Application_4_simu.ipynb

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     "data": {
      "application/javascript": [
       "$.getScript('ipython_notebook_toc.js')\n"
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       "<IPython.core.display.Javascript object>"
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   "source": [
    "%%javascript\n",
    "$.getScript('ipython_notebook_toc.js')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[comment]: <> (emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp; ### <left>ESTP-Cachan </left>  <right> ESTP-Cachan </right> )\n",
    "\n",
    "<div style=\"float: left; margin-right: 20px;font-size:20px\" ><b> ESTP-Cachan </b></div> <div style=\"float: right; margin-right: 20px;font-size:20px\" ><b> Février 2020 </b></div>\n",
    "&nbsp;\n",
    "<hr> </hr>\n",
    "\n",
    "<div style=\"text-align: center;font-size:30px;font-weight:bold\"> Application 4 : simulation </div>\n",
    "<hr> </hr>\n",
    "\n",
    "\n",
    "[comment]: <> (# <div style=\"text-align: left\"> your-text-here</div> <div style=\"text-align: right\"> your-text-here </div> <a href=\"https://en.wikipedia.org\">click me</a> we have three types of horizontal rules ---- ; _____ ; ***** and we can also do tables)  \n",
    "   "
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"float: left; margin-right: 20px;font-size:27px;font-weight:bold\" id=\"tocheading\">Table of Contents</div>\n",
    "<div style=\"height:30px\" id=\"spacebeforetoc\"> </div>\n",
    "<div style=\"font-size:20px;font-weight:bold\" id=\"toc\"></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"float: left; margin-right: 20px;font-size:27px;font-weight:bold\" id=\"tocheading\">Importer les packages (obligatoire)</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercice 1: simulation loi Binomiale"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le but de cette application est de simuler des réalisations de plusieurs variables aléatoires à partir d'une loi uniforme sur $[0,1]$.\n",
    "\n",
    "On commence donc par présenter comment simuler des réalisations d'une loi uniforme sur $[0,1]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.45141147 0.819125   0.50877184 0.753212   0.93427769 0.27312071]\n"
     ]
    }
   ],
   "source": [
    "nb_var = 6 ## nombre de réalisations de la loi uniforme.\n",
    "u = np.random.rand(nb_var)\n",
    "print(u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Soit $X$ une variable aléatoire suivant une loi de Bernoulli de paramètre $p \\in [0,1]$. Proposer un algorithme pour simuler $X$ et implémenter une fonction ``Simu_Bernoulli`` qui simule $n$ réalisations de $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient les 6 réalisations suivantes pour X : [0 0 1 0 0 1] \n"
     ]
    }
   ],
   "source": [
    "def Simu_Bernoulli(p, n):\n",
    "    u = np.random.rand(n)\n",
    "    return (u <= p).astype('int32')\n",
    "p = 0.5\n",
    "n = 6\n",
    "print(\" On obtient les %i réalisations suivantes pour X : %s \" %(n,Simu_Bernoulli(p, n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Soit $Y$ une variable aléatoire suivant une loi Binomiale de paramètre $n \\in \\mathbb{N}^*$ et $p \\in [0,1]$. En utilisant le résultat de la première question, implémenter une fonction ``Simu_Bino`` qui prend $n$ réalisations de $X$ et qui renvoie une réalisation de $Y$. Tester la."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour Y : 3 \n"
     ]
    }
   ],
   "source": [
    "def Simu_Bino(X):\n",
    "    return X.sum()\n",
    "p = 0.5\n",
    "n = 6\n",
    "print(\" On obtient la réalisation suivante pour Y : %s \" %(Simu_Bino(Simu_Bernoulli(p, n))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. On considère le jeu suivant: vous lancez $1000$ fois une pièce de monnaie non-biaisée en l'air et à chaque fois que vous obtenez face vous remportez $10$ centimes. Grâce à la question précédente, écrire une fonction ``Gain_moyen`` qui permet d'estimer votre gain moyen. Comparer l'estimation à la valeur théorique. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " Le gain estimé est 50.009 alors que le gain théorique vaut 50.00\n"
     ]
    }
   ],
   "source": [
    "def Gain_moyen():\n",
    "    res = 0\n",
    "    for i in range(500):\n",
    "        nb_pile = Simu_Bino(Simu_Bernoulli(0.5, 1000)) \n",
    "        res += nb_pile*0.1    \n",
    "    return res/500\n",
    "gain_approx = Gain_moyen()\n",
    "gain_theo = 0.5*1000*0.1\n",
    "print(\" Le gain estimé est %3.3f alors que le gain théorique vaut %3.2f\" %(gain_approx,gain_theo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4. On reprend le même jeu. En utilisant la question 2, écrire la fonction ``Plus_que_52`` qui estime la probabilité de gagner plus que $52$ euros. Que remarquez-vous?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " Le gain estimé est 0.094.\n"
     ]
    }
   ],
   "source": [
    "def Plus_que_52():\n",
    "    res = 0.\n",
    "    for i in range(500):\n",
    "        nb_pile = Simu_Bino(Simu_Bernoulli(0.5, 1000)) \n",
    "        res += ((nb_pile*0.1) >= 52)\n",
    "    return res/500.\n",
    "proba_approx = Plus_que_52()\n",
    "print(\" Le gain estimé est %3.3f.\" %proba_approx)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercice 2: simulation loi Géométrique"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le but de cet exercice est d'apprendre à simuler une variable aléatoire $X$ suivant une loi de Géométrique de paramètre $p \\in [0,1]$. En utilisant que $X$ correspond aux nombre de répétitions nécessaires d'une variable de Bernoulli pour obtenir un succès, proposer un algorithme ``Simu_Geo1`` qui simule une réalisation de $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour Y : 2 \n"
     ]
    }
   ],
   "source": [
    "def Simu_Geo1(p):\n",
    "    count = 1\n",
    "    u = np.random.rand() ## uniforme\n",
    "    b = u<= p ## bernoulli\n",
    "    while b == 0:\n",
    "        u = np.random.rand() ## uniforme\n",
    "        b = u<= p ## bernoulli  \n",
    "        count += 1\n",
    "    return count\n",
    "p = 0.5\n",
    "print(\" On obtient la réalisation suivante pour Y : %s \" % Simu_Geo1(p) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Soit $U$ une variable uniforme sur $[0,1]$. Montrer que \n",
    "$$\n",
    "X' = 1 + \\lfloor \\cfrac{\\ln(U)}{\\ln(1-p)} \\rfloor,\n",
    "$$\n",
    "suit une loi de Géométrique de paramètre $p$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. En utilisant la question précédente, proposer une fonction ``Simu_Geo2`` qui produit $n$ réalisations de $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient les 6 réalisations suivantes pour X : [2 1 1 1 1 1] \n"
     ]
    }
   ],
   "source": [
    "def Simu_Geo2(p, n):\n",
    "    u = np.random.rand(n)\n",
    "    return 1 + (np.log(u)/np.log(1-p)).astype('int32')\n",
    "p = 0.5\n",
    "n = 6\n",
    "print(\" On obtient les %i réalisations suivantes pour X : %s \" %(n,Simu_Geo2(p, n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4. Le vecteur $e_0$ ci-dessous contient $200$ réalisations d'une variable $X$ suivant une loi Géométrique dont on ne connait pas le paramètre $p \\in [0,1]$. Ecrire une fonction ``Estim_p`` qui permet d'estimer $p$ à partir de $e_0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "e_0 = np.array([5,7,1,3,1,2,3,4,2,3,7,1,4,2,4,3,5,7,4,2,2,8,3,4,14,7,1,2,2,2,1,1,7,1,5,5,4,2,1,3,6,2,1,1,\n",
    "                3,9,3,8,4,1,2,1,1,1,4,6,2,10,3,1,3,1,4,5,2,4,2,4,4,1,1,1,6,1,2,2,3,4,12,3,1,2,4,1,1,2,4,2,\n",
    "                5,2,1,3,2,1,5,1,5,5,6,2,2,3,2,1,2,8,3,1,10,1,1,1,3,1,1,3,6,1,5,4,3,2,2,1,1,3,1,5,11,1,1,1,\n",
    "                2,13,4,4,3,13,1,1,5,2,1,1,1,10,4,2,3,1,2,2,3,7,1,1,2,1,3,1,2,3,1,1,1,2,4,6,1,5,15,2,2,2,5,\n",
    "                1,1,4,3,2,5,2,1,2,3,9,1,1,1,4,2,5,7,3,1,6,1,2,1,7])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " La valeur estimée pour p est 0.310.\n"
     ]
    }
   ],
   "source": [
    "def Estim_p(e_0):\n",
    "    return 1/e_0.mean()\n",
    "estimated_p = Estim_p(e_0)\n",
    "print(\" La valeur estimée pour p est %3.3f.\" % estimated_p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercice 3: simulation loi de Poisson"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Dans cet exercice, on souhaite simuler une variable $N$ suivant une loi de Poisson de paramètre $ \\lambda > 0$. Soit $(Y_i)_{i\\geq 1}$ des variables Exponentielles indépendantes de paramètre $\\lambda$. Montrer que \n",
    "$$\n",
    "N_1 = \\inf\\{i ; \\; \\sum_{j \\leq i +1} Y_j \\geq 1 \\},\n",
    "$$\n",
    "suit une loi de Poisson de paramètre $\\lambda$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. En utilisant que $- \\cfrac{\\ln(U)}{\\lambda} $ suit une loi Exponentielle de paramètre $\\lambda$ pour toute variable $U$ sur $[0,1]$. Montrer que \n",
    "$$\n",
    "N_2 = \\inf\\{i ; \\; \\prod_{j \\leq i +1} U_j \\leq e^{-\\lambda} \\},\n",
    "$$\n",
    "suit une loi de Poisson de paramètre $\\lambda$ avec $(U_i)_{i\\geq 1} $ des variables indépendantes toutes de loi uniforme sur $[0,1]$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. En utilisant la question précédente, écrire une fonction ``Simu_Poisson`` qui simule une réalisation de $N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour N : 0 \n"
     ]
    }
   ],
   "source": [
    "def Simu_Poisson(lambda_0):\n",
    "    count = 0\n",
    "    prod_cum = np.random.rand()\n",
    "    seuil = np.exp(-lambda_0)\n",
    "    while prod_cum > seuil:\n",
    "        prod_cum *= np.random.rand()\n",
    "        count += 1\n",
    "    return count\n",
    "lambda_0 = 1\n",
    "print(\" On obtient la réalisation suivante pour N : %s \" % Simu_Poisson(lambda_0) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercice 4: simulation loi uniforme"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. On veut simuler, dans cet exercice, une variable $X$ suivant une loi uniforme sur $\\{0,\\ldots,m-1\\}$ avec $m \\in \\mathbb{N}^*$. Pour ce faire, on s'aide d'une variable $U$ uniforme sur $[0,1]$.\n",
    "    1. Montrer que \n",
    "$$\n",
    "X_1 = \\lfloor m U \\rfloor,\n",
    "$$\n",
    "suit la même loi que $X$. \n",
    "    2. En utilisant la question précédente, proposer une fonction ``Simu_U_m`` qui simule $n$ réalisations de $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient les 6 réalisations suivantes pour X : [1 0 0 2 1 4] \n"
     ]
    }
   ],
   "source": [
    "def Simu_U_m(m,n):\n",
    "    u = np.random.rand(n)\n",
    "    return (m * u).astype('int32')\n",
    "m = 7\n",
    "n = 6\n",
    "print(\" On obtient les %i réalisations suivantes pour X : %s \" %(n,Simu_U_m(m,n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. On souhaite maintenant simuler une variable $Y$ suivant une loi uniforme sur l'ensemble $\\{a,\\ldots,b\\}$ avec $a < b \\in \\mathbb{Z}$. Pour ce faire, on utilise une variable $U$ uniforme sur $[0,1]$.\n",
    "    1. Montrer que \n",
    "$$\n",
    "X_1 = a + \\lfloor (b-a+1) \\times U \\rfloor,\n",
    "$$\n",
    "suit la même loi que $Y$. \n",
    "    2. En utilisant la question précédente, proposer une fonction ``Simu_U_ab`` qui simule $n$ réalisations de $Y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient les 6 réalisations suivantes pour X : [ 5 -3 -1  4  4  3] \n"
     ]
    }
   ],
   "source": [
    "def Simu_U_ab(a,b,n):\n",
    "    u = np.random.rand(n)\n",
    "    return a + ((b-a+1) * u).astype('int32')\n",
    "a,b = -3,7\n",
    "n = 6\n",
    "print(\" On obtient les %i réalisations suivantes pour X : %s \" %(n,Simu_U_ab(a,b, n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. On aimerait enfin simuler une variable $Z$ suivant une loi uniforme sur $[a,b]$ avec $a < b \\in \\mathbb{R}$. Pour ce faire, on recourt encore une fois à une variable $U$ uniforme sur $[0,1]$.\n",
    "    1. Montrer que \n",
    "$$\n",
    "Z_1 = a +  (b-a) \\times U ,\n",
    "$$\n",
    "suit la même loi que $Z$. \n",
    "    2. En utilisant la question précédente, proposer une fonction ``Simu_U_abr`` qui simule $n$ réalisations de $Z$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient les 6 réalisations suivantes pour X : [ 1.386 -1.414  4.963  3.968  0.306 -1.591] \n"
     ]
    }
   ],
   "source": [
    "def Simu_U_abr(a,b,n):\n",
    "    u = np.random.rand(n)\n",
    "    return a + ((b-a) * u)\n",
    "a,b = -3,7\n",
    "n = 6\n",
    "print(\" On obtient les %i réalisations suivantes pour X : %s \" %(n,Simu_U_abr(a,b, n).round(3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercice 5: simulation d'une loi discrète"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. L'objectif de cet exercice est de simuler une variable discrète $D$ à valeurs dans un ensemble fini $\\{x_0,\\ldots,x_m\\}$ tel que \n",
    "$$\n",
    "\\mathbb{P}[X =  x_i ] = p_i, \\qquad  \\forall i \\in \\{0,\\ldots, m\\}.\n",
    "$$\n",
    "\n",
    "\n",
    "* Soit $U$ une variable aléatoire uniforme sur $[0,1]$. Montrer que\n",
    "\\begin{align*}\n",
    "D_1 & = \\sum_{i \\leq m} x_i \\mathbf{1}_{p_1+ \\ldots + p_{i-1} < U \\leq p_1+ \\ldots + p_i}\\\\\n",
    "    & = x_0\\mathbf{1}_{U \\leq p_0} + x_1 \\mathbf{1}_{p_0 < U \\leq p_0 + p_1}+ \\ldots + x_i \\mathbf{1}_{p_1 + \\ldots + p_{i-1} < U \\leq p_1+ \\ldots + p_i} + \\ldots x_m \\mathbf{1}_{p_1 + \\ldots + p_{m-1} < U \\leq 1},\n",
    "\\end{align*}\n",
    "admet la même distribution que $D$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Déduire de la question précédente une fonction ``Simu_D`` qui simule une réalisations de $D$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour D : 0.40 \n"
     ]
    }
   ],
   "source": [
    "def Simu_D(x,p):\n",
    "    '''\n",
    "    x : the vector of values \n",
    "    p : the vector of probabilities\n",
    "    '''\n",
    "    count = 0\n",
    "    u = np.random.rand()\n",
    "    cum_sum = p[count]\n",
    "    while u > cum_sum:\n",
    "        count += 1\n",
    "        cum_sum += p[count]\n",
    "    return x[count]\n",
    "x = [0, 0.4, 2.4, 32, 47, 50]\n",
    "p = [0.1, 0.2, 0.1 , 0.2, 0.3, 0.1]\n",
    "print(\" On obtient la réalisation suivante pour D : %3.2f \" % Simu_D(x,p) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. La méthode utilisée ressemble-t-elle à autre technique de simulation? Quelle est la relation entre les deux?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercice 6: inversion de la fonction de répartition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le but de cet exercice est de simuler plusieurs variables aléatoires en utilisant la méthode d'inversion de la fonction de répartition. Pour ce faire, on se donne une variable uniforme $U$.\n",
    "\n",
    "1. Rappeler le principe de la méthode d'inversion de la fonction de répartition\n",
    "\n",
    "2. **Simulation d'une loi exponentielle:** Soit $X$ une variable aléatoire suivant une loi Exponentielle de paramètre $\\lambda > 0 $.\n",
    "    1. Donner la fonction de répartition de $X$.\n",
    "    2. Calculer l'inverse de la fonction de répartition.\n",
    "    3. En utilisant la méthode d'inversion de la fonction de répartition, montrer que\n",
    "$$\n",
    "X_1 = -\\cfrac{\\ln(U)}{\\lambda},\n",
    "$$\n",
    "       admet la même distribution que $X$. \n",
    "    4. Programmer une fonction ``Simu_Expo`` qui génère $n$ réalisations de $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient les 6 réalisations suivantes pour X : [1.000e-03 5.180e-01 1.068e+00 7.790e-01 6.980e-01 1.530e-01] \n"
     ]
    }
   ],
   "source": [
    "def Simu_Expo(lambda_0,n):\n",
    "    u = np.random.rand(n)\n",
    "    return - np.log(u)/lambda_0\n",
    "lambda_0 = 3\n",
    "n = 6\n",
    "print(\" On obtient les %i réalisations suivantes pour X : %s \" %(n,Simu_Expo(lambda_0,n).round(3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. **Simulation d'une loi de Cauchy:** Soit $Y$ une variable aléatoire suivant une loi de Cauchy, i.e. la densité de $X$ vaut \n",
    "$$\n",
    "f_X(x) = \\cfrac{c}{\\pi (x^2 + c^2)}, \\quad \\forall x \\in \\mathbb{R},\n",
    "$$\n",
    "avec $c > 0 $.\n",
    "    1. Donner la fonction de répartition de $X$.\n",
    "    2. Calculer l'inverse de la fonction de répartition.\n",
    "    3. En utilisant la méthode d'inversion de la fonction de répartition, montrer que\n",
    "$$\n",
    "Y_1 = c \\tan(\\pi U),\n",
    "$$\n",
    "       admet la même distribution que $Y$. \n",
    "    4. Programmer une fonction ``Simu_Cauchy`` qui génère $n$ réalisations de $Y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient les 6 réalisations suivantes pour X : [ 1.684  0.586 -4.437 -8.483 -0.256  1.081] \n"
     ]
    }
   ],
   "source": [
    "def Simu_Cauchy(c,n):\n",
    "    u = np.random.rand(n)\n",
    "    return c * np.tan( np.pi * u )\n",
    "c = 1\n",
    "n = 6\n",
    "print(\" On obtient les %i réalisations suivantes pour X : %s \" %(n,Simu_Cauchy(c,n).round(3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. **Simulation d'une loi de Pareto:** Soit $Z$ une variable aléatoire suivant une loi de Pareto, i.e. la densité de $Z$ vaut \n",
    "$$\n",
    "f_Z(x) = \\cfrac{\\theta}{x^{1+\\theta}}, \\quad \\forall x \\geq 1,\n",
    "$$\n",
    "avec $\\theta > 0 $.\n",
    "    1. Donner la fonction de répartition de $X$.\n",
    "    2. Calculer l'inverse de la fonction de répartition.\n",
    "    3. En utilisant la méthode d'inversion de la fonction de répartition, montrer que\n",
    "$$\n",
    "Z_1 = U^{-1/\\theta},\n",
    "$$\n",
    "       admet la même distribution que $Z$. \n",
    "    4. Programmer une fonction ``Simu_Pareto`` qui génère $n$ réalisations de $Z$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient les 6 réalisations suivantes pour X : [ 2.948 20.628  6.284  1.086  2.463  2.199] \n"
     ]
    }
   ],
   "source": [
    "def Simu_Pareto(theta,n):\n",
    "    u = np.random.rand(n)\n",
    "    return ( u**(-1/theta) )\n",
    "theta = 1\n",
    "n = 6\n",
    "print(\" On obtient les %i réalisations suivantes pour X : %s \" %(n,Simu_Pareto(theta,n).round(3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercice 7: simulation par la méthode de rejet"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dans cet exercice, on veut simuler certaines variables aléatoires en utilisant la méthode de rejet. Pour ce faire, on se donne une variable uniforme $U$.\n",
    "\n",
    "1. **Simulation d'une variable:** Soit $X$ une variable aléatoire suivant une loi $\\Gamma(3/2,1)$, i.e. la densité de $X$ vaut \n",
    "$$\n",
    "f_X(x) = \\cfrac{2}{\\sqrt{\\pi}} x^{1/2} e^{-x}, \\quad \\forall x \\geq 0.\n",
    "$$\n",
    "On veut simuler cette variable par la méthode de rejet en utilisant une variable Exponentielle $Y$ dont la densité vaut \n",
    "$$\n",
    "f_Y(x) = \\lambda e^{- \\lambda x}, \\quad \\forall x \\geq 0,\n",
    "$$\n",
    "avec $\\lambda > 0 $.\n",
    "    1. Montrer que\n",
    "    $$\n",
    "    c(\\lambda) = \\sup_{x \\geq 0} \\cfrac{f_X(x)}{f_Y(x)} = \\left\\{\n",
    "    \\begin{array}{ll}\n",
    "    + \\infty & \\text{ si } \\lambda> 1,\\\\\n",
    "    \\cfrac{2e^{-1/2}}{\\sqrt{2 \\pi \\lambda^2 (1-\\lambda)} }& \\text{ si } \\lambda \\in ]0,1].\n",
    "    \\end{array}\n",
    "    \\right.\n",
    "    $$\n",
    "    2. Trouver $\\lambda_m$ qui minimise $c(\\lambda)$. \n",
    "    3. Déduire de la question précédente la loi Exponentielle à utiliser pour simuler $X$ et écrire fonction ``Simu_Rejet`` qui simule une réalisation de $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour X : 0.87 \n"
     ]
    }
   ],
   "source": [
    "def f_x(x): \n",
    "    return 2 * x**(1/2) * np.exp(-x) / (np.sqrt(np.pi ))\n",
    "\n",
    "def f_y(x): \n",
    "    return 2/3 * np.exp(-2/3 * x)\n",
    "\n",
    "def Simu_Rejet(f_x,f_y,c,gen_x):\n",
    "    '''\n",
    "    f_x : densité de X\n",
    "    f_x : densité de Y \n",
    "    c : constante de la méthode de rejet\n",
    "    gen_x :  fonction qui génère X\n",
    "    '''\n",
    "    u = np.random.rand()\n",
    "    y = gen_x()\n",
    "    while c * u * f_x(y) > f_y(y):\n",
    "        u = np.random.rand()\n",
    "        y = gen_x()\n",
    "    return y\n",
    "\n",
    "lambda_star = 2/3\n",
    "gen_x = lambda : Simu_Expo(lambda_star,1)[0]\n",
    "c = 3**(3/2) * np.exp(-1/2) / (np.sqrt( 2 * np.pi ))\n",
    "print(\" On obtient la réalisation suivante pour X : %3.2f \" % Simu_Rejet(f_x,f_y,c,gen_x) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. **Simulation d'une loi Gaussienne:** Soit $Z$ une variable aléatoire suivant une loi normale centrée réduite, i.e. la densité de $Z$ vaut \n",
    "$$\n",
    "f_Z(x) = \\cfrac{1}{\\sqrt{2\\pi}} e^{-x^2/2}, \\quad \\forall x \\in \\mathbb{R}.\n",
    "$$\n",
    "On veut simuler cette variable par la méthode de rejet en utilisant la loi double exponentielle $W$ dont la densité vaut \n",
    "$$\n",
    "f_W(x) = \\cfrac{1}{2} e^{- |x|}, \\quad \\forall x \\in \\mathbb{R}.\n",
    "$$\n",
    "    1. Montrer que\n",
    "    $$\n",
    "    c = \\sup_{x \\in \\mathbb{R}} \\cfrac{f_Z(x)}{f_W(x)} = \\sqrt{\\cfrac{2e}{\\pi}}.\n",
    "    $$\n",
    "    2. Soit $E$ une variable exponentielle de paramètre $\\lambda =1$ et $B$ une variable de Bernoulli de paramètre $ p= 1/2$. Montrer que\n",
    "    $$\n",
    "    W_1 = (2 B - 1) \\times E ,\n",
    "    $$\n",
    "    suit la même loi que $W$.\n",
    "    3. En utilisant les questions précédentes, proposer une fonction ``Simu_Rejet_G`` qui simule une réalisation de $Z$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour Z : 0.32 \n"
     ]
    }
   ],
   "source": [
    "def f_z(x): \n",
    "    return  np.exp(-x*x/2) / (np.sqrt(2 * np.pi ))\n",
    "\n",
    "def f_w(x): \n",
    "    return 0.5*np.exp(-abs(x))\n",
    "\n",
    "def gen_w():\n",
    "    b = Simu_Bernoulli(0.5, 1)[0]\n",
    "    e = Simu_Expo(1,1)[0]\n",
    "    return (2*b -1) * e\n",
    "    \n",
    "def Simu_Rejet_G():\n",
    "    c = np.sqrt( 2 * np.exp(1)/ np.pi )\n",
    "    y = Simu_Rejet(f_z,f_w,c,gen_w)\n",
    "    return y\n",
    "print(\" On obtient la réalisation suivante pour Z : %3.2f \" % Simu_Rejet_G() )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercice 8: simulation d'un vecteur Gaussien"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dans cet exercice, on présente une méthode pour simuler des vecteurs Gaussiens. \n",
    "\n",
    "1. **Simulation de plusieurs Gaussiennes indépendantes:** On souhaite simuler $X = (X_1,X_2)$ deux variables Gaussiennes indépendantes. Pour ce faire, on utilise $R$ une variable exponentielle $1/2$ de paramètre et $\\theta$ une variable uniforme sur $[0,2 \\pi]$.\n",
    "    1. Montrer que\n",
    "    $$\n",
    "    \\big( \\sqrt(R) \\times \\cos(\\theta), \\sqrt{R} \\times \\sin(\\theta) \\big),\n",
    "    $$\n",
    "    suit la même loi que $X$.\n",
    "    2. Déduire de la question précédente, une fonction ``Simu_G2`` qui produit $2n$ réalisations de $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient les 6 réalisations suivantes pour X : [ 2.056  0.064 -0.784  0.776 -0.708 -2.199] \n"
     ]
    }
   ],
   "source": [
    "def Simu_G2(n):\n",
    "    r = Simu_Expo(0.5,n)\n",
    "    theta  = Simu_U_abr(0,2 * np.pi,n)\n",
    "    x_1 = np.sqrt(r) * np.cos(theta)\n",
    "    x_2 = np.sqrt(r) * np.sin(theta)\n",
    "    return np.concatenate([x_1 , x_2])\n",
    "n= 3\n",
    "print(\" On obtient les %i réalisations suivantes pour X : %s \" % (2*n,Simu_G2(n).round(3)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. **Simulation d'un vecteur Gaussien 1:** Soit $m \\in \\mathbb{N}^*$. On souhaite simuler un vecteur Gaussien $X = (X_1,\\ldots,X_m)$ de moyenne $m \\in \\mathbb{R}^d$ et de matrice de covariance $\\Sigma = (\\mathrm{Cov}(X_i,X_j))_{1\\leq i,j \\leq m}$ inversible, i.e. la densité de $X$ vaut \n",
    "$$\n",
    "f_X(x) = \\cfrac{1}{\\sqrt{2 \\pi}^{m/2} \\sqrt{det(\\Sigma)}} e^{- (x - m)^{\\top} \\Sigma^{-1} (x-m)/2 }, \\qquad \\forall x =  (x_1,\\ldots,x_m) \\in \\mathbb{R}^m.\n",
    "$$\n",
    "Pour ce faire, on suppose qu'on sait simuler $Y = (Y_1,\\ldots,Y_m) $ des variables gaussiennes centrées réduites indépendantes.\n",
    "    1. Soit $A$ une matrice et $b$ un vecteur. Montrer que\n",
    "    $$\n",
    "    b + A Y,\n",
    "    $$\n",
    "    est un vecteur Gaussien de moyenne $b$ et de matrice de covariance $A A^{\\top}$.\n",
    "    2. Montrer que $\\Sigma$ est symétrique positive, i.e. \n",
    "    $$\n",
    "    \\Sigma_{i,j} = \\Sigma_{j,i}, \\qquad x^{\\top } \\Sigma x \\geq 0, \\quad \\forall x \\in \\mathbb{R}^m.\n",
    "     $$\n",
    "    En déduire l'existence d'une matrice $L$ tel que \n",
    "    $$\n",
    "    \\Sigma =  L L^{\\top}.\n",
    "    $$\n",
    "    3. En utilisant la question précédente, écrire une fonction ``Simu_VG`` qui produit une réalisation de $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour X : [0.136 0.343 5.204] \n"
     ]
    }
   ],
   "source": [
    "def Simu_VG(m1, Sigma1):\n",
    "    n = m1.shape[0]\n",
    "    if n%2 == 0:\n",
    "        y = Simu_G2(int(n//2))\n",
    "    else:\n",
    "        y = Simu_G2(int((n+1)//2))[:n]\n",
    "    L1 = np.linalg.cholesky(Sigma1)\n",
    "    return m1 + np.dot(L1,y)\n",
    "m1 = np.array([1, - 2, 4])\n",
    "Sigma1  = np.array([[ 1, -1, 0],\n",
    "                    [ -1, 5, 6],\n",
    "                    [ 0,  6, 10]])\n",
    "print(\" On obtient la réalisation suivante pour X : %s \" % Simu_VG(m1, Sigma1).round(3) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. **Simulation d'un vecteur Gaussien 2:** Soit $\\Delta > 0$ un pas de temps et $m \\in \\mathbb{N}^*$. On observe aux instants $t_0,\\ldots,t_m$ tel que \n",
    "$$\n",
    "t_i = i \\times \\Delta, \\qquad \\forall i \\in \\{0,\\ldots,m\\},\n",
    "$$\n",
    "des variables $W_{t_0},\\ldots, W_{t_m}$ où $W_{t_0} = 0$ et le vecteur $W = (W_{t_1},\\ldots, W_{t_m}) $ est Gaussien de moyenne nulle et de matrice de covariance \n",
    "$$\n",
    "\\Sigma_{i,j} = \\mathrm{Cov}(W_{t_i},W_{t_j}) = \\cfrac{1}{2} \\big(t_i^{2 H} + t_j^{2H} - |t_i - t_j|^{2H} \\big),\n",
    "$$\n",
    "avec $H \\in (0,1)$.\n",
    "    1. On se place dans le cas $H = 1/2$. Montrer que \n",
    "    $$\n",
    "    \\mathrm{Cov}\\big(W_{t_{i+1}} - W_{t_{i}},W_{t_{j+1}} - W_{t_{j}}\\big) = \n",
    "    \\left\\{\n",
    "    \\begin{array}{ll}\n",
    "    0 & \\text{ si } i \\ne j, \\\\\n",
    "    \\sqrt{t_{i+1} - t_{i}} & \\text{ si } i = j.\n",
    "    \\end{array}\n",
    "    \\right.\n",
    "    $$\n",
    "    2. Ecrire une fonction ``Simu_DeltaW`` qui simule une réalisation du vecteur $\\Delta W = (W_{t_1},\\ldots, W_{t_m} - W_{t_{m-1}}) $. \n",
    "    3. En utilisant les questions précédentes, proposer une fonction ``Simu_W1`` qui génère une réalisation de $W$ en utilisant $\\Delta W$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour $\\Delta W$ : [ 0.025 -0.291 -0.23  -0.345 -0.056  0.501 -0.504  0.241  0.352 -0.56 ] \n"
     ]
    }
   ],
   "source": [
    "def Simu_DeltaW(m, delta):\n",
    "    if m%2 == 0:\n",
    "        y = Simu_G2(int(m//2))\n",
    "    else:\n",
    "        y = Simu_G2(int((m+1)//2))[:m]  \n",
    "    return np.sqrt(delta) * y\n",
    "m = 10\n",
    "delta = 0.1\n",
    "print(r\" On obtient la réalisation suivante pour $\\Delta W$ : %s \" % Simu_DeltaW(m, delta).round(3) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour $W$ : [0.643 0.836 0.592 0.914 0.418 0.492 0.302 0.523 0.963 1.329] \n"
     ]
    }
   ],
   "source": [
    "def Simu_W1(m, delta):\n",
    "    deltaw = Simu_DeltaW(m, delta)\n",
    "    return deltaw.cumsum()\n",
    "m = 10\n",
    "delta = 0.1\n",
    "print(r\" On obtient la réalisation suivante pour $W$ : %s \" % Simu_W1(m, delta).round(3) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. **Simulation d'un vecteur Gaussien 3:** On se replace dans le même cadre que celui de la question 2 et on suppose cette fois que $H$ peut être différent de $1/2$.\n",
    "    1. Ecrire une fonction ``Construire_Sigma`` qui renvoie la matrice de covariance de $W = (W_{t_1},\\ldots, W_{t_m} - W_{t_{m-1}}) $. \n",
    "    2. En utilisant la questions précédente, programmer une fonction ``Simu_W2`` qui génère une réalisation de $W$.\n",
    "    3. On simule ci-dessous plusieurs trajectoires pour différentes valeurs de $H$. Quel est l'effet de H?    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la matrice de covariance suivant : [[0.025 0.038 0.047]\n",
      " [0.038 0.076 0.098]\n",
      " [0.047 0.098 0.146]] \n"
     ]
    }
   ],
   "source": [
    "def Construire_Sigma(m,delta,h):\n",
    "    Cov_mat = np.zeros((m,m))\n",
    "    ### time_stamps\n",
    "    t = delta*np.arange(m+1)\n",
    "    t_power2h = t**(2*h)\n",
    "    for i in range(m): \n",
    "        for j in range(i+1): \n",
    "            temp = 0.5 * (t_power2h[i+1] + t_power2h[j+1] - t_power2h[i-j])\n",
    "            Cov_mat[i,j] = temp\n",
    "            Cov_mat[j,i] = temp\n",
    "    return Cov_mat\n",
    "m = 3\n",
    "delta = 0.1\n",
    "h = 0.8\n",
    "print(r\" On obtient la matrice de covariance suivant : %s \" % Construire_Sigma(m,delta,h).round(3) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour X : [ 0.082  0.047  0.112 -0.125 -0.105 -0.336 -0.567 -0.672 -0.772 -0.666] \n"
     ]
    }
   ],
   "source": [
    "def Simu_W2(m, delta, h):\n",
    "    m1 = np.zeros(m) \n",
    "    Sigma1 = Construire_Sigma(m,delta,h)\n",
    "    return Simu_VG(m1, Sigma1)\n",
    "m = 10\n",
    "delta = 0.1\n",
    "h = 0.8\n",
    "print(\" On obtient la réalisation suivante pour X : %s \" % Simu_W2(m, delta, h).round(3) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le code ci-dessous permet d'observer les trajectoires et de répondre à la dernière question."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "\n",
      "\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 100\n",
    "delta = 0.01\n",
    "h_values = np.arange(0.3,1,0.2)\n",
    "W_values = []\n",
    "for h in h_values:\n",
    "    W_values.append(Simu_W2(m, delta, h))\n",
    "W_values = np.array(W_values)\n",
    "\n",
    "#### Tracer les graphes\n",
    "t = delta*np.arange(m+1)\n",
    "fig = plt.figure( figsize = (12,8))\n",
    "count = 0\n",
    "for h in h_values:\n",
    "    print()\n",
    "    y = np.concatenate([[0],W_values[count]])\n",
    "    plt.plot(t, y, label = \" h : \" + str(h.round(2)))\n",
    "    count += 1\n",
    "plt.grid()\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "# Exercice 9: simulation par composition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le but de cet exercice est de simuler une variable $X$ qui s'écrit \n",
    "$$\n",
    "X = p_1 X_1 + \\ldots + p_i X_i + \\ldots p_m X_m, \\qquad p_i \\geq 0, \\quad \\forall i \\in \\{1,\\ldots,m\\}, \\qquad \\sum_{i} p_i = 1,\n",
    "$$\n",
    "où $X_1,\\ldots,X_m$ sont des variables Gaussiennes de moyennes $m_1,\\ldots, m_m$ et de variance $\\sigma^2_1,\\ldots,\\sigma^2_m$.\n",
    "\n",
    "On note $Z$ la variables qui prend les valeurs $\\{1,\\ldots,m\\}$ avec les probabilités\n",
    "$$\n",
    "\\mathbb{P}[Z = i ] = p_i, \\qquad \\forall i \\in \\{1,\\ldots,m\\}.\n",
    "$$\n",
    "1. Montrer que \n",
    "$$\n",
    "X' = X_{Z},\n",
    "$$\n",
    "suit la même loi que $X$. \n",
    "2. En utilisant la question précédente, proposer une fonction ``Simu_Melange`` qui simule une réalisations de $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour Z : 5.19 \n"
     ]
    }
   ],
   "source": [
    "def Simu_Melange(p,ms,sigmas):\n",
    "    m = p.shape[0]\n",
    "    x = np.arange(m)\n",
    "    z = Simu_D(x,p)\n",
    "    y = ms[z] + sigmas[z] * Simu_G2(1)[0]\n",
    "    return y\n",
    "p = np.array([0.1, 0.2, 0.7])\n",
    "ms = np.array([1, 2, 3])\n",
    "sigmas = np.array([1, 1, 1])\n",
    "print(\" On obtient la réalisation suivante pour Z : %3.2f \" % Simu_Melange(p,ms,sigmas) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercice 10: arrondi de monnaie au Canada"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En 2012, le Canada a décidé d'arrêter la production et simulation des pièces de $1$ cent. Toutefois il est possible de proposer des prix au cent près, la règle en vigueur pour un paiement en espèce est alors un arrondi au multiple de cinq cents le plus proche.\n",
    "\n",
    "<center>\n",
    "<img src=\"./figures/PennyGraph02_f.jpg\" style=\"width: 600px;\"/>\n",
    "    <i>(Source : Ministère des Finances Canada)</i>\n",
    "</center>\n",
    "\n",
    "**Attention :** La règle essentielle pour ce qui suit est que lorsqu'un paiement concerne plusieurs produits, l'arrondi se fait sur la somme finale :\n",
    "\n",
    "<center>\n",
    "<img src=\"./figures/PennyGraph01_f.jpg\" style=\"width: 600px;\"/>\n",
    "    <i>(Source : Ministère des Finances Canada)</i>\n",
    "</center>\n",
    "\n",
    "\n",
    "* **On se pose la question suivante :**\n",
    "\n",
    "    * **Question:** Est-ce que de manière générale cette règle est plutôt favorable ou défavorable aux clients? (Est-ce que l'on gagne plus ou moins souvent $1$ ou $2$ cents que l'on en perd?)\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "&nbsp;    \n",
    "    \n",
    "* **Modélisation:** Pour répondre à cette question, on introduit le modèle simple suivant. On suppose que tous les clients se comportent de la même manière. Pour un client qui achète $N \\in \\mathbb{N}^*$ produits, on note $R_i$ l'arrondi du panier de notre client après l'achat de $i$ articles et on s'intéresse à $R_N$ l'arrondi du panier à la fin des courses. On remarque \n",
    "    * Si $R_N\\in\\{1,2,6,7\\}$, l'arrondi est favorable au client.\n",
    "    * Si $R_N\\in\\{3,4,8,9\\}$, l'arrondi est favorable au supermarché.\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "* On suppose que \n",
    "$$\n",
    "P_{i,j} = \\mathbb{P}[R_{n+1} = i |R_n = j] = p_{\\big(i-j\\big)\\;\\text{mod}[10]} \n",
    "$$\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "* Une étude empirique a montré que :\n",
    "$$\n",
    "p_0 = 0.071, \\quad\n",
    "p_1 = 0.087, \\quad\n",
    "p_2 = 0.087, \\quad\n",
    "p_3 = 0.024, \\quad\n",
    "p_4 = 0.071, \n",
    "$$\n",
    "$$\n",
    "\\quad p_5 = 0.055, \\quad\n",
    "p_6 = 0.047, \\quad\n",
    "p_7 = 0.346, \\quad\n",
    "p_8 = 0.047, \\quad\n",
    "p_9 = 0.165. \\quad\n",
    "$$\n",
    "\n",
    "&nbsp;\n",
    "\n",
    "1. Dans cette première partie, on suppose que $N> 0$ est un entier fixé. \n",
    "    1. Quelle est le profit $g(R_N)$ que réalise un client grâce à cette règle en fonction de $R_N$?\n",
    "    3. Pour chaque client, on a $R_0 = 0$. Proposer une fonction `Simu_R` qui simule une réalisation de $R_N$.\n",
    "    4. En simulant plusieurs réalisations de la variable $R_N$, programmer une fonction ``Gain_estim_client`` qui estime le gain moyen d'un client. Cette politique est-elle plus en faveur des clients ou non?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " On obtient la réalisation suivante pour R_N : 0.00 \n"
     ]
    }
   ],
   "source": [
    "def Simu_R(N,p):\n",
    "    r = 0\n",
    "    x = np.arange(10)\n",
    "    for i in range(N):\n",
    "        p_bis = np.concatenate([p[10-r:],p[:10-r]])\n",
    "        r = Simu_D(x,p_bis)\n",
    "    return r\n",
    "N = 5\n",
    "p = np.array([0.01, 0.1, 0.1, 0.01, 0.05, \n",
    "     0.01, 0.01, 0.6, 0.01, 0.1])\n",
    "print(\" On obtient la réalisation suivante pour R_N : %3.2f \" % Simu_R(N,p) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " Le gain estimé est 0.310.\n"
     ]
    }
   ],
   "source": [
    "def g(r):\n",
    "    if (r == 1) or (r == 2) or (r == 6) or (r == 7):\n",
    "        return 10\n",
    "    elif (r == 3) or (r == 4) or (r == 8) or (r == 9):\n",
    "        return -10\n",
    "    else:\n",
    "        return 0\n",
    "    \n",
    "def Gain_estim_client(g):\n",
    "    res = 0.\n",
    "    for i in range(1000):\n",
    "        r = Simu_R(N,p)\n",
    "        res += g(r)\n",
    "    return res/1000.\n",
    "\n",
    "estimated_gain = Gain_estim_client(g)\n",
    "print(\" Le gain estimé est %3.3f.\" % estimated_gain )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Dans cette partie, on suppose toujours que $N> 0$ est un entier fixé.  \n",
    "    1. Ecrire une fonction ``Contruire_mat_trans`` qui renvoie la matrice de transition $P = (P_{i,j})_{1 \\leq i,j \\leq n}$, i.e.\n",
    "    $$\n",
    "    P =\n",
    "    \\phantom{\\begin{matrix} 0 \\end{matrix}}\n",
    "    \\begin{matrix}\n",
    "    0\\\\ 1 \\\\ 2 \\\\ \\\\ \\vdots \\\\ \\\\ \\\\ \\\\ \\\\ 9\n",
    "    \\end{matrix}\n",
    "    \\begin{pmatrix}\n",
    "    p_0 & p_1 & p_2 & \\dots & & & p_9\\\\\n",
    "    p_9 & p_0 & p_1 & \\dots & & & p_8\\\\\n",
    "    p_8 & p_9 & p_0 & \\dots & & & p_7\\\\\n",
    "     & & & & & & \\\\\n",
    "     & & & & & & \\\\\n",
    "     & &  & \\vdots & & & \\\\\n",
    "      & & & & & & \\\\\n",
    "     & & & & & & \\\\\n",
    "    p_2 & p_3 & p_4  & \\dots & & & p_1\\\\\n",
    "    p_1 & p_2 & p_3   &\\dots & & & p_0\\\\\n",
    "    \\end{pmatrix}.\n",
    "    $$\n",
    "    2. Montrer que le profit moyen d'un client est \n",
    "    $$\n",
    "    \\mathbb{E}[G_N] = e_0 \\times P^N \\times g, \\quad e_0 = \\big(1,0,\\ldots,0\\big), \\quad g = \\big(g(0),\\ldots,g(1)\\big).\n",
    "    $$\n",
    "    3. En utilisant la question précédente, écrire une fonction ``Gain_client`` qui renvoie le gain moyen du client."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " La matrice de transition P est : \n",
      " [[0.01 0.1  0.1  0.01 0.05 0.01 0.01 0.6  0.01 0.1 ]\n",
      " [0.1  0.01 0.1  0.1  0.01 0.05 0.01 0.01 0.6  0.01]\n",
      " [0.01 0.1  0.01 0.1  0.1  0.01 0.05 0.01 0.01 0.6 ]\n",
      " [0.6  0.01 0.1  0.01 0.1  0.1  0.01 0.05 0.01 0.01]\n",
      " [0.01 0.6  0.01 0.1  0.01 0.1  0.1  0.01 0.05 0.01]\n",
      " [0.01 0.01 0.6  0.01 0.1  0.01 0.1  0.1  0.01 0.05]\n",
      " [0.05 0.01 0.01 0.6  0.01 0.1  0.01 0.1  0.1  0.01]\n",
      " [0.01 0.05 0.01 0.01 0.6  0.01 0.1  0.01 0.1  0.1 ]\n",
      " [0.1  0.01 0.05 0.01 0.01 0.6  0.01 0.1  0.01 0.1 ]\n",
      " [0.1  0.1  0.01 0.05 0.01 0.01 0.6  0.01 0.1  0.01]] \n"
     ]
    }
   ],
   "source": [
    "def Contruire_mat_trans(p):\n",
    "    ## Covariance Matrix\n",
    "    mat_trans = np.zeros((10,10))\n",
    "    for r in range(10): # i = 2\n",
    "        p_bis = np.concatenate([p[10-r:],p[:10-r]])\n",
    "        mat_trans[r,:] = p_bis\n",
    "    return mat_trans\n",
    "print(\" La matrice de transition P est : \\n %s \" % Contruire_mat_trans(p).round(3) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.30949158\n"
     ]
    }
   ],
   "source": [
    "def Gain_client(p,N):\n",
    "    P = Contruire_mat_trans(p)\n",
    "    P_N = np.linalg.matrix_power(P,N)\n",
    "    g_N = np.zeros(10)\n",
    "    for r in range(10):\n",
    "        g_N[r] = g(r)\n",
    "    return np.dot(P_N[0,:],g_N)\n",
    "print(Gain_client(p,N))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. Dans cette partie, on suppose que le nombre $N$ achetés par un client suit une loi de Poisson de paramètre $\\lambda = 10$ indépendante de $R_N$. En simulant plusieurs réalisations de $R_N$, écrire une fonction ``Gain_estimN_client`` qui estime le gain de l'agent dans ce cas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " Le gain estimé est -0.300.\n"
     ]
    }
   ],
   "source": [
    "def Gain_estimN_client(lambda_0,g):\n",
    "    res = 0.\n",
    "    for i in range(1000):\n",
    "        N = Simu_Poisson(lambda_0)\n",
    "        r = Simu_R(N,p)\n",
    "        res += g(r)\n",
    "    return res/1000.\n",
    "lambda_0 = 20\n",
    "estimated_gain = Gain_estimN_client(lambda_0,g)\n",
    "print(\" Le gain estimé est %3.3f.\" % estimated_gain )"
   ]
  }
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