{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# TP1 : variables aléatoires, lois et théorèmes limites." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# I. Rappels de commandes Python" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pour commencer, voici quelques manipulations pour prendre en main python.\n", "Pour voir le résultat des commandes indiquées dans une cellule, il faut l'exécuter, en appuyant sur la flèche $\\blacktriangleright$ en haut ou en appuyant sur \"Maj+Entrée\"." ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "1+1" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 1+1\n", "# quand on veut écrire quelque chose qui ne sera pas exécuté, on met un # devant... \n", "1+2" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "hello\n" ] } ], "source": [ "print('hello')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pour utiliser l'aide de python au sujet d'une commande, on peut taper `help(commande)`." ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Help on built-in function round in module builtins:\n", "\n", "round(number, ndigits=None)\n", " Round a number to a given precision in decimal digits.\n", " \n", " The return value is an integer if ndigits is omitted or None. Otherwise\n", " the return value has the same type as the number. ndigits may be negative.\n", "\n" ] } ], "source": [ "help(round)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1.433" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "round(1.43267,3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Nous utiliserons trois bibliothèques dans ce TP : numpy (pour la gestion des vecteurs, la génération de variables aléatoires), scipy stats pour les différentes lois et matplotlib les graphiques. Il faut importer ces bibliothèques pour pouvoir les utiliser :" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import scipy.stats as st\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On peut maintenant utiliser les fonctions de ces bibliothèques (on indique la bibliothèque utilisée avec son abréviation) :" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([0., 0., 0., 0.])" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.zeros(4)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# II. Diagrammes en bâtons de lois usuelles discrètes" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. Diagramme en bâtons de la loi $\\mathcal{B}(10, 0.25)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pour tracer un diagramme en bâtons de la loi $\\mathcal{B}(10,0.25)$ il faut préparer deux vecteurs : \n", "\n", "- le premier contient la liste des abscisses. Ici, nous avons une v.a. qui prend les valeurs $0,\\ldots,10$. Pour créer ce vecteur, on peut utiliser la commande ``np.arange(11)``.\n", "\n", "- le second contient la liste des hauteurs des bâtons. Ici, nous voulons la liste des $P(X=k)$, pour $k=0,...,10$. La fonction ``stats.binom.pmf(k, 10, 0.25)`` permet de calculer cette probabilité. Mieux, si on remplace $k$ par un vecteur, alors la fonction calcule cette probabilité pour toutes les valeurs du vecteur, coordonnées par coordonnées.\n", "\n", "Ainsi, on peut calculer toutes les probabilités $P(X=k)$ pour $k=0,..,10$, d'un seul coup. Testons.\n" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10])" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.arange(11)" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "np.float64(0.05631351470947266)" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "st.binom.pmf(0, 10, 0.25)" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "np.float64(0.1877117156982421)" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "st.binom.pmf(1, 10, 0.25)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([5.63135147e-02, 1.87711716e-01, 2.81567574e-01, 2.50282288e-01,\n", " 1.45998001e-01, 5.83992004e-02, 1.62220001e-02, 3.08990479e-03,\n", " 3.86238098e-04, 2.86102295e-05, 9.53674316e-07])" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "st.binom.pmf(np.arange(11), 10, 0.25)" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "image/png": "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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "k = np.arange(11)\n", "P = st.binom.pmf(k, 10, 0.25)\n", "\n", "plt.bar(k, P)\n", "\n", "plt.xlabel(\"$k$\")\n", "plt.ylabel(\"$P(X=k)$\")\n", "plt.title(\"Loi binomiale $B(10, 0.25)$\")\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Rappel : si $X$ de loi binomiale $\\mathcal{B}(n,p)$, $X$ représente le nombre de succès dans un schéma de Bernouilli (une épreuve de Bernoulli est une expérience aléatoire qui n'a que deux issues possibles, que l'on note SUCCES et ECHEC), constitué de n répétitions indépendantes de la même épreuve de Bernoulli. Le paramètre $p$ représente la probabilité d'un succès à cette épreuve de Bernoulli.\n", "\n", "On a, pour tout $k=0,\\ldots,n$,\n", "$$P(X=k)=\\binom{n}{k} p^k (1-p)^{n-k}.$$\n", "\n", "La moyenne d'une v.a. qui suit la loi binomiale $\\mathcal{B}(10,0.25)$ vaut $2.5$. C'est cohérent avec le tracé. Par ailleurs, ce sont les valeurs $2$ et $3$ qui sont le plus souvent observées (leur probabilité est la plus élevée)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. Diagrammes en bâtons d'autres lois" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Tracer les digrammes en bâtons des lois suivantes." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi binomiale $\\mathcal{B}(10,0.5)$**" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi binomiale $\\mathcal{B}(100,0.25)$**" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi de Poisson $\\mathcal{P}(2)$**" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi de Poisson $\\mathcal{P}(10)$**" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi géométrique $\\mathcal{G}(0.75)$**" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi géométrique $\\mathcal{G}(0.25)$**" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi uniforme $\\mathcal{U}(\\{1,2,3,4,5,6\\})$**" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## III. Fonction de répartition de la lois usuelles discrètes" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1. Fonction de répartition de la loi $\\mathcal{B}(10,0.25)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "La fonction de répartition d'une v.a. $X$ est la fonction qui à $t\\in \\mathbb{R}$ associe $F_X(t)=P(X\\leq t)$.\n", "Pour une loi discrète, cette fonction est une fonction en escalier, croissante, qui tend vers 0 en $-\\infty$ et vers 1 en $+\\infty$. Un saut en un point $t$ a pour hauteur $P(X=t)$.\n", "\n", "En python, dans le module stats de scipy, on trouve toutes les fonctions de répartition des lois usuelles, sous le nom : ``st.binom.cdf``, ``st.poisson.cdf``... les différents paramètres de ces fonctions sont listés dans ``help(st.binom)``, ``help(st.poisson)``...\n", "\n", "Pour tracer des fonction en escalier en python on peut utiliser l'option ``drawstyle='steps-post'`` de ``plt.plot``." ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "image/png": "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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "k = np.arange(-1, 12)\n", "F = st.binom.cdf(k, 10, 0.25)\n", "\n", "plt.plot(k, F, drawstyle='steps-post')\n", "\n", "plt.xlabel(\"$t$\")\n", "plt.ylabel(\"$F_X(t)$\")\n", "plt.grid()\n", "plt.title(\"Fonction de répartition de la loi $B(10, 0.25)$\")\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2. Fonctions de répartitions d'autres lois dicrètes." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi binomiale $\\mathcal{B}(10,0.5)$**" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi binomiale $\\mathcal{B}(100,0.25)$**" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi de Poisson $\\mathcal{P}(2)$**" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Loi géométrique $\\mathcal{G}(0.25)$**" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## III. Approximation de la loi binomiale par la loi de Poisson." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Rappel : si $\\lambda>0$ et $(p_n)_{n\\in \\mathbb{N}}$ satisfait $np_n \\rightarrow \\infty$ lorsque $n\\rightarrow \\infty$, \n", "alors pour tout $k\\in \\mathbb{N}$, \n", "$$\n", "\\binom{n}{k} p_n^k (1-p_n)^{n-k} \\underset{n\\rightarrow \\infty}{\\longrightarrow} e^{-\\lambda} \\frac{\\lambda^k}{k!}.\n", "$$\n", "Pour illuster cette convergence, tracer sur le même graphique le diagramme en bâtons de la loi $\\mathcal{B}(n,p_n)$ et de la loi $\\mathcal{P}(\\lambda)$ pour $\\lambda = 2$, $p_n=\\frac{\\lambda}{n}$ et diférentes valeurs de $n$.\n", "\n", "On pourra utiliser l'option ``width`` de ``plt.bar`` pour superposer deux diagrammes en bâtons et la commande ``plt.legend()``.\n" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## IV. Loi des grands nombres." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "La loi des grands nombres stipule que, pour une suite de variables aléatoires indépendantes, de même loi $(X_n)_{n\\geq 1}$ d'espérance $m$ et de variance finie, la moyenne empirique\n", "$$S_n =\\frac{X_1+\\ldots+X_n}{n}$$\n", "est proche de $m$ lorsque $n$ devient grand. \n", "\n", "Plus formellement, pour tout $\\varepsilon >0$,\n", "$$\\lim_{n\\rightarrow +\\infty} \\mathbb{P}\\left\n", "(\\left\\vert M_n - m\\right\\vert \\geq \\varepsilon\\right) =0.$$\n", "\n", "Afin d'illustrer ce résultat simuler un vecteur de $N$ variables aléatoires indépendantes $(X_1,\\ldots, X_N)$ de loi géométrique de paramètre $p=0.25$ (on pourra utiliser la fonction ``np.random.geometric``) et tracer $M_n$ en fonction de $n$ (on pourra utiliser la fonction ``np.cumsum``)." ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Faire de même en simulant des variables $X_i$ de loi de Poisson $\\mathcal{P}(3)$, et de loi binomiale $\\mathcal{B}(5,0.1)$. " ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [], "source": [ "#\n", "#\n", "#" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## V. Théorème de Moivre Laplace" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Le théorème de Moivre-Laplace donne une approximation de la loi binomiale par la loi normale.\n", "\n", "Plus formellement, pour tout $a, b \\in \\mathbb{R}$ avec $a