simulation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quelques expérimentation de simulation probabiliste\n",
    "\n",
    "Nous allons faire ensemble quelques expérimentations pour nous faire la main."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Importons d'abord les modules `random` et `numpy.random`, ainsi que `numpy` et `matplotlib`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random as rd\n",
    "import numpy.random as npr\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tirage d'entiers équiprobables entre 1 et 6\n",
    "\n",
    "On veut simuler le tirage de N dés équilibrés, et afficher les fréquences d'apparition des différentes faces."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Commençons par utiliser le module `random` et effectuons une boucle simulant les N tirages :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lancer_des(N):\n",
    "    compteurs = [0]*6\n",
    "    for k in range(1000000):\n",
    "        i = rd.randint(1,6)\n",
    "        compteurs[i-1] += 1\n",
    "        frequences = [x/N for x in compteurs]\n",
    "    plt.bar(range(1,7),frequences)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "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": [
    "lancer_des(1000000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Utilisons maintenant `numpy` :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lancer_des_numpy(N):\n",
    "    x = npr.randint(1,7,N)\n",
    "    compteurs = [np.count_nonzero(x == i) for i in range(1,7)]\n",
    "    frequences = [x/N for x in compteurs]\n",
    "    plt.bar(range(1,7),frequences)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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": [
    "lancer_des_numpy(1000000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## La loi de Bernoulli de paramètre $p$\n",
    "\n",
    "Une variable aléatoire suit la loi de Bernoulli de paramètre $p$ si elle vaut 1 avec probabilité $p$ et 0 avec probabilité $1-p$.\n",
    "\n",
    "On tire un nombre aléatoire uniforme dans $x\\in [0,1]$, et la variable aléatoire vaut 1 si $x<p$ et 0 sinon.\n",
    "\n",
    "Programmons ceci en faisant une boucle python :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bernoulli(p,N):\n",
    "    \"\"\"\n",
    "    Fabrique un tableau numpy de taille N, contenant N tirages de \n",
    "    Bernoulli de paramètre p\n",
    "    \"\"\"\n",
    "    x = npr.rand(N)\n",
    "    for i in range(N):\n",
    "        if x[i] < p:\n",
    "            x[i] = 1\n",
    "        else:\n",
    "            x[i] = 0\n",
    "    return x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Une petite astuce : si $X$ suit une loi uniforme sur $[0,1]$, alors la partie entière $\\lfloor p+X \\rfloor$ suit la loi de Bernoulli de paramètre $p$.\n",
    "\n",
    "Ceci permet d'utiliser facilement `numpy` :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bernoulli2(p,N):\n",
    "    x = npr.rand(N)\n",
    "    return np.floor(p+x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On a appris que  l'espérance et la variance de la loi binomiale valent respectivement $p$ et $p(1-p)$. \n",
    "\n",
    "Vérifions-le en calculant la moyenne emprique et la variance empirique d'un grand échantillon :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Moyenne : 0.3332621\n",
      "Variance : 0.22219847270359\n"
     ]
    }
   ],
   "source": [
    "p = 1/3\n",
    "N = 10000000\n",
    "x = bernoulli2(p,N)\n",
    "print(\"Moyenne :\", np.mean(x))  # on attend 1/3\n",
    "print(\"Variance :\", np.var(x))  # on attend 2/9"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## La loi de Gauss ou loi normale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = npr.normal(0,1,100000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L'histogramme normé (option `denisty = True`) de répartitition de ce vecteur Gaussien"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([5.37907168e-05, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,\n",
       "        1.61372150e-04, 3.76535018e-04, 3.22744301e-04, 1.12960505e-03,\n",
       "        1.72130294e-03, 3.81914089e-03, 5.32528096e-03, 1.08119341e-02,\n",
       "        1.57068893e-02, 2.47437297e-02, 3.95899676e-02, 5.67492062e-02,\n",
       "        7.83192836e-02, 1.08388294e-01, 1.49914728e-01, 1.92678348e-01,\n",
       "        2.28502965e-01, 2.74225074e-01, 3.19032741e-01, 3.50177566e-01,\n",
       "        3.75405412e-01, 3.96007257e-01, 3.95469350e-01, 3.88691720e-01,\n",
       "        3.64916223e-01, 3.22582929e-01, 2.83423287e-01, 2.47222134e-01,\n",
       "        1.95314093e-01, 1.56477195e-01, 1.23395904e-01, 8.53658675e-02,\n",
       "        6.38495808e-02, 4.50766207e-02, 2.95311035e-02, 1.74819830e-02,\n",
       "        1.14036320e-02, 6.72383960e-03, 4.24946663e-03, 1.88267509e-03,\n",
       "        1.12960505e-03, 9.14442185e-04, 4.30325734e-04, 5.37907168e-05,\n",
       "        1.61372150e-04, 1.61372150e-04]),\n",
       " array([-4.85844257, -4.67253689, -4.4866312 , -4.30072552, -4.11481983,\n",
       "        -3.92891415, -3.74300847, -3.55710278, -3.3711971 , -3.18529141,\n",
       "        -2.99938573, -2.81348005, -2.62757436, -2.44166868, -2.25576299,\n",
       "        -2.06985731, -1.88395162, -1.69804594, -1.51214026, -1.32623457,\n",
       "        -1.14032889, -0.9544232 , -0.76851752, -0.58261184, -0.39670615,\n",
       "        -0.21080047, -0.02489478,  0.1610109 ,  0.34691658,  0.53282227,\n",
       "         0.71872795,  0.90463364,  1.09053932,  1.276445  ,  1.46235069,\n",
       "         1.64825637,  1.83416206,  2.02006774,  2.20597342,  2.39187911,\n",
       "         2.57778479,  2.76369048,  2.94959616,  3.13550184,  3.32140753,\n",
       "         3.50731321,  3.6932189 ,  3.87912458,  4.06503026,  4.25093595,\n",
       "         4.43684163]),\n",
       " <a list of 50 Patch objects>)"
      ]
     },
     "execution_count": 10,
     "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": [
    "plt.hist(x,50,density=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L'histogramme cumulatife (option `cumulative = True`). On trouve la fonction de répartition de la loi de probabilités."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([1.0000e-05, 1.0000e-05, 1.0000e-05, 1.0000e-05, 4.0000e-05,\n",
       "        1.1000e-04, 1.7000e-04, 3.8000e-04, 7.0000e-04, 1.4100e-03,\n",
       "        2.4000e-03, 4.4100e-03, 7.3300e-03, 1.1930e-02, 1.9290e-02,\n",
       "        2.9840e-02, 4.4400e-02, 6.4550e-02, 9.2420e-02, 1.2824e-01,\n",
       "        1.7072e-01, 2.2170e-01, 2.8101e-01, 3.4611e-01, 4.1590e-01,\n",
       "        4.8952e-01, 5.6304e-01, 6.3530e-01, 7.0314e-01, 7.6311e-01,\n",
       "        8.1580e-01, 8.6176e-01, 8.9807e-01, 9.2716e-01, 9.5010e-01,\n",
       "        9.6597e-01, 9.7784e-01, 9.8622e-01, 9.9171e-01, 9.9496e-01,\n",
       "        9.9708e-01, 9.9833e-01, 9.9912e-01, 9.9947e-01, 9.9968e-01,\n",
       "        9.9985e-01, 9.9993e-01, 9.9994e-01, 9.9997e-01, 1.0000e+00]),\n",
       " array([-4.85844257, -4.67253689, -4.4866312 , -4.30072552, -4.11481983,\n",
       "        -3.92891415, -3.74300847, -3.55710278, -3.3711971 , -3.18529141,\n",
       "        -2.99938573, -2.81348005, -2.62757436, -2.44166868, -2.25576299,\n",
       "        -2.06985731, -1.88395162, -1.69804594, -1.51214026, -1.32623457,\n",
       "        -1.14032889, -0.9544232 , -0.76851752, -0.58261184, -0.39670615,\n",
       "        -0.21080047, -0.02489478,  0.1610109 ,  0.34691658,  0.53282227,\n",
       "         0.71872795,  0.90463364,  1.09053932,  1.276445  ,  1.46235069,\n",
       "         1.64825637,  1.83416206,  2.02006774,  2.20597342,  2.39187911,\n",
       "         2.57778479,  2.76369048,  2.94959616,  3.13550184,  3.32140753,\n",
       "         3.50731321,  3.6932189 ,  3.87912458,  4.06503026,  4.25093595,\n",
       "         4.43684163]),\n",
       " <a list of 50 Patch objects>)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXQAAAD4CAYAAAD8Zh1EAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8QZhcZAAANSklEQVR4nO3dX4wd51nH8e+vbkMvmjYSXgTyn9oSrlQrVAosbqVcNJAUOWlk31TURkEtRPVNDa2aglKCQhVu0kaiBNVQrFD1D6WWW/50VVy5BYJAiFTetEmEbVKtTKi3LoqbhACqIFg8XOxxdLo56zPrnN3Zfff7kSydmXk15/FY+unxOzPvSVUhSVr/XtF3AZKkyTDQJakRBrokNcJAl6RGGOiS1IhX9vXFmzdvrh07dvT19ZK0Lj366KPfq6qpUcd6C/QdO3YwOzvb19dL0rqU5F+XOuaUiyQ1wkCXpEYY6JLUCANdkhphoEtSIwx0SWrE2EBP8skkTyf5pyWOJ8nvJZlL8kSSn5x8mZKkcbp06J8C9l7h+K3ArsGfQ8AfvPyyJEnLNTbQq+rvgGevMGQ/8Jla8AhwXZIfm1SBkqRuJvGm6Bbg/ND2/GDfdxcPTHKIhS6e7du3T+CrJa13O+7+y75LWHVP3f/2FTnvJAI9I/aN/BmkqjoKHAWYnp72p5KkBm3EgF4rJvGUyzywbWh7K3BhAueVJC3DJDr0GeBwkmPAm4Hnq+ol0y2S2mInvvaMDfQknwduAjYnmQd+C3gVQFV9AjgB3AbMAd8HfmmlipW0ugzt9WVsoFfVwTHHC3jvxCqSJF0V3xSVpEb09gMXktYOp1baYIcuSY0w0CWpEU65SBuIUytts0OXpEYY6JLUCANdkhphoEtSI7wpKjXIm58bkx26JDXCQJekRhjoktQIA12SGmGgS1IjfMpFWsd8mkXD7NAlqREGuiQ1wkCXpEYY6JLUCANdkhphoEtSI3xsUVoHfDxRXdihS1IjDHRJaoSBLkmNMNAlqREGuiQ1wkCXpEYY6JLUCANdkhphoEtSIzq9KZpkL/AgsAl4qKruX3R8O/Bp4LrBmLur6sSEa5Wa5xuhejnGduhJNgFHgFuB3cDBJLsXDftN4HhV3QAcAH5/0oVKkq6sy5TLHmCuqs5V1QvAMWD/ojEFvHbw+XXAhcmVKEnqokugbwHOD23PD/YN+zBwR5J54ATwK6NOlORQktkksxcvXryKciVJS+kS6BmxrxZtHwQ+VVVbgduAzyZ5ybmr6mhVTVfV9NTU1PKrlSQtqUugzwPbhra38tIplTuB4wBV9Y/Aq4HNkyhQktRNl0A/BexKsjPJNSzc9JxZNObbwM0ASd7IQqA7pyJJq2hsoFfVJeAwcBI4y8LTLKeT3Jdk32DYXcB7kjwOfB54d1UtnpaRJK2gTs+hD54pP7Fo371Dn88AN062NEnScvimqCQ1wkCXpEb4I9FSD3zFXyvBDl2SGmGgS1IjDHRJaoSBLkmNMNAlqREGuiQ1wkCXpEYY6JLUCANdkhphoEtSIwx0SWqEa7lIK8g1W7Sa7NAlqREGuiQ1wkCXpEYY6JLUCANdkhphoEtSIwx0SWqEgS5JjTDQJakRBrokNcJAl6RGGOiS1AgDXZIa4WqL0gS4qqLWAjt0SWqEgS5JjTDQJakRnQI9yd4kTyaZS3L3EmN+PsmZJKeT/Mlky5QkjTP2pmiSTcAR4G3APHAqyUxVnRkaswv4EHBjVT2X5EdWqmBJ0mhdOvQ9wFxVnauqF4BjwP5FY94DHKmq5wCq6unJlilJGqdLoG8Bzg9tzw/2DXsD8IYk/5DkkSR7J1WgJKmbLs+hZ8S+GnGeXcBNwFbg75NcX1X//gMnSg4BhwC2b9++7GIlSUvr0qHPA9uGtrcCF0aM+VJV/W9V/QvwJAsB/wOq6mhVTVfV9NTU1NXWLEkaoUugnwJ2JdmZ5BrgADCzaMxfAD8DkGQzC1Mw5yZZqCTpysYGelVdAg4DJ4GzwPGqOp3kviT7BsNOAs8kOQM8DPxaVT2zUkVLkl6q01ouVXUCOLFo371Dnwv4wOCPJKkHvikqSY0w0CWpES6fKy2Dy+RqLbNDl6RGGOiS1AgDXZIaYaBLUiMMdElqhIEuSY0w0CWpEQa6JDXCQJekRhjoktQIA12SGmGgS1IjDHRJaoSBLkmNcPlcaRGXyNV6ZYcuSY0w0CWpEQa6JDXCQJekRhjoktQIA12SGmGgS1IjDHRJaoSBLkmNMNAlqREGuiQ1wkCXpEYY6JLUCANdkhphoEtSIzqth55kL/AgsAl4qKruX2LcO4AvAD9dVbMTq1JaAa57rtaM7dCTbAKOALcCu4GDSXaPGHct8KvA1yddpCRpvC5TLnuAuao6V1UvAMeA/SPG/TbwUeC/J1ifJKmjLoG+BTg/tD0/2PeiJDcA26rqy1c6UZJDSWaTzF68eHHZxUqSltYl0DNiX714MHkF8DHgrnEnqqqjVTVdVdNTU1Pdq5QkjdUl0OeBbUPbW4ELQ9vXAtcDf5vkKeAtwEyS6UkVKUkar0ugnwJ2JdmZ5BrgADBz+WBVPV9Vm6tqR1XtAB4B9vmUiyStrrGBXlWXgMPASeAscLyqTie5L8m+lS5QktRNp+fQq+oEcGLRvnuXGHvTyy9LkrRcvikqSY0w0CWpEQa6JDXCQJekRhjoktSITk+5SOuZqypqo7BDl6RGGOiS1AgDXZIaYaBLUiMMdElqhIEuSY0w0CWpEQa6JDXCQJekRhjoktQIA12SGmGgS1IjDHRJaoSrLaoZrqqojc4OXZIaYaBLUiMMdElqhIEuSY0w0CWpEQa6JDXCQJekRhjoktQIA12SGmGgS1IjDHRJaoRruWjdcc0WabROHXqSvUmeTDKX5O4Rxz+Q5EySJ5L8dZLXT75USdKVjA30JJuAI8CtwG7gYJLdi4Z9E5iuqjcBXwQ+OulCJUlX1qVD3wPMVdW5qnoBOAbsHx5QVQ9X1fcHm48AWydbpiRpnC6BvgU4P7Q9P9i3lDuBr4w6kORQktkksxcvXuxepSRprC6BnhH7auTA5A5gGnhg1PGqOlpV01U1PTU11b1KSdJYXZ5ymQe2DW1vBS4sHpTkFuAe4K1V9T+TKU+S1FWXDv0UsCvJziTXAAeAmeEBSW4A/hDYV1VPT75MSdI4YwO9qi4Bh4GTwFngeFWdTnJfkn2DYQ8ArwG+kOSxJDNLnE6StEI6vVhUVSeAE4v23Tv0+ZYJ1yVJWibfFNWa5Ruh0vK4loskNcJAl6RGGOiS1AgDXZIaYaBLUiMMdElqhIEuSY0w0CWpEb5YpN75ApE0GXboktQIA12SGmGgS1IjDHRJaoSBLkmNMNAlqREGuiQ1wufQtWp83lxaWXboktQIA12SGmGgS1IjDHRJaoQ3RTVx3vyU+mGHLkmNMNAlqREGuiQ1wkCXpEZ4U1RXzZuf0tpihy5JjTDQJakRTrloLKdWpPXBDl2SGmGHrhfZiUvrW6dAT7IXeBDYBDxUVfcvOv5DwGeAnwKeAd5ZVU9NtlRNgqEttWtsoCfZBBwB3gbMA6eSzFTVmaFhdwLPVdWPJzkAfAR450oUrG4Mbmnj6dKh7wHmquocQJJjwH5gOND3Ax8efP4i8PEkqaqaYK0bmgEtaZwugb4FOD+0PQ+8eakxVXUpyfPADwPfGx6U5BBwaLD5X0mevJqie7CZRX+XDcrr4DW4zOvwMq5BPvKyvvf1Sx3oEugZsW9x591lDFV1FDja4TvXlCSzVTXddx198zp4DS7zOqzNa9DlscV5YNvQ9lbgwlJjkrwSeB3w7CQKlCR10yXQTwG7kuxMcg1wAJhZNGYGeNfg8zuAv3H+XJJW19gpl8Gc+GHgJAuPLX6yqk4nuQ+YraoZ4I+AzyaZY6EzP7CSRfdg3U0TrRCvg9fgMq/DGrwGsZGWpDb46r8kNcJAl6RGGOjLlOSDSSrJ5r5r6UOSB5L8c5Inkvx5kuv6rmm1JNmb5Mkkc0nu7rue1ZZkW5KHk5xNcjrJ+/quqU9JNiX5ZpIv913LZQb6MiTZxsISCN/uu5YefQ24vqreBHwL+FDP9ayKoSUwbgV2AweT7O63qlV3Cbirqt4IvAV47wa8BsPeB5ztu4hhBvryfAz4dUa8NLVRVNVXq+rSYPMRFt5L2AheXAKjql4ALi+BsWFU1Xer6huDz//JQpht6beqfiTZCrwdeKjvWoYZ6B0l2Qd8p6oe77uWNeSXga/0XcQqGbUExoYMM4AkO4AbgK/3W0lvfpeF5u7/+i5kmOuhD0nyV8CPjjh0D/AbwM+tbkX9uNJ1qKovDcbcw8J/wT+3mrX1qNPyFhtBktcAfwq8v6r+o+96VluS24Gnq+rRJDf1Xc8wA31IVd0yan+SnwB2Ao8ngYVphm8k2VNV/7aKJa6Kpa7DZUneBdwO3LyB3gjusgRG85K8ioUw/1xV/Vnf9fTkRmBfktuAVwOvTfLHVXVHz3X5YtHVSPIUMF1VG261ucGPnfwO8Naquth3PatlsEbRt4Cbge+wsCTGL1TV6V4LW0VZ6GY+DTxbVe/vu561YNChf7Cqbu+7FnAOXcv3ceBa4GtJHkvyib4LWg2DG8GXl8A4CxzfSGE+cCPwi8DPDv7tHxt0qVoj7NAlqRF26JLUCANdkhphoEtSIwx0SWqEgS5JjTDQJakRBrokNeL/AYU1FCAOyxkWAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(x,50,cumulative=True,density=True)"
   ]
  },
  {
   "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}