Simulation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il faut commencer par importer des bibliothèques (qui contiennent les fonctions utiles, comme les fonctions pour générer un nombre pris uniforomément dans $[0,1]$, pour tracer des graphes, etc.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy.random as rd\n",
    "from math import *\n",
    "import matplotlib.pyplot as plt\n",
    "import ipywidgets as widgets"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## La fonction random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La fonction \"random( )\" (de la bibliothèque numpy.random) génère un nombre *pseudo-aléatoire*, qui ressemble beaucoup à un nombre pris dans l'intervalle $[0,1]$ de manière aléatoire. Des appels succesifs à cette fonction sont des tirages *indépendants* de nombres pris dans l'intervalle $[0,1]$ de manière aléatoire.\n",
    "\n",
    "On dit que la fonction random( ) *génère* des variables aléatoires de loi $\\mathcal{U}(0,1)$.\n",
    "\n",
    "Le programme suivant imprime 10 appels à la fonction random( ). Vous pouvez l'exécuter plusieurs fois pour voir que les nombres renvoyés sont différents."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7885523085127445\n",
      "0.2389233961762086\n",
      "0.4346364196380408\n",
      "0.6553252983222685\n",
      "0.461555932553837\n",
      "0.6519329691412274\n",
      "0.617558554496986\n",
      "0.5075912630037495\n",
      "0.8646036238016096\n",
      "0.7212590274371726\n"
     ]
    }
   ],
   "source": [
    "for i in range(10):\n",
    "    print(rd.random())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut dessiner un histogramme des résultats de 100 appels à la fonction random( ). La hauteur des barres correspond au nombre de fois que la fonction random( ) renvoie un nombre dans l'intervalle donné par la barre.\n",
    "\n",
    "Vous pouvez exécuter le programme suivant plusieurs fois: vous obtiendrez des histogrammes différents, car les 100 appels à la fonction random( ) donnent des résultats aléatoires (il y a de grandes chances que les 100 appels suivants donneront des résultats différents)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXAAAAD4CAYAAAD1jb0+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjMsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+AADFEAAAN7klEQVR4nO3dbYyl5V3H8e+vO9AWpYLuohuWZcFQIiEmkBOlNmm1lGZFAr4gBhIq1dVNa1rrUwoNL2r0jfjQqpFYJy1CLdJarHZDrBYpBG0AHZ4Xtg9Igd0tdIdg0dgo0P59MQfdTHfn3HPOPefstfP9JCdz7vvcc67/NefMb++5zn1dm6pCktSeV826AEnSeAxwSWqUAS5JjTLAJalRBrgkNWpumo1t3Lixtm3bNs0mJal5991333NVtWn5/qkG+LZt21hYWJhmk5LUvCRPHWq/QyiS1CgDXJIaZYBLUqMMcElqlAEuSY0ywCWpUSMDPMn1SQ4k2X2Ix34jSSXZuDblSZIOp8sZ+A3A9uU7k5wCXAA83XNNkqQORgZ4Vd0FPH+Ihz4EvA9wQXFJmoGxxsCTXAzsr6qHOhy7M8lCkoXFxcVxmpu5zVu2kmTqt81bts6665KOYKueSp/kOOAa4G1djq+qeWAeYDAYNHm2/uz+vZx61a1Tb/epay+aepuS2jHOGfgPAqcBDyV5EtgC3J/kB/osTJK0slWfgVfVI8BJr2wPQ3xQVc/1WJckaYQulxHeDNwNnJlkX5Ida1+WJGmUkWfgVXX5iMe39VaNJKkzZ2JKUqMMcElqlAEuSY0ywCWpUQa4JDXKAJekRhngktQoA1ySGmWAS1KjDHB9h1ktn+sSutLqrHoxKx39ZrV8LriErrQanoFLUqMMcElqlAEuSY0ywCWpUQa4JDXKAJekRhngktQoA1ySGmWAS1Kjuvyv9NcnOZBk90H7fi/JF5M8nORvkpywtmVKkpbrcgZ+A7B92b7bgLOr6oeBLwPv77kuSdIIIwO8qu4Cnl+273NV9fJw8x5gyxrUJklaQR9j4D8PfPZwDybZmWQhycLi4mIPza0jG46ZyYqAmq5Zrf7oyo/tm2g1wiTXAC8DNx3umKqaB+YBBoNBTdLeuvOtl2ayKqArAk7XrFZ/9HVu39gBnuRK4CLg/KoymCVpysYK8CTbgauAN1fVN/stSZLURZfLCG8G7gbOTLIvyQ7gT4DjgduSPJjkw2tcpyRpmZFn4FV1+SF2f3QNapEkrYIzMSWpUQa4JDXKAJekRhngktQoA1ySGmWAS1KjDHBJapQBLkmNMsAlqVEGuCQ1ygCXpEYZ4JLUKANckhplgEtSowxwSWqUAS5JjTLAJalRBrgkNcoAl6RGGeCS1CgDXJIaNTLAk1yf5ECS3Qft+94ktyX5yvDriWtbpiRpuS5n4DcA25ftuxq4varOAG4fbkuSpmhkgFfVXcDzy3ZfAtw4vH8j8NM91yVJGmHcMfDvr6pnAIZfTzrcgUl2JllIsrC4uDhmc5KOJpu3bCXJ1G9zr37tTNpNwuYtW3v/Oc71/ozLVNU8MA8wGAxqrduTdOR7dv9eTr3q1qm3+9S1F82k3Vfa7tu4Z+BfT7IZYPj1QH8lSZK6GDfAdwFXDu9fCXymn3IkSV11uYzwZuBu4Mwk+5LsAH4HuCDJV4ALhtuSpCkaOQZeVZcf5qHze65FkrQKzsSUpEYZ4JLUKANckhplgEtSowxwSWqUAS5JjTLAJalRBrgkNcoAl6RGGeCS1CgDXJIaZYBLUqMMcElqlAEuSY0ywCWpUQa4JDXKAJekRhngktQoA1ySGmWAS1KjJgrwJL+a5NEku5PcnOQ1fRUmSVrZ2AGe5GTgl4FBVZ0NbAAu66swSdLKJh1CmQNem2QOOA742uQlSZK6GDvAq2o/8PvA08AzwAtV9bnlxyXZmWQhycLi4uL4lWp92HAMSaZ+27xl66x7Lq3a3LjfmORE4BLgNOAbwKeSXFFVHz/4uKqaB+YBBoNBTVCr1oNvvcSpV9069WafuvaiqbcpTWqSIZS3Al+tqsWqegn4NPBj/ZQlSRplkgB/GjgvyXFJApwP7OmnLEnSKJOMgd8L3ALcDzwyfK75nuqSJI0w9hg4QFV9APhAT7VIklbBmZiS1CgDXJIaZYBLUqMMcElqlAEuSY0ywCWpUQa4JDXKAJekRhngktSoiWZiTtPmLVt5dv/eWZeho9VwGdt1ZT32+SjTTIA/u3/vTJYZBZcaXRdmtIwtzPD9tR77fJRxCEWSGmWAS1KjDHBJapQBLkmNMsAlqVEGuCQ1ygCXpEYZ4JLUKANckhplgEtSoyYK8CQnJLklyReT7Enyhr4KkyStbNK1UP4I+PuqujTJscBxPdQkSepg7ABP8jrgTcA7AKrqReDFfsqSJI0yyRDK6cAi8OdJHkjykSTftfygJDuTLCRZWFxcnKA5SdLBJgnwOeBc4E+r6hzgv4Crlx9UVfNVNaiqwaZNmyZoTpJ0sEkCfB+wr6ruHW7fwlKgS5KmYOwAr6pngb1JzhzuOh94rJeqJEkjTXoVynuAm4ZXoDwB/NzkJUmSupgowKvqQWDQUy2SpFVwJqYkNcoAl6RGGeCS1CgDXJIaZYBLUqMMcElqlAEuSY0ywCWpUQa4JDXKAJekRhngktQoA1ySGmWAS1KjDHBJapQBLkmNMsAlqVEGuCQ1ygCXpEYZ4JLUKANckho1cYAn2ZDkgSS39lGQJKmbPs7A3wvs6eF5JEmrMFGAJ9kC/BTwkX7KkSR1NekZ+B8C7wO+3UMtkqRVGDvAk1wEHKiq+0YctzPJQpKFxcXFcZuTJC0zyRn4G4GLkzwJfAJ4S5KPLz+oquaralBVg02bNk3QnCTpYGMHeFW9v6q2VNU24DLg81V1RW+VSZJW5HXgktSouT6epKruBO7s47kkSd14Bi5JjTLAJalRBrgkNcoAl6RGGeCS1CgDXJIaZYBLUqMMcElqlAEuSY0ywCWpUQa4JDXKAJekRhngktQoA1ySGmWAS1KjDHBJapQBLkmNMsAlqVEGuCQ1ygCXpEYZ4JLUqLEDPMkpSe5IsifJo0ne22dhkqSVzU3wvS8Dv15V9yc5HrgvyW1V9VhPtUmSVjD2GXhVPVNV9w/v/yewBzi5r8IkSSvrZQw8yTbgHODeQzy2M8lCkoXFxcU+mpMk0UOAJ/lu4K+BX6mq/1j+eFXNV9WgqgabNm2atDlJ0tBEAZ7kGJbC+6aq+nQ/JUmSupjkKpQAHwX2VNUH+ytJktTFJGfgbwTeDrwlyYPD24U91SVJGmHsywir6p+B9FiLJGkVnIkpSY0ywCWpUQa4JDXKAJekRhngktQoA1ySGmWAS1KjDHBJapQBLkmNMsAlqVEGuCQ1ygCXpEYZ4JLUKANckhplgEtSowxwSWqUAS5JjTLAJalRBrgkNcoAl6RGGeCS1KiJAjzJ9iRfSvJ4kqv7KkqSNNrYAZ5kA3Ad8JPAWcDlSc7qqzBJ0somOQP/EeDxqnqiql4EPgFc0k9ZkqRRUlXjfWNyKbC9qn5huP124Eer6t3LjtsJ7Bxungl8aZVNbQSeG6vItq3Hftvn9WE99hkm6/epVbVp+c65CYrJIfZ9x78GVTUPzI/dSLJQVYNxv79V67Hf9nl9WI99hrXp9yRDKPuAUw7a3gJ8bbJyJEldTRLg/wqckeS0JMcClwG7+ilLkjTK2EMoVfVykncD/wBsAK6vqkd7q+z/jT380rj12G/7vD6sxz7DGvR77A8xJUmz5UxMSWqUAS5JjTpiAnzUtPwkr07yyeHj9ybZNv0q+9Whz7+W5LEkDye5Pcmps6izb12XYEhyaZJK0vwlZ136nORnhq/3o0n+cto19q3D+3trkjuSPDB8j184izr7lOT6JAeS7D7M40nyx8OfycNJzp2owaqa+Y2lD0H/DTgdOBZ4CDhr2TG/BHx4eP8y4JOzrnsKff4J4Ljh/Xe13ueu/R4edzxwF3APMJh13VN4rc8AHgBOHG6fNOu6p9DneeBdw/tnAU/Ouu4e+v0m4Fxg92EevxD4LEvzaM4D7p2kvSPlDLzLtPxLgBuH928Bzk9yqMlErRjZ56q6o6q+Ody8h6Vr7VvXdQmG3wZ+F/jvaRa3Rrr0+ReB66rq3wGq6sCUa+xblz4X8Lrh/e/hKJhHUlV3Ac+vcMglwMdqyT3ACUk2j9vekRLgJwN7D9reN9x3yGOq6mXgBeD7plLd2ujS54PtYOlf7taN7HeSc4BTqurWaRa2hrq81q8HXp/kC0nuSbJ9atWtjS59/k3giiT7gL8D3jOd0mZqtb/3K5pkKn2fukzL7zR1vyGd+5PkCmAAvHlNK5qOFfud5FXAh4B3TKugKejyWs+xNIzy4yz9pfVPSc6uqm+scW1rpUufLwduqKo/SPIG4C+Gff722pc3M73m2JFyBt5lWv7/HZNkjqU/uVb6U+VI12kpgiRvBa4BLq6q/5lSbWtpVL+PB84G7kzyJEvjhLsa/yCz6/v7M1X1UlV9laVF386YUn1roUufdwB/BVBVdwOvYWnBp6NZr0uQHCkB3mVa/i7gyuH9S4HP1/BTgUaN7PNwKOHPWArv1sdEX7Fiv6vqharaWFXbqmobS2P/F1fVwmzK7UWX9/ffsvShNUk2sjSk8sRUq+xXlz4/DZwPkOSHWArwxalWOX27gJ8dXo1yHvBCVT0z9rPN+lPbZZ/OfpmlT66vGe77LZZ+eWHpxf0U8DjwL8Dps655Cn3+R+DrwIPD265Z1zyNfi879k4avwql42sd4IPAY8AjwGWzrnkKfT4L+AJLV6g8CLxt1jX30OebgWeAl1g6294BvBN450Gv83XDn8kjk763nUovSY06UoZQJEmrZIBLUqMMcElqlAEuSY0ywCWpUQa4JDXKAJekRv0vXfWKHApnccIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist([rd.random() for i in range(100)],bins=10,edgecolor='black')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Passons à 1000 appels à la fonction random( ), toujours représentés sous la forme d'un histogramme."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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JDTLcJalB/wdmo9Ok5DD8MAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist([rd.random() for i in range(1000)],bins=10,edgecolor='black')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Et pour 10000 appels..."
   ]
  },
  {
   "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": [
    "plt.hist([rd.random() for i in range(10000)],bins=10,edgecolor='black')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On voir apparaître la forme de la densité de la loi uniforme! C'est normal, s'il y a a beaucoup d'appels à la fonction random( ), la proportion de fois où la valeur renvoyée tombera dans un intervalle $[a,b]$ sera proche de la probabilité $P(X\\in [a,b])$. Il s'agit de la loi des grands nombres."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Des fonctions simples de la fonction random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On va maintenant utiliser la fonction random( ) pour créer des variables aléatoires plus générales."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### La différence entre random( )$^2$ et random( )$\\times$random( )\n",
    "\n",
    "Il y a une différence entre appeler la fonction random( ) et la mettre au carré\n",
    "et appeler deux fois la fonction random( ) et faire le produit! En effet, dans le premier cas, on obtient $X^2$\n",
    "où $X\\sim \\mathcal{U}(0,1)$\n",
    "et dans le deuxième cas on obtient $X\\times Y$\n",
    "où $X\\sim \\mathcal{U}(0,1)$, $Y\\sim \\mathcal{U}(0,1)$ sont indépendantes. Il s'agit de deux variables aléatoires qui n'ont pas la même loi!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercice: \n",
    "1. Soit $X$ une variable aléatoire de loi $\\mathcal{U}(0,1)$. Montrer que $X^2$ admet pour densité $\\frac{1}{2\\sqrt{x}} \\mathbf{1}_{]0,1[}(x)$.\n",
    "2. Soient $X$ et $Y$ deux variables aléatoires indépendantes, toutes les deux de loi $\\mathcal{U}(0,1)$. Montrer que $Z= X Y$ admet pour densité $(-\\ln z) \\mathbf{1}_{]0,1[}(z)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Traçons un histogramme de 10000 appels à la fonction random( )$^2$, et comparons-le à la densité trouvée dans l'exercice ci-dessus pour $X^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "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": [
    "plt.title('Histogramme de 10000 appels à random^2')\n",
    "plt.hist([rd.random()**2 for i in range(10000)],bins=20,edgecolor='black',density=True)\n",
    "plt.plot([i*0.01 for i in range(1,100)],[0.5/sqrt(i*0.01) for i in range(1,100)])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Traçons un histogramme de 10000 appels à la fonction random( )$\\times$random( ), et comparons-le à la densité trouvée dans l'exercice ci-dessus pour $Z=X Y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "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": [
    "plt.title('Histogramme de 10000 appels à random*random')\n",
    "plt.hist([rd.random()*rd.random() for i in range(10000)],bins=20,edgecolor='black',density=True)\n",
    "plt.plot([i*0.01 for i in range(1,100)],[-log(i*0.01) for i in range(1,100)])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Obtenir une variable aléatoire de Bernoulli, une binomiale...\n",
    "\n",
    "#### Variable aléatoire de Bernoulli\n",
    "On peut aussi utiliser la fonction random( ) pour générer une variable aléatoire de Bernoulli de paramètre $p$. Il suffit de créer une épreuve dont la probabilitee de succès vaut $p$. L'épreuve est la suivante: si random( )$<p$, alors c'est un succès; si random( )$\\geq p$, alors c'est un échec."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Bernoulli(p):\n",
    "    if rd.random()<p:\n",
    "        return 1\n",
    "    else:\n",
    "        return 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Voyons ce que donne 200 appels successifs à la fonction Bernoulli($p$). Vous pouvez changer la valeur de $p$ dans la première ligne. On obtient une suite de 0 et de 1 aléatoire."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 1 0 "
     ]
    }
   ],
   "source": [
    "p=0.3\n",
    "for i in range(200):\n",
    "    print(Bernoulli(p), end =\" \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Variable aléatoire binomiale\n",
    "Pour générer une variable aléatoire binomiale de paramètres $n,p$, il suffit d'utiliser la fonction Bernoulli que l'on vient de créer: pour rappel, une  variable aléatoire binomiale de paramètres $n,p$ est une somme de $n$ variables aléatoires de Bernoulli de paramètre $p$ indépendantes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Binomiale(n,p):\n",
    "    x=0\n",
    "    for i in range(n):\n",
    "        x=x+Bernoulli(p)\n",
    "    return x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Voyons ce que donne 200 appels successifs à la fonction Binomiale($n,p$). Vous pouvez changer les valeurs de $n$ et $p$ dans la première ligne. On obtient une suite de 0 et de 1 aléatoire."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11 7 10 9 7 4 8 4 8 7 9 8 12 4 7 7 9 8 7 10 8 12 8 6 7 11 8 9 6 5 8 11 10 10 11 16 7 9 10 10 8 9 9 9 10 9 6 10 8 6 8 7 10 9 6 8 7 8 5 3 5 10 6 8 5 10 7 10 6 10 9 8 6 8 7 11 8 13 9 4 4 8 7 7 2 9 9 7 5 8 6 9 6 13 9 6 10 7 10 5 6 9 7 8 10 9 10 6 9 7 9 10 7 8 5 9 10 6 10 12 6 8 7 8 8 6 4 8 8 9 10 6 11 7 5 9 11 8 8 9 9 6 7 9 8 9 6 8 7 8 3 10 7 8 7 12 6 10 9 11 11 8 6 6 10 8 8 8 10 10 8 8 6 7 4 8 8 9 7 8 7 6 8 11 8 8 6 9 11 7 8 12 11 6 7 7 5 9 7 6 "
     ]
    }
   ],
   "source": [
    "n=20\n",
    "p=0.4\n",
    "for i in range(200):\n",
    "    print(Binomiale(n,p), end =\" \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercice: Variable aléatoire géométrique\n",
    "Compléter le programme suivant, qui permet de générer une variable aléatoire géométrique: pour rappel, \n",
    "une variable aléatoire géométrique de paramètre $p$ est l'instant de premier succès parmi des épreuves indépendantes de probabilité de succès $p$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Geom(p):\n",
    "    x=1\n",
    "    while Bernoulli(p)==0:  ### Bernoulli(p)==0 correspond à un échec lors de l'épreuve\n",
    "        ??\n",
    "    return x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 "
     ]
    }
   ],
   "source": [
    "p=0.5\n",
    "for i in range(200):\n",
    "    print(Geom(p), end =\" \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quelques variables aléatoires à densité: exponentielle et loi normale\n",
    "\n",
    "#### Variable aléatoire exponentielle\n",
    "\n",
    "Exercice: montrer que si $X\\sim \\mathcal{U}(0,1)$, alors $- \\frac{1}{a}\\ln X$ suit une loi exponentielle de paramètre $a$. \n",
    "\n",
    "On peut utiliser ce fait pour générer une variable aléatoire exponentielle de paramètre $a$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Expo(a):\n",
    "    return -log(rd.random())/a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Imprimons $50$ appels à la fonction Expo(a). Vous pouvez modifier la valeur de $a$ pour voir l'effet que cela a sur la suite (aléatoire) de nombre renvoyés."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.46226854479841517 3.5703559277709793 0.008893938255642046 0.07390587794632676 3.6692817573389775 1.7638647157594673 1.4721825396690578 0.6857697203671755 0.5235565160084555 0.28115163277304084 0.630204682252933 0.05226146133481815 5.062819641032205 3.0245748628865843 4.820369990633732 2.580989907455623 2.721838821594953 6.921880825419322 0.9720955074696924 0.26177802911362125 2.6021925836013624 0.22289802545834259 8.587973936109735 4.975915906770565 0.1283688420443145 0.3731832520944285 4.340469113377698 1.3674647168479646 0.6871910525875512 0.9671837807589271 1.2877006644776066 4.182004336488164 1.4766011522961395 0.25916710276253435 4.957816530921072 1.352149529357208 0.28074013012965426 1.6586340718571566 8.75166857050309 0.7210701788121311 0.11447430256270538 0.12185383441512572 2.5252773037713028 0.2362284721889002 0.36238434644073186 0.02232819017878788 0.012973566407349757 0.5062341204447646 0.8028896187870233 0.2836458120863107 "
     ]
    }
   ],
   "source": [
    "a=0.5\n",
    "for i in range(50):\n",
    "    print(Expo(a), end=\" \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Traçons maintenant un histogramme de $k=10000$ appels à la fonction Expo(a), et comparons-le à la densité de la loi exponentielle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "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": [
    "a=0.5\n",
    "k=10000\n",
    "plt.title('Histogramme de {} appels à la fonction Expo'.format(k))\n",
    "plt.hist([Expo(a) for i in range(k)],bins=20,edgecolor='black',density=True)\n",
    "plt.plot([i*0.01 for i in range(1,int(600/a))],[a*exp(-a*i*0.01) for i in range(1,int(600/a))])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Variable aléatoire normale\n",
    "\n",
    "Exercice (difficile): Montrer que si $U\\sim \\mathcal{U}(0,1)$ et $V\\sim \\mathcal{U}(0,1)$ sont indépendantes, \n",
    "alors $Z:= \\sqrt{- 2\\ln U} \\times \\sin (2\\pi V)$ suit une loi $\\mathcal{N}(0,1)$.\n",
    "\n",
    "On peut utiliser ce fait pour générer une variable aléatoire de loi normale."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Normale():\n",
    "    return sqrt(-2*log(rd.random()))*sin(2*pi*rd.random())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Traçons maintenant un histogramme de $k=10000$ appels à la fonction Normale(), et comparons-le à la densité de la loi normale."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "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": [
    "k=10000\n",
    "plt.title('Histogramme de {} appels à la fonction Normale'.format(k))\n",
    "plt.hist([Normale() for i in range(k)],bins=20,edgecolor='black',density=True)\n",
    "plt.plot([i*0.01 for i in range(-400,400)],[1/sqrt(2*pi)*exp(-0.5*(i*0.01)**2) for i in range(-400,400)])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quelques exemples supplémentaires\n",
    "\n",
    "Une variable aléatoire est dite de Cauchy si elle a pour densité $\\frac{1}{\\pi} \\frac{1}{1+x^2}$.\n",
    "\n",
    "Exercice: Montrer que si $U\\sim \\mathcal{U}(0,1)$, alors $\\tan(\\pi (U-\\frac12))$ suit une loi de Cauchy. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Cauchy():\n",
    "    return tan(pi*(rd.random()-0.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Traçons maintenant un histogramme de $k=10000$ appels à la fonction Normale(), et comparons-le à la densité de la loi normale."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "scrolled": true
   },
   "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": [
    "k=10000\n",
    "plt.title('Histogramme de {} appels à la fonction Cauchy'.format(k))\n",
    "plt.hist([Cauchy() for i in range(10000)],range=(-6,6),bins=20,edgecolor='black',density=True)\n",
    "plt.plot([i*0.01 for i in range(-600,600)],[1/pi*1/(1+(i*0.01)**2) for i in range(-600,600)])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quelques exercices pour s'entraîner"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercice\n",
    "Soit $X$ une variable aléatoire de loi $\\mathcal{U}(0,1)$.\n",
    "1. Déterminer la densité de $Y=\\cos(\\pi X)$.\n",
    "2. Réaliser un histogramme de 10000 appels à la fonction $\\cos$(pi*random( )), et comparer avec la densité trouvée."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercice\n",
    "Soient $X,Y$ deux variables aléatoires indépendantes de loi $\\mathcal{U}(0,1)$.\n",
    "1. Déterminer la densité de $X+Y$.\n",
    "2. Réaliser un histogramme de 10000 appels à la fonction random( )+random( ), et comparer avec la densité trouvée."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercice\n",
    "Soient $X,Y$ deux variables aléatoires indépendantes de loi exponentialle de paramètre $1$.\n",
    "1. Déterminer la densité de $Z=XY$.\n",
    "2. Réaliser un histogramme de 10000 appels à la fonction Expo(1)*Expo(1), et comparer avec la densité trouvée."
   ]
  },
  {
   "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}