{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bernoulli trial "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What's a Bernoulli trial? A coin-flip! \n",
    "\n",
    "How do we simulate a coin flip? With random number generators. \n",
    "\n",
    "Using numpy for instance:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0, 1, 0, 0, 0, 1, 1, 0, 1])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "(np.random.rand(10) > .5).astype(np.int)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also have coin flips that are heads with a given probability $p$. To have a biased coin with probability 70%, we can do this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 1, 1, 1, 1, 0, 1, 1, 0])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(np.random.rand(10) > (1 - .7)).astype(np.int)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Binomial distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "... is obtained when we do $n$ coin flips with probability $p$ and count the number of heads."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(np.random.rand(20) > (1 - .3)).sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's one sample of a Binomial distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we do this a lot, we can see what the distribution looks like in histogram form."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([8.4000e-04, 7.3000e-03, 2.6920e-02, 7.2100e-02, 1.2802e-01,\n",
       "        1.8034e-01, 1.9138e-01, 1.6548e-01, 1.1478e-01, 6.6280e-02,\n",
       "        2.9000e-02, 1.1920e-02, 4.2200e-03, 1.1400e-03, 2.4000e-04,\n",
       "        0.0000e+00, 4.0000e-05, 0.0000e+00, 0.0000e+00, 0.0000e+00]),\n",
       " array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9., 10., 11., 12.,\n",
       "        13., 14., 15., 16., 17., 18., 19., 20.]),\n",
       " <a list of 20 Patch objects>)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "plt.style.use('ggplot')\n",
    "\n",
    "samples = [(np.random.rand(20) > (1 - .3)).sum() for _ in range(50000)]\n",
    "plt.hist(samples, range=(0, 20), bins=20, density=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, the mean value we expect is $n p$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "20 * .3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Poisson "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What happens if we now do something like $np$ is fixed but $n$ gets large?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def simulate_binomial(n, p):\n",
    "    return (np.random.rand(n) > (1 - p)).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "ntimesp = 10.\n",
    "for n in [20, 50, 100, 500, 1000, 5000]:\n",
    "    p = ntimesp / n\n",
    "    samples = [simulate_binomial(n, p) for _ in range(5000)]\n",
    "    plt.hist(samples, range=(0, 20), bins=20, histtype='step', label=f'n={n}, p={p}', density=True, cumulative=True)\n",
    "plt.legend(loc='upper left');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's compare that with samples from the Poisson distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "samples = np.random.poisson(lam=ntimesp, size=5000)\n",
    "plt.hist(samples, range=(0, 20), bins=20, histtype='step', label=f'n={n}, p={p}', density=True, cumulative=True);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The exponential distribution "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Well, how do we get to the exponential distribution? It needs a mean waiting time..."
   ]
  }
 ],
 "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}