From 6c44e233387ca427f063c5c806190c61f54fcfa4 Mon Sep 17 00:00:00 2001 From: Rose-Delamare Date: Sun, 8 Dec 2024 15:22:59 +0100 Subject: [PATCH] =?UTF-8?q?=E2=9A=A1=20compliated=20exo1?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- module2-exo1.md | 45 ++++++++++++++++++ .../unnamed-chunk-3-1.png | Bin 0 -> 132365 bytes 2 files changed, 45 insertions(+) create mode 100644 module2-exo1.md create mode 100644 module2-exo1_files/figure-markdown_strict/unnamed-chunk-3-1.png diff --git a/module2-exo1.md b/module2-exo1.md new file mode 100644 index 0000000..e8ab1e6 --- /dev/null +++ b/module2-exo1.md @@ -0,0 +1,45 @@ +## En demandant à la lib maths + +Mon ordinateur m’indique que *π* vaut *approximativement* + + pi + + ## [1] 3.141593 + +## En utilisant la méthode des aiguilles du Buffon + +Mais calculé avec la méthode des `aiguilles du Buffon`, on obtiendrait +comme **approximation** : + + set.seed(42) + N = 100000 + x = runif(N) + theta = pi/2*runif(N) + 2/(mean(x+sin(theta)>1)) + + ## [1] 3.14327 + +## Avec un argument “fréquentiel” de surface + +Sinon, une méthode plus simple à comprendre et ne faisant pas intervenir +d’appel à la fonction sinus se base sur le fait que si *X* ∼ *U*(0,1) et +*Y* ∼ *U*(0,1) alors *P*\[*X*2+*Y*2≤1\] = *π*/4 +(voir [méthode de Monte Carlo sur +Wikipedia](https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80)). +Le code suivant illustre ce fait : + + set.seed(42) + N = 1000 + df = data.frame(X = runif(N), Y = runif(N)) + df$Accept = (df$X**2 + df$Y**2 <=1) + library(ggplot2) + ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw() + +![](module2-exo1_files/figure-markdown_strict/unnamed-chunk-3-1.png) Il +est alors aisé d’obtenir une approximation (pas terrible) de *π* en +comptant combien de fois, en moyenne, *X*2 + *Y*2 +est inférieur à 1 : + + 4*mean(df$Accept) + + ## [1] 3.156 diff --git 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