{ "cells": [ { "cell_type": "markdown", "execution_count": null, "id": "7a4792f4-64c4-4e9e-8b85-fa9db6528669", "metadata": { "tags": [] }, "source": [ "# Série2 : Correction \n", " ## Exercice 1: \n", "\n", "Nous avons à notre disposition les données d'échec ou de réussite de l'examen de cette matière de 2020. Nous allons nous intérésser en particulier à deux variables binaires de ces données:\n", "- La section: **S** $\\in\\{\\text{GM},\\text{EL}\\}$\n", "- Le succes à l'examen: **A** $\\in\\{0,1\\}$\n", "\n", "Dans le cadre des probabilités conditionnelles, avec deux variables binaires, une erreur qui revient souvent est de penser que:\n", "\\begin{equation*} \\mathbb{P}\\left(\\textbf{A} = 1 | \\textbf{S} = \\text{GM}\\right) + \\mathbb{P}\\left(\\textbf{A} = 1 | \\textbf{S} = \\text{EL}\\right) = 1 \\end{equation*}\n", "Néanmoins, cela est généralement faux. Nous allons donc nous appliquer à montrer que cela est faux dans notre cas, ainsi que de voir quel serait la version correcte de cette équation." ] }, { "cell_type": "code", "execution_count": null, "id": "aaf4ca73-3a40-46f9-9061-bfa0484e5e5e", "metadata": { "tags": [] }, "outputs": [], "source": [ "library(dplyr)\n", "\n", "data <- read.csv(\"Success_ProbaStat.csv\")" ] }, { "cell_type": "markdown", "execution_count": null, "id": "6926d7d4-9a32-4b3e-a3cc-1850307c6bf9", "metadata": {}, "source": [ "1\\) Synthétiser ces données de façon à avoir le nombre d'échec ou de réussite par section." ] }, { "cell_type": "code", "execution_count": null, "id": "1caec49b-584a-477e-b520-9892466503c2", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\t\n", "\t\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\n", "
A data.frame: 4 × 4
SectionSuccesnprop
<chr><int><int><dbl>
EL0 110.04526749
EL1 380.15637860
GM0 320.13168724
GM11620.66666667
\n" ], "text/latex": [ "A data.frame: 4 × 4\n", "\\begin{tabular}{llll}\n", " Section & Succes & n & prop\\\\\n", " & & & \\\\\n", "\\hline\n", "\t EL & 0 & 11 & 0.04526749\\\\\n", "\t EL & 1 & 38 & 0.15637860\\\\\n", "\t GM & 0 & 32 & 0.13168724\\\\\n", "\t GM & 1 & 162 & 0.66666667\\\\\n", "\\end{tabular}\n" ], "text/markdown": [ "\n", "A data.frame: 4 × 4\n", "\n", "| Section <chr> | Succes <int> | n <int> | prop <dbl> |\n", "|---|---|---|---|\n", "| EL | 0 | 11 | 0.04526749 |\n", "| EL | 1 | 38 | 0.15637860 |\n", "| GM | 0 | 32 | 0.13168724 |\n", "| GM | 1 | 162 | 0.66666667 |\n", "\n" ], "text/plain": [ " Section Succes n prop \n", "1 EL 0 11 0.04526749\n", "2 EL 1 38 0.15637860\n", "3 GM 0 32 0.13168724\n", "4 GM 1 162 0.66666667" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########### Solution R: ##########\n", "\n", "Prop <- data %>% count(Section, Succes) %>% mutate(prop = prop.table(n))\n", "Prop" ] }, { "cell_type": "markdown", "execution_count": null, "id": "d9161036-666c-4a0c-bb59-bb0d7850d479", "metadata": {}, "source": [ "2\\) Montrer expérimentalement que $ \\mathbb{P}\\left(\\textbf{A} = 1 | \\textbf{S} = \\text{GM}\\right) + \\mathbb{P}\\left(\\textbf{A} = 1 | \\textbf{S} = \\text{EL}\\right) \\neq 1 $:" ] }, { "cell_type": "code", "execution_count": null, "id": "2bd104d0-9038-4424-8da2-3247fbfc665b", "metadata": {}, "outputs": [ { "data": { "text/html": [ "1.61056175047339" ], "text/latex": [ "1.61056175047339" ], "text/markdown": [ "1.61056175047339" ], "text/plain": [ "[1] 1.610562" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########### Solution R: ##########\n", "\n", "# Probabilité empirique d'être en GM ou en EL\n", "prop_section <- data %>% count(Section) %>% mutate(prop = prop.table(n))\n", "# Probabilité empirique de réussir l'examen en fonction de la section GM ou en EL\n", "prop_succes <- Prop %>% filter(Succes == 1) %>% select(Section,prop)\n", "# Probabilité empirique de ne pas réussir l'examen en fonction de la section GM ou en EL\n", "prop_fail <- Prop %>% filter(Succes == 0) %>% select(Section,prop)\n", "\n", "# Calcul de cette somme de probabilité\n", "prob_sum <- prop_succes[1,2]/prop_section[1,3] + prop_succes[2,2]/prop_section[2,3]\n", "prob_sum" ] }, { "cell_type": "markdown", "execution_count": null, "id": "edd35636-33c2-455e-a25a-2a3ca81eabe2", "metadata": {}, "source": [ "3\\) Cette erreur est plutôt commune et vient d'une confusion avec une autre équation qui elle est correcte:\n", "\\begin{equation*}\\mathbb{P}\\left(\\textbf{A} = 1 | \\textbf{S} = \\text{s}\\right) + \\mathbb{P}\\left(\\textbf{A} = 0 | \\textbf{S} = \\text{s}\\right) = 1, \\forall s\\in\\{\\text{GM},\\text{EL}\\}\n", "\\end{equation*}\n", "Vérifier expérimentalement que cette équation est correcte. \n", "**Solution**: \n", "Ce résultat est vrai de manière générale, il découle directement de la définition de la probabilité conditionnel. Sauriez-vous montrer rigoureusement pourquoi cela est vrai ?\n" ] }, { "cell_type": "code", "execution_count": null, "id": "ab5db9c2-3893-4652-93d2-647d9ac979c6", "metadata": {}, "outputs": [ { "data": { "text/html": [ "1" ], "text/latex": [ "1" ], "text/markdown": [ "1" ], "text/plain": [ "[1] 1" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "1" ], "text/latex": [ "1" ], "text/markdown": [ "1" ], "text/plain": [ "[1] 1" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "########### Solution R: ##########\n", "\n", "EL <- prop_succes[1,2]/prop_section[1,3] + prop_fail[1,2]/prop_section[1,3]\n", "GM <- prop_succes[2,2]/prop_section[2,3] + prop_fail[2,2]/prop_section[2,3]\n", "\n", "EL\n", "GM" ] }, { "cell_type": "markdown", "execution_count": null, "id": "1b8143f5-afac-42a4-b23b-370e94e8c789", "metadata": { "tags": [] }, "source": [ "## Exercice 2: \n", "\n", "Nous allons considérer quatre fonctions définies de l'ensemble $\\left\\{0, 1, 2, ..., 10\\right\\}$ dans $\\mathbb{R}$. Le but de cet exercice est de vérifier lesquels de ces fonctions sont des fonctions de masses." ] }, { "cell_type": "code", "execution_count": null, "id": "5fe3683d-2412-4fe7-8c41-fe8d450a8f6f", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/html": [ "0.25" ], "text/latex": [ "0.25" ], "text/markdown": [ "0.25" ], "text/plain": [ "[1] 0.25" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "source(\"fonction.r\")\n", "\n", "X = c(0,1,2,3,4,5,6,7,8,9,10)\n", "\n", "# Vous pouvez accéder à ces quatres fonctions, f1,f2,f3,f4, de la manière suivante\n", "f1(2)" ] }, { "cell_type": "markdown", "execution_count": null, "id": "53c6c17e-85a1-4536-a45f-76818cce5bf4", "metadata": {}, "source": [ "1\\) La fonction ```f1``` est-elle une fonction de masse ?\n", "\n", "**Solution**: Oui, la fonction de ```f1``` est une fonction de masse.\n" ] }, { "cell_type": "code", "execution_count": null, "id": "d32c2069-fc18-4ca1-924e-0ce288c74b75", "metadata": { "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1] \"La fonction f1 est donc une fonction de masse\"\n" ] } ], "source": [ "########### Solution R: ##########\n", "\n", "# Vérifier si f1 est une fonction de masse\n", "masse <- TRUE\n", "sum_ <- 0\n", "for (x in X){\n", " sum_ <- sum_ + f1(x)\n", " if (f1(x)<0){\n", " masse = FALSE\n", " sprintf(\"f1(%i) = %f < 0\", x,f1(x))\n", " }\n", "}\n", "\n", "if (abs(sum_-1)>1e-06){\n", " masse = FALSE\n", "}\n", "\n", "if (masse){\n", " print(\"La fonction f1 est donc une fonction de masse\")\n", "} else{\n", " print(\"La fonction f1 n'est donc pas une fonction de masse\")\n", "}" ] }, { "cell_type": "markdown", "execution_count": null, "id": "fbebfc7a-0fae-4df4-90ff-80585761176a", "metadata": {}, "source": [ "2\\) La fonction ```f2``` est-elle une fonction de masse ?\n", "\n", "**Solution**: Non, la fonction de ```f2``` n'est pas une fonction de masse car elle ne respecte pas la condition $\\sum_i f_2(x_i) = 1$\n" ] }, { "cell_type": "code", "execution_count": null, "id": "f7b501ce-df58-4d20-9fbc-0626c0e9a7d7", "metadata": { "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1] \"La fonction f2 n'est donc pas une fonction de masse\"\n" ] } ], "source": [ "########### Solution R: ##########\n", "\n", "# Vérifier si f2 est une fonction de masse\n", "masse <- TRUE\n", "sum_ <- 0\n", "for (x in X){\n", " sum_ <- sum_ + f2(x)\n", " if (f2(x)<0){\n", " masse = FALSE\n", " sprintf(\"f2(%i) = %f < 0\", x,f2(x))\n", " }\n", "}\n", "\n", "if (abs(sum_-1)>1e-06){\n", " masse = FALSE\n", "}\n", "\n", "if (masse){\n", " print(\"La fonction f2 est donc une fonction de masse\")\n", "} else{\n", " print(\"La fonction f2 n'est donc pas une fonction de masse\")\n", "}" ] }, { "cell_type": "markdown", "execution_count": null, "id": "28f56f47-a823-469d-a4c3-4da6ccb871a1", "metadata": {}, "source": [ "3\\) La fonction ```f3``` est-elle une fonction de masse ?\n", "\n", "**Solution**: Oui, la fonction de ```f3``` est une fonction de masse.\n" ] }, { "cell_type": "code", "execution_count": null, "id": "456b3f7b-6ff7-45f0-8035-99618911e4a0", "metadata": { "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1] \"La fonction f3 est donc une fonction de masse\"\n" ] } ], "source": [ "########### Solution R: ##########\n", "\n", "# Vérifier si f2 est une fonction de masse\n", "masse <- TRUE\n", "sum_ <- 0\n", "for (x in X){\n", " sum_ <- sum_ + f3(x)\n", " if (f3(x)<0){\n", " masse = FALSE\n", " sprintf(\"f3(%i) = %f < 0\", x,f3(x))\n", " }\n", "}\n", "\n", "if (abs(sum_-1)>1e-06){\n", " masse = FALSE\n", "}\n", "\n", "if (masse){\n", " print(\"La fonction f3 est donc une fonction de masse\")\n", "} else{\n", " print(\"La fonction f3 n'est donc pas une fonction de masse\")\n", "}" ] }, { "cell_type": "markdown", "execution_count": null, "id": "0b6824bf-e5e7-4908-b911-bb03eec572ea", "metadata": {}, "source": [ "4\\) La fonction ```f4``` est-elle une fonction de masse ?\n", "\n", "**Solution**: Non, la fonction de ```f4``` n'est pas une fonction de masse car elle ne respecte pas la condition $\\sum_i f_4(x_i) = 1$ et on a que $f4(9) < 0$ ce qui n'est pas possible, par définition, pour une fonction de masse.\n" ] }, { "cell_type": "code", "execution_count": null, "id": "ceae07e5-fab1-493c-8491-cddc0783996a", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1] \"La fonction f4 n'est donc pas une fonction de masse\"\n" ] } ], "source": [ "########### Solution R: ##########\n", "\n", "# Vérifier si f4 est une fonction de masse\n", "masse <- TRUE\n", "sum_ <- 0\n", "for (x in X){\n", " sum_ <- sum_ + f4(x)\n", " if (f4(x)<0){\n", " masse = FALSE\n", " sprintf(\"f4(%i) = %f < 0\", x,f4(x))\n", " }\n", "}\n", "\n", "if (abs(sum_-1)>1e-06){\n", " masse = FALSE\n", "}\n", "\n", "if (masse){\n", " print(\"La fonction f4 est donc une fonction de masse\")\n", "} else{\n", " print(\"La fonction f4 n'est donc pas une fonction de masse\")\n", "}" ] }, { "cell_type": "markdown", "execution_count": null, "id": "c413482b-1037-416b-9c43-d226ed8708af", "metadata": { "tags": [] }, "source": [ "## Exercice 3: \n", "\n", "Le but de cet exercice est de déterminer si les fonctions suivantes peuvent être des fonctions de répartitions." ] }, { "cell_type": "code", "execution_count": null, "id": "73eb96f8-4fd9-40e7-ab1a-6dcdfa81cce4", "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": 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lPM77ygcweBZHm1gKSsqiE9ptxe5sVPygQky6sFJGVAKhoHkkdAUgakonEgeQQk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\ngFRhQAKSG5AqDEhAcgNShQEJSG5AqjAgAckNSBUGJCC5AanCgAQ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"text/plain": [ "plot without title" ] }, "metadata": { "image/png": { "height": 420, "width": 420 } }, "output_type": "display_data" } ], "source": [ "library(ggplot2)\n", "library(patchwork)\n", "\n", "data1 <- data.frame(x=c(-10.,0.,0.,10.,10.,20.,20.,30.,30.,40.,40.,50.,50.,60.,60.,70.,70.,80.,80.,90.,90.,100.,100.,120.,120.),\n", " y=c(0.,0.,0.1,0.1,0.19,0.19,0.25,0.25,0.34,0.34,0.45,0.45,0.5,0.5,0.57,0.57,0.61,0.61,0.63,0.63,0.65,0.65,0.67,0.67,0.67))\n", "data2 <- data.frame(x=0.1*c(-10.,0.,0.,10.,10.,20.,20.,30.,30.,35.,35.,50.,50.,54.,54.,70.,70.,83.,83.,90.,90.,115.,115.,120.,120.),\n", " y=c(0.,0.,0.05,0.05,0.09,0.09,0.15,0.15,0.34,0.34,0.39,0.39,0.47,0.47,0.67,0.67,0.86,0.86,0.89,0.89,0.91,0.91,1.,1.,1.))\n", "data3 <- data.frame(x=c(-10.,0.,0.,10.,10.,20.,20.,30.,30.,35.,35.,50.,50.,54.,54.,70.,70.,83.,83.,90.,90.,115.,115.,120.,120.),\n", " y=c(0.,0.,0.05,0.05,0.09,0.09,0.15,0.15,0.34,0.34,0.39,0.39,0.47,0.47,0.67,0.67,0.86,0.86,0.84,0.84,0.91,0.91,1.,1.,1.))\n", "\n", "f1 <- ggplot(data1,aes(x,y)) + geom_line() + labs(title = \"Figure 1\") + theme(plot.title = element_text(hjust = 0.5))\n", "f2 <- ggplot(data2,aes(x,y)) + geom_line() + labs(title = \"Figure 2\") + theme(plot.title = element_text(hjust = 0.5))\n", "f3 <- ggplot(data3,aes(x,y)) + geom_line() + labs(title = \"Figure 3\") + theme(plot.title = element_text(hjust = 0.5))\n", "\n", "(f1 + f2) / f3 " ] }, { "cell_type": "markdown", "execution_count": null, "id": "147fb203-de61-40c9-8cab-e9eb7b8a1ee0", "metadata": {}, "source": [ "1\\) La Figue 1 est-elle le plot d'une fonction de répartition ?\n", "\n", "**Solution**: Non, car $ F(x) = \\Pr (X \\leq x) $ implique que $ \\lim_{x\\to\\infty} F(x) = 1 $. \n", "\n", "2\\) La Figue 2 est-elle le plot d'une fonction de répartition ?\n", "\n", "**Solution**: Oui. \n", "\n", "3\\) La Figue 3 est-elle le plot d'une fonction de répartition ?\n", "\n", "**Solution**: Non, car la fonction doit être non décroissante.\n" ] }, { "cell_type": "markdown", "execution_count": null, "id": "67434dea-3bc5-4331-acd7-632dfe21f7a8", "metadata": { "tags": [] }, "source": [ "## Exercice 4: Lien entre une variable de Bernoulli et une variable géométrique\n", "\n", "Considérons un archer, qui vise le milieu de la cible. Il tire autant de flèche que nécessaire et s'arrête une fois qu'une d'entre elle frappe le centre de la cible. On suppose qu'il parvient à atteindre son but avec une probabilité $p$ à chaque flèche et que chaque tir est indépendant des autres.\n", "- Le fait que le $i^{ème}$ tir atteigne le centre de la cible peut alors être modéliser par la variable aléatoire $X_i \\sim Bern(p)$. \n", "- De même, le nombre de flèches nécéssaires avant d'atteindre le centre de la cible peut être modéliser par la variable aléatoire $ S = \\min\\left\\{{i\\in\\mathbb{N} | X_i = 1}\\right\\}$, qui suit une loi géométrique de paramètre $p$." ] }, { "cell_type": "markdown", "execution_count": null, "id": "141b7ec6-82ef-4670-8a0f-aa06bcf403b7", "metadata": {}, "source": [ "1\\) Dans question nous allons créer une fonction ```bern``` qui génère aléatoirement une variable de Bernouilli pour un paramètre $p$ donné en argument. \n", "a\\) En utilisant la fonction ```runif``` qui permet de générer des réalisations de variables aléatoires uniformes, et la fonction indicatrice $\\mathbf{1}_{\\{U1-\\frac{2p}{3}\\}\\}}\\end{equation*} \n", "\n", "On pourrait encore en envisager une infinité, le point important est la mesure (la taille) de l'intervalle sur lequel la fonction indicatrice vaut $1$ soit égal à $p$.\n" ] }, { "cell_type": "markdown", "execution_count": null, "id": "181ae162-997f-48b9-9c2c-883c520df4fd", "metadata": {}, "source": [ "2\\) En utilisant la fonction ```bern```, écrire une fonction ```geom``` qui simule la génération d'une variable aléatoire $S = \\min\\left\\{{i\\in\\mathbb{N} | X_i = 1}\\right\\}$, où $X_i \\sim Bern(p)$." ] }, { "cell_type": "code", "execution_count": null, "id": "4f80a109-6d5b-4668-a8d0-5ec8e4089e7f", "metadata": { "tags": [] }, "outputs": [], "source": [ "geom <- function(p){\n", " if (p<1e-06){\n", " stop(\"La probabilité de succès doit être supérieure à zéro.\")\n", " }\n", " \n", " \n", " S <- 1\n", " X <- bern(p,1)\n", " \n", " while (X != 1){\n", " S <- S + 1\n", " X <- bern(p,1)\n", " }\n", " S\n", " \n", "}" ] }, { "cell_type": "markdown", "execution_count": null, "id": "b0939f97-0f38-48cb-b602-de06681f2682", "metadata": {}, "source": [ "3\\) Nous allons maintenant estimer quelques probabilités liées à notre variable aléatoire $S$: \n", "a\\) Grâce à un plot et une valeure numérique, estimer la valeure de $\\mathbb{P}(S = 1)$ ($n = 10000$ devrait être suffisant pour une bonne estmination). \n", "\n", "**Solution**: \n", "Observe que la probabilité que notre estimation semble converger vers $p = 0.3$, qui est justement la vrai valeur de $\\mathbb{P}(S = 1)$ (pour $p = 0.3$).\n" ] }, { "cell_type": "code", "execution_count": null, "id": "5c4839fa-6099-4dd1-9092-21ad500dd3f3", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/html": [ "'On estime donc cette probabilité par : 0.292500'" ], "text/latex": [ "'On estime donc cette probabilité par : 0.292500'" ], "text/markdown": [ "'On estime donc cette probabilité par : 0.292500'" ], "text/plain": [ "[1] \"On estime donc cette probabilité par : 0.292500\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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kGniRuvpV67Lae+cRs3Z+I47s7NC/Zl6xl3Il2SdSzBMMqwEV3L03fW67aphrHB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"text/plain": [ "plot without title" ] }, "metadata": { "image/png": { "height": 420, "width": 420 } }, "output_type": "display_data" } ], "source": [ "n <- 10000\n", "p <- 0.3\n", "\n", "# Générer n observations de S\n", "df <- data.frame(S = sapply(p*rep(1,n), FUN = geom))\n", "\n", "# Proposer une estimation de P(S = 1)\n", "prob1 <- sum(df$S == rep(1,n))/n\n", "sprintf(\"On estime donc cette probabilité par : %f\",prob1)\n", "\n", "# Faire un plot montrant la convergence de cette probabilité\n", "df$prob <- cumsum((df$S == rep(1,n)))/(1:n)\n", "df$iter <- 1:n\n", "\n", "fig <-ggplot(df,aes(x=iter,y=prob)) + geom_line() + \n", " labs(title=\"Convergence de l'estimation de P(S=1)\", \n", " x = 'Nombre de variable générée',y=\"Estimation de la probabilité\") +\n", " theme(plot.title = element_text(hjust = 0.5, size = 16))\n", "fig" ] }, { "cell_type": "markdown", "execution_count": null, "id": "80dfbd98-0600-46c2-8a9c-595aabfbe2ab", "metadata": {}, "source": [ "b\\) De la même manière que la question précédente, estimer la probabilité $\\mathbb{P}(S>3)$.\n", " \n", "**Solution**: \n", "Observe que la probabilité que notre estimation semble converger vers une valeure proche de $0.34$, qui est justement proche de la vrai valeur de $\\mathbb{P}(S > 3) = (1-p)^3 = 0.343$ (pour $p = 0.3$). \n" ] }, { "cell_type": "code", "execution_count": null, "id": "954df319-97c5-4f55-8b77-53d15259dafb", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/html": [ "'On estime donc cette probabilité par : 0.338800'" ], "text/latex": [ "'On estime donc cette probabilité par : 0.338800'" ], "text/markdown": [ "'On estime donc cette probabilité par : 0.338800'" ], "text/plain": [ "[1] \"On estime donc cette probabilité par : 0.338800\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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VUlmvioYUiYF+2m6ocfqX6r5UvKBQ9+02Byd8s5GFl2vfipsfL4RmpvmyUCzLhr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\nil4Fg1IziCu4UaR3BvZ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"text/plain": [ "plot without title" ] }, "metadata": { "image/png": { "height": 420, "width": 420 } }, "output_type": "display_data" } ], "source": [ "n <- 10000\n", "p <- 0.3\n", "\n", "# Générer n observations de S\n", "df <- data.frame(S = sapply(p*rep(1,n), FUN = geom))\n", "\n", "# Proposer une estimation de P(S = 1)\n", "prob2 <- sum(df$S > 3.*rep(1,n))/n\n", "sprintf(\"On estime donc cette probabilité par : %f\",prob2)\n", "\n", "# Faire un plot montrant la convergence de cette probabilité\n", "df$prob <- cumsum((df$S > 3*rep(1,n)))/(1:n)\n", "df$iter <- 1:n\n", "\n", "fig <-ggplot(df,aes(x=iter,y=prob)) + geom_line() + \n", " labs(title=\"Convergence de l'estimation de P(S>1)\", \n", " x = 'Nombre de variable générée',y=\"Estimation de la probabilité\") +\n", " theme(plot.title = element_text(hjust = 0.5, size = 16))\n", "fig" ] }, { "cell_type": "markdown", "execution_count": null, "id": "3f6aafab-e9fa-42c7-814a-551a970ab050", "metadata": {}, "source": [ "c\\) Enfin, pour des valeures $m$ et $t$ fixées, vérifier que $\\mathbb{P}(S > t + m| S > t) = \\mathbb{P}(S > m)$.\n", "\n", "**Solution**: \n", "On observe donc ces deux valeurs convergent bien l'une vers l'autre, et que chacune convergent vers la valeurs $\\mathbb{P}(S > t + m| S > t) = \\mathbb{P}(S > m) = (1-p)^m = 0.489$ (pour $p = 0.3$).\n" ] }, { "cell_type": "code", "execution_count": null, "id": "3b2575d3-f7f4-41af-8834-218df3283ff2", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/html": [ "'On estime donc la probabilité P(S > t + m | S > t) par : 0.493775'" ], "text/latex": [ "'On estime donc la probabilité P(S > t + m \\textbar{} S > t) par : 0.493775'" ], "text/markdown": [ "'On estime donc la probabilité P(S > t + m | S > t) par : 0.493775'" ], "text/plain": [ "[1] \"On estime donc la probabilité P(S > t + m | S > t) par : 0.493775\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "'On estime donc la probabilité P(S > m) par : 0.487300'" ], "text/latex": [ "'On estime donc la probabilité P(S > m) par : 0.487300'" ], "text/markdown": [ "'On estime donc la probabilité P(S > m) par : 0.487300'" ], "text/plain": [ "[1] \"On estime donc la probabilité P(S > m) par : 0.487300\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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AWdrRcS/BuDODmkecY5z564n9qFThYy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\niv+qtaPfEPEHSDJAkiU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"text/plain": [ "plot without title" ] }, "metadata": { "image/png": { "height": 420, "width": 420 } }, "output_type": "display_data" } ], "source": [ "n <- 20000\n", "p <- 0.3\n", "\n", "t <- 4\n", "m <- 2\n", "\n", "# Générer deux ensemble de n observations de S pour estimer chacunes de ces \n", "# deux probabilités avec deux ensembles distinct\n", "df <- data.frame(S1 = sapply(p*rep(1,n), FUN = geom),S2 = sapply(p*rep(1,n), FUN = geom))\n", "\n", "# Proposer une estimation de P(S > t + m | S > t)\n", "prob3_1 <- sum(df$S1 > (t+m)*rep(1,n))/n\n", "prob3_2 <- sum(df$S2 > t*rep(1,n))/n\n", "prob3 <- prob3_1/prob3_2\n", "sprintf(\"On estime donc la probabilité P(S > t + m | S > t) par : %f\",prob3)\n", "\n", "# Proposer une estimation de P(S > m)\n", "prob3_ <- sum(df$S2 > m*rep(1,n))/n\n", "sprintf(\"On estime donc la probabilité P(S > m) par : %f\",prob3_)\n", "\n", "# Faire un plot montrant la convergence de cette probabilité\n", "p_sup_t <- cumsum((df$S1 > t*rep(1,n)))\n", "ind <- (p_sup_t > 0)\n", "\n", "df_ <- data.frame(iter = (1:n)[ind], \n", " diff = abs(cumsum((df$S1 > (t+m)*rep(1,n)))[ind]/cumsum((df$S1 > t*rep(1,n)))[ind]-cumsum((df$S2 > m*rep(1,n)))[ind]/(1:n)[ind]))\n", "\n", "fig <-ggplot(df_,aes(x=iter,y=diff)) + geom_line() + \n", " labs(title=\"Convergence de la différence entre les deux estimations\", \n", " x = 'Nombre de variable générée',y=\"Différence absolue en les deux estimations\") +\n", " theme(plot.title = element_text(hjust = 0.5, size = 16))\n", "fig" ] }, { "cell_type": "markdown", "execution_count": null, "id": "f322e097-de2a-4313-9cf3-dc7bfcb1f6d0", "metadata": { "tags": [] }, "source": [ "## Exercice 5: \n", "\n", "1\\) Laquelle des fonctions suivantes est la densité d'une variable continue ? Reconnaissez-vous cette loi ?\n", "\n", "**Solution**: \n", "Seules les deux dernières sont des fonctions de densité d'une variable continue car les deux autres ne satisfont pas $\\int_{\\mathbb{R}} f(x) dx = 1$. La figure 1 représente la fonction de densité d'une loi uniforme $\\mathcal{U}(0,1)$.\n" ] }, { "cell_type": "code", "execution_count": null, "id": "332e9bfe-57e0-4eb6-84f5-3ca06937613a", "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": 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AUiQSQ9IJIUiASR9IBIUiASRNKj8CJxX+3bNPMTjS+GJtnq9aVFTzbOXKw7pKpe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"text/plain": [ "plot without title" ] }, "metadata": { "image/png": { "height": 420, "width": 420 } }, "output_type": "display_data" } ], "source": [ "library(ggplot2)\n", "library(patchwork)\n", "\n", "data1 <- data.frame(x=c(-1.,0.,0.,1.,1.,2.),\n", " y=c(0.,0.,2.,2.,0.,0.))\n", "data2 <- data.frame(x=c(-1.,0.25,0.25,0.75,0.75,2.),\n", " y=c(0.,0.,1.,1.,0.,0.))\n", "data3 <- data.frame(x=c(-1.,0.,0.,1.,1.,2.),\n", " y=c(0.,0.,1.,1.,0.,0.))\n", "data4 <- data.frame(x=c(-1.,-0.5,0.,0.5,1.,1.5,2.,2.5),\n", " y=c(0.,0.,1.,0.,0.,1.,0.,0.))\n", "\n", "f1 <- ggplot(data1,aes(x,y)) + geom_line() + labs(title = \"Figure 1\") + theme(plot.title = element_text(hjust = 0.5))\n", "f2 <- ggplot(data2,aes(x,y)) + geom_line() + labs(title = \"Figure 2\") + theme(plot.title = element_text(hjust = 0.5))\n", "f3 <- ggplot(data3,aes(x,y)) + geom_line() + labs(title = \"Figure 3\") + theme(plot.title = element_text(hjust = 0.5))\n", "f4 <- ggplot(data4,aes(x,y)) + geom_line() + labs(title = \"Figure 4\") + theme(plot.title = element_text(hjust = 0.5))\n", "(f1 + f2) / (f3 + f4) " ] }, { "cell_type": "markdown", "execution_count": null, "id": "e4478c91-4ec6-44bf-ad72-037a24316435", "metadata": { "tags": [] }, "source": [ "## Exercice 6: \n", "\n", "1\\) Est-ce que les fonctions suivantes sont des fonctions de répartition? Si oui, le sont-elles pour des variables continues ou discrètes ?\n", "\n", "**Solution**: \n", "- Le premier et le dernier graphique représentent des fonctions de répartition de lois discrètes.\n", "- Le deuxième et le troisième représentent des fonctions de répartition de lois continues.\n", "- Le quatrième graphique représente une fonction de répartition d'une loi qui est ni discrète, ni continue (c'est en fait une loi mixte).\n", "- La fonction représentée sur le cinquième graphique n'est pas une fonction de répartition, car elle n'est pas croissante. \n" ] }, { "cell_type": "code", "execution_count": null, "id": "836eff77-3990-453b-b2ea-05b9105badb8", "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": 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iDF6UjCQLodVrLCSQApTkcSBdL2ltl7oxzqhk82TOmXP55/Kz5sMCHlI/PuGFMV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\nSDOhnyaiBUm4tbOrQLLhSKwgjYOh6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"text/plain": [ "plot without title" ] }, "metadata": { "image/png": { "height": 420, "width": 420 } }, "output_type": "display_data" } ], "source": [ "library(ggplot2)\n", "library(patchwork)\n", "\n", "data1 <- data.frame(x=c(-10.,0.,0.,10.,10.,20.,20.,30.,30.,40.,40.,50.,50.,60.,60.,70.,70.,80.,80.,90.,90.,100.,100.,120.,120.),\n", " y=c(0.,0.,0.1,0.1,0.19,0.19,0.25,0.25,0.34,0.34,0.45,0.45,0.5,0.5,0.57,0.57,0.61,0.61,0.63,0.63,0.65,0.65,0.67,0.67,0.67))/0.67\n", "data2 <- data.frame(x=seq(-10, 10, length.out = 100),\n", " y=exp(seq(-10, 10, length.out = 100))/(1+exp(seq(-10, 10, length.out = 100))))\n", "data3 <- data.frame(x=c(-1.,0.,1.,2.),\n", " y=c(0.,0.,1.,1.))\n", "data4 <- data.frame(x=c(-1.,0,1,2,2,3,4),\n", " y=c(0.,0.,0.4,0.4,0.6,1.,1))\n", "data5 <- data.frame(x=c(-1.,0,1,2,2,3,4),\n", " y=c(0.,0.,0.4,0.,0.6,1.,1))\n", "data6 <- data.frame(x=c(-1.,0,0,1,1,2),\n", " y=c(0.,0.,0.4,0.4,1.,1))\n", "\n", "f1 <- ggplot(data1,aes(x,y)) + geom_line() + labs(title = \"Figure 1\") + theme(plot.title = element_text(hjust = 0.5))\n", "f2 <- ggplot(data2,aes(x,y)) + geom_line() + labs(title = \"Figure 2\") + theme(plot.title = element_text(hjust = 0.5))\n", "f3 <- ggplot(data3,aes(x,y)) + geom_line() + labs(title = \"Figure 3\") + theme(plot.title = element_text(hjust = 0.5))\n", "f4 <- ggplot(data4,aes(x,y)) + geom_line() + labs(title = \"Figure 4\") + theme(plot.title = element_text(hjust = 0.5))\n", "f5 <- ggplot(data5,aes(x,y)) + geom_line() + labs(title = \"Figure 5\") + theme(plot.title = element_text(hjust = 0.5))\n", "f6 <- ggplot(data6,aes(x,y)) + geom_line() + labs(title = \"Figure 6\") + theme(plot.title = element_text(hjust = 0.5))\n", "(f1 + f2 + f3) / (f4 + f5 + f5) " ] } ], "metadata": { "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": "4.2.1" } }, "nbformat": 4, "nbformat_minor": 5 }