{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simuler, avec Python ou un tableur, N échantillons de taille 𝑛 d’une variable aléatoire, d’espérance 𝜇 et d’écart type 𝜎. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "exemple : considérons la loi : \n",
    "\n",
    "| $x_i$  | 5   |10  | 20 |\n",
    "|:-- :|:-- :|:-- :|:-- :|\n",
    "| $p_i$   | 0.2| 0.7| 0.1|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculer son espérance et son écart-type :"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L'espérance vaut $\\mu = 5\\times 0.2 + 10 \\times 0.7 + 20 \\times 0.1 = 10$\n",
    "\n",
    "L'écart-type vaut $\\sigma = \\sqrt{0.2\\times 25 + 0.7\\times 0 + 0.1\\times 100} = \\sqrt{15}  \\simeq 3.9$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pour simuler un échantilon de taille 1, compléter la fonction suivante :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from random import randint\n",
    "\n",
    "def simulationUn():\n",
    "    nombreAleatoire  = randint(1,10)\n",
    "    if nombreAleatoire <= 2:\n",
    "        return 5\n",
    "    else:\n",
    "        if nombreAleatoire <= 9:\n",
    "            return 10\n",
    "        else:\n",
    "            return 20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simulationUn()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pour simuler un échantilon de taille n, compléter la fonction suivante :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from random import randint\n",
    "\n",
    "def simulation(n):\n",
    "    i = 1\n",
    "    while i <= n :\n",
    "        print( simulationUn() )\n",
    "        i = i+1\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10\n",
      "20\n",
      "10\n",
      "10\n",
      "10\n",
      "5\n",
      "10\n"
     ]
    }
   ],
   "source": [
    "simulation(7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Modifions cette fonction pour qu'elle stocke l'échantillon dans une liste :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from random import randint\n",
    "\n",
    "def simulation(n):\n",
    "    i = 1\n",
    "    L = []  # L est une liste vide\n",
    "    while i <= n :\n",
    "        L = L +  [simulationUn()]\n",
    "        i = i+1\n",
    "    return L"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[10, 10, 10, 10]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simulation(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ecrire une fonction qui calcule la moyenne des éléments d'une liste :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def moyenneListe(L):\n",
    "    m  = 0\n",
    "    for i in L:\n",
    "        m = m+i\n",
    "    return m/len(L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[10, 5, 10, 10, 10, 10, 10, 5, 5, 20, 10, 10, 5, 5, 10, 5, 10, 10, 10, 5]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "8.75"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = simulation(20)\n",
    "print(a)\n",
    "moyenneListe(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulation de N échantillons de taille n et calcul de la moyenne "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "La proportion des cas où où l'écart entre la moyenne de l'échantillon et 10 est inférieur ou égal à 2sigma/racine(n) est :  0.943\n"
     ]
    }
   ],
   "source": [
    "from math import sqrt\n",
    "N = 1000\n",
    "n = 1000\n",
    "C = 0 #compte les cas où l'écart entre la moyenne de l'échantillon et 10 est inférieur ou égal à 2*sigma/racine(n)\n",
    "for i in range(N):\n",
    "    if abs( moyenneListe(simulation(n)) - 10 ) <= 2*sqrt(15)/sqrt(n):\n",
    "        C = C + 1\n",
    "print(\"La proportion des cas où où l'écart entre la moyenne de l'échantillon et 10 est inférieur ou égal à 2sigma/racine(n) est : \", C/N )\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "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": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}