{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Générer des variables aléatoires"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## runif\n",
    "La fonction `runif` permet de générer des pseudo-variables aléatoires indépendantes entre deux bornes\n",
    "`runif(n=combien, min, max)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "3.35023581283167"
      ],
      "text/latex": [
       "3.35023581283167"
      ],
      "text/markdown": [
       "3.35023581283167"
      ],
      "text/plain": [
       "[1] 3.350236"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "runif(1,0,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.762836940586567</li>\n",
       "\t<li>0.604817938059568</li>\n",
       "\t<li>0.0636072147171944</li>\n",
       "\t<li>0.052814619615674</li>\n",
       "\t<li>0.429516698000953</li>\n",
       "\t<li>0.327015534741804</li>\n",
       "\t<li>0.425390120828524</li>\n",
       "\t<li>0.093060856917873</li>\n",
       "\t<li>0.203973314259201</li>\n",
       "\t<li>0.0420023950282484</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.762836940586567\n",
       "\\item 0.604817938059568\n",
       "\\item 0.0636072147171944\n",
       "\\item 0.052814619615674\n",
       "\\item 0.429516698000953\n",
       "\\item 0.327015534741804\n",
       "\\item 0.425390120828524\n",
       "\\item 0.093060856917873\n",
       "\\item 0.203973314259201\n",
       "\\item 0.0420023950282484\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.762836940586567\n",
       "2. 0.604817938059568\n",
       "3. 0.0636072147171944\n",
       "4. 0.052814619615674\n",
       "5. 0.429516698000953\n",
       "6. 0.327015534741804\n",
       "7. 0.425390120828524\n",
       "8. 0.093060856917873\n",
       "9. 0.203973314259201\n",
       "10. 0.0420023950282484\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 0.76283694 0.60481794 0.06360721 0.05281462 0.42951670 0.32701553\n",
       " [7] 0.42539012 0.09306086 0.20397331 0.04200240"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "runif(10,0,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "On peut insérer maintenant les valeurs générées à l'intérieur d'un vecteur"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "x<-runif(100,0,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Ou les insérer à l'intérieur d'une matrice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0.539611001</td><td>0.43703539 </td><td>0.85709912 </td><td>0.4599631  </td><td>0.97466830 </td><td>0.9150659  </td><td>0.1251427  </td><td>0.1800253  </td><td>0.63296958 </td><td>0.1277790  </td></tr>\n",
       "\t<tr><td>0.969200172</td><td>0.98869122 </td><td>0.45878880 </td><td>0.2725860  </td><td>0.58830241 </td><td>0.7808764  </td><td>0.2056756  </td><td>0.6443141  </td><td>0.82069732 </td><td>0.1994270  </td></tr>\n",
       "\t<tr><td>0.161860105</td><td>0.44683774 </td><td>0.05306833 </td><td>0.3773334  </td><td>0.75579931 </td><td>0.1784268  </td><td>0.3055089  </td><td>0.4147207  </td><td>0.43887943 </td><td>0.6040850  </td></tr>\n",
       "\t<tr><td>0.446521448</td><td>0.54748316 </td><td>0.92775225 </td><td>0.9982479  </td><td>0.54972412 </td><td>0.6394982  </td><td>0.6864430  </td><td>0.9867912  </td><td>0.03091283 </td><td>0.5808925  </td></tr>\n",
       "\t<tr><td>0.007107472</td><td>0.02445618 </td><td>0.35345890 </td><td>0.4459476  </td><td>0.77903144 </td><td>0.4392217  </td><td>0.8644656  </td><td>0.3406485  </td><td>0.56053496 </td><td>0.4633779  </td></tr>\n",
       "\t<tr><td>0.963198963</td><td>0.40298549 </td><td>0.56838054 </td><td>0.8158883  </td><td>0.03108763 </td><td>0.7979690  </td><td>0.6087287  </td><td>0.5747714  </td><td>0.07915931 </td><td>0.4372559  </td></tr>\n",
       "\t<tr><td>0.579066600</td><td>0.34913192 </td><td>0.23066770 </td><td>0.2059267  </td><td>0.41230033 </td><td>0.7082100  </td><td>0.2880309  </td><td>0.9337467  </td><td>0.77597654 </td><td>0.8853603  </td></tr>\n",
       "\t<tr><td>0.165705290</td><td>0.74011543 </td><td>0.33763606 </td><td>0.1456844  </td><td>0.56672791 </td><td>0.2197435  </td><td>0.1583491  </td><td>0.2270206  </td><td>0.06134143 </td><td>0.2939507  </td></tr>\n",
       "\t<tr><td>0.594905792</td><td>0.27719153 </td><td>0.37522597 </td><td>0.8297370  </td><td>0.49838097 </td><td>0.3884056  </td><td>0.3239845  </td><td>0.1054112  </td><td>0.80395329 </td><td>0.1680158  </td></tr>\n",
       "\t<tr><td>0.375249445</td><td>0.84355537 </td><td>0.38782100 </td><td>0.8637450  </td><td>0.70958528 </td><td>0.3343178  </td><td>0.3899539  </td><td>0.6956865  </td><td>0.90822088 </td><td>0.8057296  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{llllllllll}\n",
       "\t 0.539611001 & 0.43703539  & 0.85709912  & 0.4599631   & 0.97466830  & 0.9150659   & 0.1251427   & 0.1800253   & 0.63296958  & 0.1277790  \\\\\n",
       "\t 0.969200172 & 0.98869122  & 0.45878880  & 0.2725860   & 0.58830241  & 0.7808764   & 0.2056756   & 0.6443141   & 0.82069732  & 0.1994270  \\\\\n",
       "\t 0.161860105 & 0.44683774  & 0.05306833  & 0.3773334   & 0.75579931  & 0.1784268   & 0.3055089   & 0.4147207   & 0.43887943  & 0.6040850  \\\\\n",
       "\t 0.446521448 & 0.54748316  & 0.92775225  & 0.9982479   & 0.54972412  & 0.6394982   & 0.6864430   & 0.9867912   & 0.03091283  & 0.5808925  \\\\\n",
       "\t 0.007107472 & 0.02445618  & 0.35345890  & 0.4459476   & 0.77903144  & 0.4392217   & 0.8644656   & 0.3406485   & 0.56053496  & 0.4633779  \\\\\n",
       "\t 0.963198963 & 0.40298549  & 0.56838054  & 0.8158883   & 0.03108763  & 0.7979690   & 0.6087287   & 0.5747714   & 0.07915931  & 0.4372559  \\\\\n",
       "\t 0.579066600 & 0.34913192  & 0.23066770  & 0.2059267   & 0.41230033  & 0.7082100   & 0.2880309   & 0.9337467   & 0.77597654  & 0.8853603  \\\\\n",
       "\t 0.165705290 & 0.74011543  & 0.33763606  & 0.1456844   & 0.56672791  & 0.2197435   & 0.1583491   & 0.2270206   & 0.06134143  & 0.2939507  \\\\\n",
       "\t 0.594905792 & 0.27719153  & 0.37522597  & 0.8297370   & 0.49838097  & 0.3884056   & 0.3239845   & 0.1054112   & 0.80395329  & 0.1680158  \\\\\n",
       "\t 0.375249445 & 0.84355537  & 0.38782100  & 0.8637450   & 0.70958528  & 0.3343178   & 0.3899539   & 0.6956865   & 0.90822088  & 0.8057296  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 0.539611001 | 0.43703539  | 0.85709912  | 0.4599631   | 0.97466830  | 0.9150659   | 0.1251427   | 0.1800253   | 0.63296958  | 0.1277790   | \n",
       "| 0.969200172 | 0.98869122  | 0.45878880  | 0.2725860   | 0.58830241  | 0.7808764   | 0.2056756   | 0.6443141   | 0.82069732  | 0.1994270   | \n",
       "| 0.161860105 | 0.44683774  | 0.05306833  | 0.3773334   | 0.75579931  | 0.1784268   | 0.3055089   | 0.4147207   | 0.43887943  | 0.6040850   | \n",
       "| 0.446521448 | 0.54748316  | 0.92775225  | 0.9982479   | 0.54972412  | 0.6394982   | 0.6864430   | 0.9867912   | 0.03091283  | 0.5808925   | \n",
       "| 0.007107472 | 0.02445618  | 0.35345890  | 0.4459476   | 0.77903144  | 0.4392217   | 0.8644656   | 0.3406485   | 0.56053496  | 0.4633779   | \n",
       "| 0.963198963 | 0.40298549  | 0.56838054  | 0.8158883   | 0.03108763  | 0.7979690   | 0.6087287   | 0.5747714   | 0.07915931  | 0.4372559   | \n",
       "| 0.579066600 | 0.34913192  | 0.23066770  | 0.2059267   | 0.41230033  | 0.7082100   | 0.2880309   | 0.9337467   | 0.77597654  | 0.8853603   | \n",
       "| 0.165705290 | 0.74011543  | 0.33763606  | 0.1456844   | 0.56672791  | 0.2197435   | 0.1583491   | 0.2270206   | 0.06134143  | 0.2939507   | \n",
       "| 0.594905792 | 0.27719153  | 0.37522597  | 0.8297370   | 0.49838097  | 0.3884056   | 0.3239845   | 0.1054112   | 0.80395329  | 0.1680158   | \n",
       "| 0.375249445 | 0.84355537  | 0.38782100  | 0.8637450   | 0.70958528  | 0.3343178   | 0.3899539   | 0.6956865   | 0.90822088  | 0.8057296   | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "      [,1]        [,2]       [,3]       [,4]      [,5]       [,6]     \n",
       " [1,] 0.539611001 0.43703539 0.85709912 0.4599631 0.97466830 0.9150659\n",
       " [2,] 0.969200172 0.98869122 0.45878880 0.2725860 0.58830241 0.7808764\n",
       " [3,] 0.161860105 0.44683774 0.05306833 0.3773334 0.75579931 0.1784268\n",
       " [4,] 0.446521448 0.54748316 0.92775225 0.9982479 0.54972412 0.6394982\n",
       " [5,] 0.007107472 0.02445618 0.35345890 0.4459476 0.77903144 0.4392217\n",
       " [6,] 0.963198963 0.40298549 0.56838054 0.8158883 0.03108763 0.7979690\n",
       " [7,] 0.579066600 0.34913192 0.23066770 0.2059267 0.41230033 0.7082100\n",
       " [8,] 0.165705290 0.74011543 0.33763606 0.1456844 0.56672791 0.2197435\n",
       " [9,] 0.594905792 0.27719153 0.37522597 0.8297370 0.49838097 0.3884056\n",
       "[10,] 0.375249445 0.84355537 0.38782100 0.8637450 0.70958528 0.3343178\n",
       "      [,7]      [,8]      [,9]       [,10]    \n",
       " [1,] 0.1251427 0.1800253 0.63296958 0.1277790\n",
       " [2,] 0.2056756 0.6443141 0.82069732 0.1994270\n",
       " [3,] 0.3055089 0.4147207 0.43887943 0.6040850\n",
       " [4,] 0.6864430 0.9867912 0.03091283 0.5808925\n",
       " [5,] 0.8644656 0.3406485 0.56053496 0.4633779\n",
       " [6,] 0.6087287 0.5747714 0.07915931 0.4372559\n",
       " [7,] 0.2880309 0.9337467 0.77597654 0.8853603\n",
       " [8,] 0.1583491 0.2270206 0.06134143 0.2939507\n",
       " [9,] 0.3239845 0.1054112 0.80395329 0.1680158\n",
       "[10,] 0.3899539 0.6956865 0.90822088 0.8057296"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x<-matrix(x, 10)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Nous avons maintenant obtenu une matrice de dimension 10X10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Seed"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Comme dans SAS, nous avons appris comment générer les mêmes variables aléatoires dans un contexte de cumulation par exemple, avec la fonction seed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "set.seed(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.18488225992769</li>\n",
       "\t<li>0.702374035958201</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.18488225992769\n",
       "\\item 0.702374035958201\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.18488225992769\n",
       "2. 0.702374035958201\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.1848823 0.7023740"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "runif(2,0,1) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.573326334822923</li>\n",
       "\t<li>0.168051920365542</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.573326334822923\n",
       "\\item 0.168051920365542\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.573326334822923\n",
       "2. 0.168051920365542\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.5733263 0.1680519"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "runif(2,0,1) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.943839338840917</li>\n",
       "\t<li>0.943474958650768</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.943839338840917\n",
       "\\item 0.943474958650768\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.943839338840917\n",
       "2. 0.943474958650768\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.9438393 0.9434750"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "runif(2,0,1) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Si on remet le seed=2, on obtient alors les mêmes valeurs que nous avons obtenues auparavant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "set.seed(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.18488225992769</li>\n",
       "\t<li>0.702374035958201</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.18488225992769\n",
       "\\item 0.702374035958201\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.18488225992769\n",
       "2. 0.702374035958201\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.1848823 0.7023740"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "runif(2,0,1) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.573326334822923</li>\n",
       "\t<li>0.168051920365542</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.573326334822923\n",
       "\\item 0.168051920365542\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.573326334822923\n",
       "2. 0.168051920365542\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.5733263 0.1680519"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "runif(2,0,1) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.943839338840917</li>\n",
       "\t<li>0.943474958650768</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.943839338840917\n",
       "\\item 0.943474958650768\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.943839338840917\n",
       "2. 0.943474958650768\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.9438393 0.9434750"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "runif(2,0,1) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Sample (échantillonage)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "La fonction `sample` tire un échantillon aléatoire de $n$ variables à partir d'un ensemble de données allant de $\\left\\{1, \\dots ,N\\right\\}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>2</li>\n",
       "\t<li>8</li>\n",
       "\t<li>7</li>\n",
       "\t<li>4</li>\n",
       "\t<li>9</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 2\n",
       "\\item 8\n",
       "\\item 7\n",
       "\\item 4\n",
       "\\item 9\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 2\n",
       "2. 8\n",
       "3. 7\n",
       "4. 4\n",
       "5. 9\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 2 8 7 4 9"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample(1:10,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.467615901259705</li>\n",
       "\t<li>0.585986070567742</li>\n",
       "\t<li>0.318405627040192</li>\n",
       "\t<li>0.117259352467954</li>\n",
       "\t<li>0.808955513406545</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.467615901259705\n",
       "\\item 0.585986070567742\n",
       "\\item 0.318405627040192\n",
       "\\item 0.117259352467954\n",
       "\\item 0.808955513406545\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.467615901259705\n",
       "2. 0.585986070567742\n",
       "3. 0.318405627040192\n",
       "4. 0.117259352467954\n",
       "5. 0.808955513406545\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.4676159 0.5859861 0.3184056 0.1172594 0.8089555"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample(x,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.0845668376423419</li>\n",
       "\t<li>0.693119892384857</li>\n",
       "\t<li>0.849038900341839</li>\n",
       "\t<li>0.410260857315734</li>\n",
       "\t<li>0.884574939031154</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.0845668376423419\n",
       "\\item 0.693119892384857\n",
       "\\item 0.849038900341839\n",
       "\\item 0.410260857315734\n",
       "\\item 0.884574939031154\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.0845668376423419\n",
       "2. 0.693119892384857\n",
       "3. 0.849038900341839\n",
       "4. 0.410260857315734\n",
       "5. 0.884574939031154\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.08456684 0.69311989 0.84903890 0.41026086 0.88457494"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample(x,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$n\\leq N$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "ename": "ERROR",
     "evalue": "Error in sample.int(length(x), size, replace, prob): cannot take a sample larger than the population when 'replace = FALSE'\n",
     "output_type": "error",
     "traceback": [
      "Error in sample.int(length(x), size, replace, prob): cannot take a sample larger than the population when 'replace = FALSE'\nTraceback:\n",
      "1. sample(x, length(x) + 1)",
      "2. sample.int(length(x), size, replace, prob)"
     ]
    }
   ],
   "source": [
    "sample(x,length(x)+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Lorsque notre échantillon tiré est plus grand que notre ensemble de données, on peut utiliser un tirage avec remise:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# sample(0:1, 100, replace = T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Cette fonction fonctionne aussi sur un tirage de variable de type caractères"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>'Kentucky'</li>\n",
       "\t<li>'Montana'</li>\n",
       "\t<li>'Delaware'</li>\n",
       "\t<li>'Wyoming'</li>\n",
       "\t<li>'North Carolina'</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 'Kentucky'\n",
       "\\item 'Montana'\n",
       "\\item 'Delaware'\n",
       "\\item 'Wyoming'\n",
       "\\item 'North Carolina'\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 'Kentucky'\n",
       "2. 'Montana'\n",
       "3. 'Delaware'\n",
       "4. 'Wyoming'\n",
       "5. 'North Carolina'\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] \"Kentucky\"       \"Montana\"        \"Delaware\"       \"Wyoming\"       \n",
       "[5] \"North Carolina\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample(state.name, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "La fonction `sample` donne des probabilités égales à tous les éléments tirés d'un ensemble de données. Toutesfois, il est possible de préciser la probabilité de chaque élément tiré."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "s<-sample(1:5, 1000, replace=T, prob=c(.2,.2,.2,.2,.2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Si on utilise la fonction `table` afin de compter l'occurrence de chaque élément"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "s\n",
       "  1   2   3   4   5 \n",
       "222 187 203 183 205 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tableau <- table(s)\n",
    "tableau"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "on remarque que chaque élément à été tiré à un taux d'environ 20%"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.222"
      ],
      "text/latex": [
       "0.222"
      ],
      "text/markdown": [
       "0.222"
      ],
      "text/plain": [
       "[1] 0.222"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tableau[[1]]/sum(tableau)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Si on change les probabilités maintenant, mais **attention** la somme des probabilités doit être égale à 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "s\n",
       "  1   2   3   4   5 \n",
       "197 509 101  87 106 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "s<-sample(1:5, 1000, replace=T, prob=c(.2,.5,.1,.1,.1))\n",
    "table(s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## rnorm\n",
    "La moyenne par défaut est égale à 0 et l'écart-type=1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.106821263122198"
      ],
      "text/latex": [
       "0.106821263122198"
      ],
      "text/markdown": [
       "0.106821263122198"
      ],
      "text/plain": [
       "[1] 0.1068213"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rnorm(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "-24.9130626933775"
      ],
      "text/latex": [
       "-24.9130626933775"
      ],
      "text/markdown": [
       "-24.9130626933775"
      ],
      "text/plain": [
       "[1] -24.91306"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rnorm(1, 0, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Créons un vecteur de 100 valeurs avec moyenne=100 et écart-type=100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "x<-rnorm(100, 0, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Si on calcule la moyenne de ce vecteur;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "-5.83207843041791"
      ],
      "text/latex": [
       "-5.83207843041791"
      ],
      "text/markdown": [
       "-5.83207843041791"
      ],
      "text/plain": [
       "[1] -5.832078"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mean(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Nous avons obtenu une moyenne proche de la moyenne de nos variables aléatoires générées par la fonction `rnorm`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "La même chose maintenant pour l'écart-type"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "93.1100361171955"
      ],
      "text/latex": [
       "93.1100361171955"
      ],
      "text/markdown": [
       "93.1100361171955"
      ],
      "text/plain": [
       "[1] 93.11004"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sd(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Toutefois, si nous augmentons le nombre de variables aléatoires généré, nous sommes alors plus proches des arguments de la fonction `rnorm` que nous avons saisie"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.963181514782263"
      ],
      "text/latex": [
       "0.963181514782263"
      ],
      "text/markdown": [
       "0.963181514782263"
      ],
      "text/plain": [
       "[1] 0.9631815"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x<-rnorm(10000, 0, 100)\n",
    "mean(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "101.228973820731"
      ],
      "text/latex": [
       "101.228973820731"
      ],
      "text/markdown": [
       "101.228973820731"
      ],
      "text/plain": [
       "[1] 101.229"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sd(x)"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "4.1.2"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  },
  "name": "_merged",
  "toc": {
   "base_numbering": 1,
   "nav_menu": {
    "height": "512px",
    "width": "252px"
   },
   "number_sections": false,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": true,
   "toc_position": {
    "height": "596px",
    "left": "0px",
    "right": "1017px",
    "top": "107px",
    "width": "250px"
   },
   "toc_section_display": "block",
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}