMxLb6XOHyl2rH5RNZl0E31YOxJZh4FHLWyy4mIRZk5dwKq1HfTpS3TpR1gWwbVdGnLdGnHZJjSzxiBG4BNm+FN6jhJksZlqgfjfwNbRcQWEbEqsDdwcp/rJEmawqZ0V2pmPhAR7wZ+CqwCfC0zL30cV7nS3bEDZLq0Zbq0A2zLoJoubZku7XjcRWb2uw6SJA2Mqd6VKknShDIYJUlqGIwdImLriLiw+bkzIt7XUSYi4nP1z9BdFBHP61d9h9NjO3aNiOVNmY/1q76jiYj3R8SlEXFJRBwfEat3TF8tIk6o38n5ETGnPzUdXQ9t2TciljXfyzv7VdeRRMR7axsu7dy36vSBP06G9NCWgT1WIuJrEXFLRFzSjFsvIk6PiCvr75nDzLuglrkyIhZMXq0HXGb6M8wP5YWem4DNO8bvDpwGBLATcH6/6zrOduwKnNLv+vVQ/9nA1cAadfi7wL4dZf4e+FL9vDdwQr/rvRJt2Rf4fL/rOko7tgMuAdakvMT3X8CWHWWmxHHSY1sG9lgBdgGeB1zSjDsCWFg/LwQO7zLfesBV9ffM+nlmv9szCD/eMY5sHvDHzLy2Y/x84LgszgPWjYiNJr96PRuuHVPJDGCNiJhBOYHd2DF9PnBs/XwiMC8iYhLrNxajtWUqeA4l6O7JzAeAs4G/7SgzVY6TXtoysDLzHOBPHaPb4+FYYK8us74aOD0z/5SZtwOnA6953Co6hRiMI9sbOL7L+G5/im72pNRofIZrB8CLIuJ3EXFaRGw7mZXqVWbeAHwauA5YCizPzJ91FHv4O6knt+XA+pNZz1702BaA19XuxxMjYtMu0/vtEuClEbF+RKxJuTvsrOdUOU56aQtMgWOlsWFmLq2fbwI27FJmqnw/k85gHEb9gwF7At/rd11Wxijt+C2le3V74D+A/zeZdetVfT4yH9gC2BhYKyLe1t9ajU+PbfkRMCczn0u5ij+WAZOZlwGHAz8DfgJcCDzY10qNU49tmRLHSjdZ+k39d3ljYDAObzfgt5l5c5dpU+lP0Q3bjsy8MzNX1M8/Bp4cERtMdgV78Erg6sxclpl/AX4AvLijzMPfSe2iXAe4bVJr2ZtR25KZt2Xm/XXwq8DzJ7mOPcnMozPz+Zm5C3A78IeOIlPmOBmtLVPoWBly81C3df19S5cyU+b7mWwG4/DezPDdjycD+9S37naidIctHaZsvw3bjoh4+tBzuIh4IWV/GMQwuQ7YKSLWrPWdB1zWUeZkYOitutcDZ9Qr5UEzals6nsPt2Tl9UETE0+rvzSjP5L7dUWTKHCejtWUKHStD2uNhAXBSlzI/BV4VETNrT8ar6jj1++2fQfwB1qLs9Os04w4ADqifg/IfJP8RuBiY2+86j7Md7wYuBX4HnAe8uN91HqEtnwAupzwP+gawGvDPwJ51+uqU7uLFwG+AZ/S7zivRln9tvpczgWf3u87DtOMXlP/79HfAvC7715Q4Tnpsy8AeK5QL36XAXyjPCfejPF//OXAl5S3b9WrZucBXm3n/rh4zi4F39Lstg/Ljn4STJKlhV6okSQ2DUZKkhsEoSVLDYJQkqWEwSpLUMBglSWoYjJIkNf4/Ief9Z0eBTf0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.98765\n",
    "X = [ naive_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The generator included in `numpy.random`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "def numpy_binomial(n: int, p: float) -> int:\n",
    "    return np.random.binomial(n, p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try this out:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 0, 1, 1]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ numpy_binomial(10, 0.1) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[4, 6, 7, 6, 5]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ numpy_binomial(10, 0.5) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[9, 9, 10, 8, 9]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ numpy_binomial(10, 0.9) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot this out also."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([288., 343.,   0., 245.,   0.,  96.,  22.,   0.,   5.,   1.]),\n",
       " array([0. , 0.6, 1.2, 1.8, 2.4, 3. , 3.6, 4.2, 4.8, 5.4, 6. ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.12345.')"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.12345\n",
    "X = [ numpy_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([  2.,  10.,  46., 106., 218., 230., 211., 114.,  51.,  12.]),\n",
       " array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9., 10.]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.5.')"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.5\n",
    "X = [ naive_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([  7.,   0.,   0.,   0.,   0., 105.,   0.,   0.,   0., 888.]),\n",
       " array([ 8. ,  8.2,  8.4,  8.6,  8.8,  9. ,  9.2,  9.4,  9.6,  9.8, 10. ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.98765.')"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.98765\n",
    "X = [ naive_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## An efficient generator using the inverse transform method\n",
    "\n",
    "1. We start by computing the binomial coefficients and then the probability $\\mathbb{P}(X=k)$ for $k\\in\\{0,\\dots,n\\}$, if $X\\sim Bin(n, p)$, and,\n",
    "2. Then use this to write a generator of Binomial-distributed random values.\n",
    "3. This function is then simplified to inline all computations.\n",
    "4. We propose a fast and simple Cython implementation, to be as efficient as possible, and hopefully comparably efficient when compared against the implementation in Numpy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Explicit computation of the probabilities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "def binomial_coefficient(n: int, k: int) -> int:\n",
    "    \"\"\"From https://en.wikipedia.org/wiki/Binomial_coefficient#Binomial_coefficient_in_programming_languages\"\"\"\n",
    "    if k < 0 or k > n:\n",
    "        return 0\n",
    "    if k == 0 or k == n:\n",
    "        return 1\n",
    "    k = min(k, n - k) # take advantage of symmetry\n",
    "    c = 1\n",
    "    for i in range(k):\n",
    "        c = (c * (n - i)) / (i + 1)\n",
    "    return c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "def proba_binomial(n: int, p: float, k: int) -> float:\n",
    "    \"\"\"Compute {n \\choose k} p^k (1-p)^(n-k)\"\"\"\n",
    "    q = 1.0 - p\n",
    "    return binomial_coefficient(n, k) * p**k * q**(n-k)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### First function using the inversion method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This first function is a generic implementation of the discrete inverse transform method.\n",
    "For more details, see [the Wikipedia page](https://en.wikipedia.org/wiki/Inverse_transform_sampling).\n",
    "\n",
    "> Inverse transformation sampling takes uniform samples of a number $u$ between $0$ and $1$, interpreted as a probability, and then returns the largest number $x$ from the domain of the distribution $\\mathbb{P}(X)$ such that $\\mathbb{P}(-\\infty <X<x)\\leq u$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "# a generic function\n",
    "from typing import Callable\n",
    "\n",
    "def inversion_method(compute_proba: Callable[[int], int], xmax: int, xmin: int =0) -> int:\n",
    "    probas = [ compute_proba(x) for x in range(xmin, xmax + 1) ]\n",
    "    result = xmin\n",
    "    current_proba = 0\n",
    "    one_uniform_sample = uniform_01()\n",
    "    while current_proba <= one_uniform_sample:\n",
    "        current_proba += probas[result]\n",
    "        result += 1\n",
    "    return result - 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "def first_inversion_binomial(n: int, p: float) -> int:\n",
    "    def compute_proba(x):\n",
    "        return proba_binomial(n, p, x)\n",
    "    xmax = n\n",
    "    xmin = 0\n",
    "    return inversion_method(compute_proba, xmax, xmin=xmin)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try out."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 0, 0, 1, 1]"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ first_inversion_binomial(10, 0.1) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[5, 3, 4, 4, 5]"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ first_inversion_binomial(10, 0.5) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[10, 9, 9, 8, 8]"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ first_inversion_binomial(10, 0.9) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems to work as wanted!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simplified code of the inversion method\n",
    "\n",
    "The previous function as a few weaknesses: it stores the $n+1$ values of $\\mathbb{P}(X=k)$ before hand, it computes all of them even if the `for` loop of the inversion method stops in average before the end (in average, it takes $np$ steps, which can be much smaller than $n$ for small $p$).\n",
    "Furthermore, the computations of both the binomial coefficients and the values $p^k (1-p)^{n-k}$ is using powers and not iterative multiplications, leading to more rounding errors.\n",
    "\n",
    "We can solve all these issues by inlining all the computations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [],
   "source": [
    "def inversion_binomial(n: int, p: float) -> int:\n",
    "    if p <= 1e-10:\n",
    "        return 0\n",
    "    if p >= 1 - 1e-10:\n",
    "        return n\n",
    "    if p > 0.5:  # speed up by computing for q and then substracting\n",
    "        return n - inversion_binomial(n, 1.0 - p)\n",
    "    result = 0\n",
    "    q = 1.0 - p\n",
    "    current_proba = q**n\n",
    "    cum_proba = current_proba\n",
    "    one_uniform_sample = uniform_01()\n",
    "    while cum_proba <= one_uniform_sample:\n",
    "        current_proba *= (p * (n - result)) / (q * (result + 1))\n",
    "        cum_proba += current_proba\n",
    "        result += 1\n",
    "    return result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try out."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 1, 2, 3]"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ inversion_binomial(10, 0.1) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[4, 6, 5, 4, 4]"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ inversion_binomial(10, 0.5) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[10, 10, 7, 10, 10]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ inversion_binomial(10, 0.9) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems to work as wanted!\n",
    "\n",
    "And now the storage is indeed $O(1)$, and the computation time is $O(x)$ if the return value is $x$, so the mean computation time is $O(np)$.\n",
    "\n",
    "Note that if $p=1/2$, then $O(np) = O(n/2) = O(n)$, and thus this improved method using the inversion method is (asymptotically) as costly as the naive method (the first method which consists of summing $n$ iid samples from a Bernoulli of mean $p$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot this out also."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([261., 367.,   0., 248.,   0.,  96.,  27.,   0.,   0.,   1.]),\n",
       " array([0. , 0.6, 1.2, 1.8, 2.4, 3. , 3.6, 4.2, 4.8, 5.4, 6. ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.12345.')"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.12345\n",
    "X = [ inversion_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([  1.,  10.,  52., 115., 208., 239., 205., 110.,  49.,  11.]),\n",
       " array([0. , 0.9, 1.8, 2.7, 3.6, 4.5, 5.4, 6.3, 7.2, 8.1, 9. ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.5.')"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.5\n",
    "X = [ inversion_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([  7.,   0.,   0.,   0.,   0., 113.,   0.,   0.,   0., 880.]),\n",
       " array([ 8. ,  8.2,  8.4,  8.6,  8.8,  9. ,  9.2,  9.4,  9.6,  9.8, 10. ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.98765.')"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.98765\n",
    "X = [ inversion_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### In Cython"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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       "static PyMethodDef __pyx_mdef_46_cython_magic_beb04c003b39b7afc4e8a1cfd3d3afad_1cython_inversion_binomial = {\"cython_inversion_binomial\", (PyCFunction)(void*)(PyCFunctionWithKeywords)__pyx_pw_46_cython_magic_beb04c003b39b7afc4e8a1cfd3d3afad_1cython_inversion_binomial, METH_VARARGS|METH_KEYWORDS, 0};\n",
       "static PyObject *__pyx_pw_46_cython_magic_beb04c003b39b7afc4e8a1cfd3d3afad_1cython_inversion_binomial(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds) {\n",
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       "      __pyx_t_3 = <span class='pyx_c_api'>__Pyx_PyObject_Call</span>(__pyx_t_4, __pyx_t_9, NULL);<span class='error_goto'> if (unlikely(!__pyx_t_3)) __PYX_ERR(0, 10, __pyx_L1_error)</span>\n",
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       "    <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_4); __pyx_t_4 = 0;\n",
       "    __pyx_t_4 = <span class='py_c_api'>PyNumber_Subtract</span>(__pyx_t_2, __pyx_t_3);<span class='error_goto'> if (unlikely(!__pyx_t_4)) __PYX_ERR(0, 10, __pyx_L1_error)</span>\n",
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       "    <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_3); __pyx_t_3 = 0;\n",
       "    __pyx_r = __pyx_t_4;\n",
       "    __pyx_t_4 = 0;\n",
       "    goto __pyx_L0;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">11</span>:     <span class=\"k\">cdef</span> <span class=\"kt\">int</span> <span class=\"nf\">result</span> <span class=\"o\">=</span> <span class=\"mf\">0</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_result = 0;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">12</span>:     <span class=\"k\">cdef</span> <span class=\"kt\">double</span> <span class=\"nf\">q</span> <span class=\"o\">=</span> <span class=\"mf\">1.0</span> <span class=\"o\">-</span> <span class=\"n\">p</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_q = (1.0 - __pyx_v_p);\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">13</span>:     <span class=\"k\">cdef</span> <span class=\"kt\">double</span> <span class=\"nf\">current_proba</span> <span class=\"o\">=</span> <span class=\"n\">q</span><span class=\"o\">**</span><span class=\"n\">n</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_current_proba = pow(__pyx_v_q, ((double)__pyx_v_n));\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">14</span>:     <span class=\"k\">cdef</span> <span class=\"kt\">double</span> <span class=\"nf\">cum_proba</span> <span class=\"o\">=</span> <span class=\"n\">current_proba</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_cum_proba = __pyx_v_current_proba;\n",
       "</pre><pre class=\"cython line score-32\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">15</span>:     <span class=\"k\">cdef</span> <span class=\"kt\">double</span> <span class=\"nf\">one_uniform_sample</span> <span class=\"o\">=</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">random</span><span class=\"p\">()</span></pre>\n",
       "<pre class='cython code score-32 '>  <span class='pyx_c_api'>__Pyx_GetModuleGlobalName</span>(__pyx_t_3, __pyx_n_s_random);<span class='error_goto'> if (unlikely(!__pyx_t_3)) __PYX_ERR(0, 15, __pyx_L1_error)</span>\n",
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       "      <span class='pyx_macro_api'>__Pyx_INCREF</span>(function);\n",
       "      <span class='pyx_macro_api'>__Pyx_DECREF_SET</span>(__pyx_t_2, function);\n",
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       "  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_t_3); __pyx_t_3 = 0;\n",
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       "  <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_2); __pyx_t_2 = 0;\n",
       "  __pyx_t_10 = __pyx_<span class='py_c_api'>PyFloat_AsDouble</span>(__pyx_t_4); if (unlikely((__pyx_t_10 == (double)-1) &amp;&amp; <span class='py_c_api'>PyErr_Occurred</span>())) <span class='error_goto'>__PYX_ERR(0, 15, __pyx_L1_error)</span>\n",
       "  <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_4); __pyx_t_4 = 0;\n",
       "  __pyx_v_one_uniform_sample = __pyx_t_10;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">16</span>:     <span class=\"k\">while</span> <span class=\"n\">cum_proba</span> <span class=\"o\">&lt;</span> <span class=\"n\">one_uniform_sample</span><span class=\"p\">:</span></pre>\n",
       "<pre class='cython code score-0 '>  while (1) {\n",
       "    __pyx_t_1 = ((__pyx_v_cum_proba &lt; __pyx_v_one_uniform_sample) != 0);\n",
       "    if (!__pyx_t_1) break;\n",
       "</pre><pre class=\"cython line score-5\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">17</span>:         <span class=\"n\">current_proba</span> <span class=\"o\">*=</span> <span class=\"p\">(</span><span class=\"n\">p</span> <span class=\"o\">*</span> <span class=\"p\">(</span><span class=\"n\">n</span> <span class=\"o\">-</span> <span class=\"n\">result</span><span class=\"p\">))</span> <span class=\"o\">/</span> <span class=\"p\">(</span><span class=\"n\">q</span> <span class=\"o\">*</span> <span class=\"p\">(</span><span class=\"n\">result</span> <span class=\"o\">+</span> <span class=\"mf\">1</span><span class=\"p\">))</span></pre>\n",
       "<pre class='cython code score-5 '>    __pyx_t_10 = (__pyx_v_p * (__pyx_v_n - __pyx_v_result));\n",
       "    __pyx_t_11 = (__pyx_v_q * (__pyx_v_result + 1));\n",
       "    if (unlikely(__pyx_t_11 == 0)) {\n",
       "      <span class='py_c_api'>PyErr_SetString</span>(PyExc_ZeroDivisionError, \"float division\");\n",
       "      <span class='error_goto'>__PYX_ERR(0, 17, __pyx_L1_error)</span>\n",
       "    }\n",
       "    __pyx_v_current_proba = (__pyx_v_current_proba * (__pyx_t_10 / __pyx_t_11));\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">18</span>:         <span class=\"n\">cum_proba</span> <span class=\"o\">+=</span> <span class=\"n\">current_proba</span></pre>\n",
       "<pre class='cython code score-0 '>    __pyx_v_cum_proba = (__pyx_v_cum_proba + __pyx_v_current_proba);\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">19</span>:         <span class=\"n\">result</span> <span class=\"o\">+=</span> <span class=\"mf\">1</span></pre>\n",
       "<pre class='cython code score-0 '>    __pyx_v_result = (__pyx_v_result + 1);\n",
       "  }\n",
       "</pre><pre class=\"cython line score-3\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">20</span>:     <span class=\"k\">return</span> <span class=\"n\">result</span></pre>\n",
       "<pre class='cython code score-3 '>  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_r);\n",
       "  __pyx_t_4 = <span class='pyx_c_api'>__Pyx_PyInt_From_int</span>(__pyx_v_result);<span class='error_goto'> if (unlikely(!__pyx_t_4)) __PYX_ERR(0, 20, __pyx_L1_error)</span>\n",
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       "  __pyx_t_4 = 0;\n",
       "  goto __pyx_L0;\n",
       "</pre></div></body></html>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%cython --annotate\n",
    "\n",
    "import random\n",
    "\n",
    "def cython_inversion_binomial(int n, double p) -> int:\n",
    "    if p <= 1e-9:\n",
    "        return 0\n",
    "    if p >= 1 - 1e-9:\n",
    "        return n\n",
    "    if p > 0.5:  # speed up by computing for q and then substracting\n",
    "        return n - cython_inversion_binomial(n, 1.0 - p)\n",
    "    cdef int result = 0\n",
    "    cdef double q = 1.0 - p\n",
    "    cdef double current_proba = q**n\n",
    "    cdef double cum_proba = current_proba\n",
    "    cdef double one_uniform_sample = random.random()\n",
    "    while cum_proba < one_uniform_sample:\n",
    "        current_proba *= (p * (n - result)) / (q * (result + 1))\n",
    "        cum_proba += current_proba\n",
    "        result += 1\n",
    "    return result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try out."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 1, 1, 0]"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ cython_inversion_binomial(10, 0.1) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[7, 9, 4, 4, 5]"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ cython_inversion_binomial(10, 0.5) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[9, 7, 10, 9, 4]"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[ cython_inversion_binomial(10, 0.9) for _ in range(5) ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems to work as wanted!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot this out also."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([279.,   0., 371.,   0., 227.,   0.,  97.,   0.,  20.,   6.]),\n",
       " array([0. , 0.5, 1. , 1.5, 2. , 2.5, 3. , 3.5, 4. , 4.5, 5. ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.12345.')"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.12345\n",
    "X = [ cython_inversion_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([  2.,   6.,  37., 105., 205., 234., 244., 126.,  33.,   8.]),\n",
       " array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9., 10.]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.5.')"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 1000\n",
    "n = 10\n",
    "p = 0.5\n",
    "X = [ cython_inversion_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "(array([  7.,   0.,   0.,   0.,   0.,  99.,   0.,   0.,   0., 894.]),\n",
       " array([ 8. ,  8.2,  8.4,  8.6,  8.8,  9. ,  9.2,  9.4,  9.6,  9.8, 10. ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, '1000 samples from a Binomial distribution with n = 10 and p = 0.98765.')"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "inversion_binomialm = 1000\n",
    "n = 10\n",
    "p = 0.98765\n",
    "X = [ cython_inversion_binomial(n, p) for _ in range(m) ]\n",
    "plt.figure()\n",
    "plt.hist(X)\n",
    "plt.title(f\"{m} samples from a Binomial distribution with n = {n} and p = {p}.\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical experiments to check time cost of the different versions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function __main__.naive_binomial(n:int, p:float) -> int>"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.first_inversion_binomial(n:int, p:float) -> int>"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.inversion_binomial(n:int, p:float) -> int>"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<function _cython_magic_beb04c003b39b7afc4e8a1cfd3d3afad.cython_inversion_binomial>"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.numpy_binomial(n:int, p:float) -> int>"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "naive_binomial\n",
    "first_inversion_binomial\n",
    "inversion_binomial\n",
    "cython_inversion_binomial\n",
    "numpy_binomial"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the `%timeit` magic to check the (mean) computation time of all the previously mentioned functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "21.8 µs ± 832 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "320 µs ± 9.6 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n",
      "2.45 µs ± 154 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n",
      "201 ns ± 31 ns per loop (mean ± std. dev. of 7 runs, 10000000 loops each)\n",
      "2.4 µs ± 307 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit naive_binomial(n, 0.123456)\n",
    "%timeit first_inversion_binomial(n, 0.123456)\n",
    "%timeit inversion_binomial(n, 0.123456)\n",
    "%timeit cython_inversion_binomial(n, 0.123456)\n",
    "%timeit numpy_binomial(n, 0.123456)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Apparently, our `cython` method is faster than the function from `numpy`!\n",
    "\n",
    "We also check that our first naive implementation of the inversion method was suboptimal, as announced, because of its pre computation of all the values of $\\mathbb{P}(X=k)$.\n",
    "However, we check that the naive method, using the sum of $n$ binomial samples, is as comparably efficient to the pure-Python inversion-based method (for this small $n=100$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "21.6 µs ± 1.38 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "311 µs ± 16.8 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n",
      "7.86 µs ± 168 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n",
      "309 ns ± 5.56 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n",
      "2.08 µs ± 319 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit naive_binomial(n, 0.5)\n",
    "%timeit first_inversion_binomial(n, 0.5)\n",
    "%timeit inversion_binomial(n, 0.5)\n",
    "%timeit cython_inversion_binomial(n, 0.5)\n",
    "%timeit numpy_binomial(n, 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "21 µs ± 629 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "298 µs ± 6.24 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n",
      "777 ns ± 22.1 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n",
      "208 ns ± 7.2 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n",
      "1.52 µs ± 56 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit naive_binomial(n, 0.987654)\n",
    "%timeit first_inversion_binomial(n, 0.987654)\n",
    "%timeit inversion_binomial(n, 0.987654)\n",
    "%timeit cython_inversion_binomial(n, 0.987654)\n",
    "%timeit numpy_binomial(n, 0.987654)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's quite awesome to see that our inversion-based method is more efficient that the numpy function, both in the pure-Python and the Cython versions!\n",
    "But it's weird, as the numpy function is... based on the inversion method, and itself written in C!\n",
    "\n",
    "> See the source code, [numpy/distributions.c line 426](https://github.com/numpy/numpy/blob/7c41164f5340dc998ea1c04d2061f7d246894955/numpy/random/mtrand/distributions.c#L426) (on the 28th February 2019, commit 7c41164).\n",
    "\n",
    "But the trick is that the implementation in numpy uses the inversion method (running in $\\Omega(np)$) if $pn < 30$, and a method denoted \"BTPE\" otherwise.\n",
    "I need to work on this method! The BTPE algorithm is much more complicated, and it is described in the following paper:\n",
    "\n",
    "> Kachitvichyanukul, V.; Schmeiser, B. W. (1988). \"Binomial random variate generation\". Communications of the ACM. 31 (2): 216–222. [doi:10.1145/42372.42381](https://doi.org/10.1145%2F42372.42381).\n",
    "\n",
    "> See the source code, [numpy/distributions.c line 263](https://github.com/numpy/numpy/blob/7c41164f5340dc998ea1c04d2061f7d246894955/numpy/random/mtrand/distributions.c#L263) (on the 28th February 2019, commit 7c41164)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Checking that sampling from $Bin(n,p)$ requires a time $\\Omega(n)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "22.3 µs ± 1.26 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "4.83 µs ± 570 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n",
      "412 ns ± 70.9 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n",
      "1.95 µs ± 173 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "n = 100\n",
    "%timeit naive_binomial(n, random.random())\n",
    "%timeit inversion_binomial(n, random.random())\n",
    "%timeit cython_inversion_binomial(n, random.random())\n",
    "%timeit numpy_binomial(n, random.random())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "228 µs ± 3.58 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n",
      "44 µs ± 2.41 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "859 ns ± 21.2 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n",
      "1.83 µs ± 201 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "n = 1000\n",
    "%timeit naive_binomial(n, random.random())\n",
    "%timeit inversion_binomial(n, random.random())\n",
    "%timeit cython_inversion_binomial(n, random.random())\n",
    "%timeit numpy_binomial(n, random.random())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.62 ms ± 238 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "n = 10000\n",
    "%timeit naive_binomial(n, random.random())\n",
    "%timeit inversion_binomial(n, random.random())\n",
    "%timeit cython_inversion_binomial(n, random.random())\n",
    "%timeit numpy_binomial(n, random.random())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, our inversion method (no matter the implementation) runs in $O(n)$ (for $p$ in average $1/2$ in the trials above).\n",
    "But numpy's implementation is using the BTPE method, which runs in $O(1)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "\n",
    "- So I was write, for the inversion method the computation time is in average $O(np)$ so it is $\\Omega(n)$ and cannot be $O(1)$.\n",
    "- But there has been many algorithms proposed in the literature which achieves a $O(1)$ running time, and the state-of-the-art algorithm is the BTPE method by V. Kachitvichyanukul & B. W. Schmeiser (from 1988). It is implemented in numpy, for the cases when $np > 30$ (that is, as soon as $n>60$ for $p=1/2$).\n",
    "- So the authors of [[Perturbed-History Exploration in Stochastic Multi-Armed Bandits, by Branislav Kveton, Csaba Szepesvari, Mohammad Ghavamzadeh, Craig Boutilier, 26 Feb 2019, arXiv:1902.10089]](https://arxiv.org/abs/1902.10089) were correct, it can indeed cost $O(1)$ time to generate the sum of $t$ samples from $Bern(1/2)$ (that is, a sample from $Bin(t, 1/2)$)."
   ]
  }
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