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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + À propos du calcul de \(\pi\) @@ -256,15 +256,15 @@ for the JavaScript code in this tag.

Table des matières

-
-

1 En demandant à la lib maths

+
+

1 En demandant à la lib maths

Mon ordinateur m'indique que \(\pi\) vaut approximativement @@ -282,8 +282,8 @@ Mon ordinateur m'indique que \(\pi\) vaut approximativement

-
-

2 En utilisant la méthode des aiguilles de Buffon

+
+

2 En utilisant la méthode des aiguilles de Buffon

Mais calculé avec la méthode des aiguilles de Buffon, on obtiendrait comme approximation : @@ -299,20 +299,20 @@ theta = pi/2*runif(N)

+
 [1] 3.14327
 
 
-
-

3 Avec un argument "fréquentiel" de surface

+
+

3 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 \(P[X^2+Y^2 \leq 1]=\pi/4\) (voir méthode -de Monte Carlo sur Wikipedia). Le code suivant illustre ce fait : +\(X \sim U(0,1)\) et \(Y \sim U(0,1)\) alors \(P[X^2+Y^2 \leq 1] = \pi/4\) (voir méthode de Monte Carlo sur Wikipedia). Le code suivant illustre ce fait :

@@ -333,7 +333,7 @@ ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + t

Il est alors aisé d'obtenir une approximation (pas terrible) de \(\pi\) en -comptant combien de fois, en moyenne, \(X^2+Y^2\) est inférieur à 1 : +comptant combien de fois, en moyenne, \(X^2 + Y^2\) est inférieur à 1 :

@@ -349,9 +349,8 @@ comptant combien de fois, en moyenne, \(X^2+Y^2\) est inférieur à 1 :
-

Date: 10/06/2020

Auteur: Julien EMMANUEL

-

Created: 2020-06-10 Wed 15:07

+

Created: 2020-06-10 Wed 15:23

Validate

diff --git a/module2/exo1/toy_document_orgmode_R_fr.org b/module2/exo1/toy_document_orgmode_R_fr.org index b596f94..2494f9a 100644 --- a/module2/exo1/toy_document_orgmode_R_fr.org +++ b/module2/exo1/toy_document_orgmode_R_fr.org @@ -1,14 +1,13 @@ #+TITLE: À propos du calcul de $\pi$ #+AUTHOR: Julien EMMANUEL -#+DATE: 10/06/2020 #+LANGUAGE: fr -# #+PROPERTY: header-args :eval never-export +#+PROPERTY: header-args :session :exports both * En demandant à la lib maths Mon ordinateur m'indique que $\pi$ vaut /approximativement/ -#+begin_src R :results output :exports both +#+begin_src R :results output :session *R* :exports both pi #+end_src @@ -19,7 +18,7 @@ pi Mais calculé avec la *méthode* des [[https://fr.wikipedia.org/wiki/Aiguille_de_Buffon][aiguilles de Buffon]], on obtiendrait comme *approximation* : -#+begin_src R :results output :exports both +#+begin_src R :results output :session *R* :exports both set.seed(42) N = 100000 x = runif(N) @@ -28,16 +27,16 @@ theta = pi/2*runif(N) #+end_src #+RESULTS: +: : [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 $P[X^2+Y^2 \leq 1]=\pi/4$ (voir [[https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80][méthode -de Monte Carlo sur Wikipedia]]). Le code suivant illustre ce fait : +$X \sim U(0,1)$ et $Y \sim U(0,1)$ alors $P[X^2+Y^2 \leq 1] = \pi/4$ (voir [[https://fr.wikipedia.org/wiki/M%25C3%25A9thode_de_Monte-Carlo#D%25C3%25A9termination_de_la_valeur_de_%25CF%2580][méthode de Monte Carlo sur Wikipedia]]). Le code suivant illustre ce fait : -#+begin_src R :results output graphics :file figure_pi_mc1.png :session *R* :exports both +#+begin_src R :results output graphics :file figure_pi_mc1.png :exports both :width 600 :height 400 :session *R* set.seed(42) N = 1000 df = data.frame(X = runif(N), Y = runif(N)) @@ -50,11 +49,11 @@ ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + t [[file:figure_pi_mc1.png]] Il est alors aisé d'obtenir une approximation (pas terrible) de $\pi$ en -comptant combien de fois, en moyenne, $X^2+Y^2$ est inférieur à 1 : +comptant combien de fois, en moyenne, $X^2 + Y^2$ est inférieur à 1 : #+begin_src R :results output :session *R* :exports both 4*mean(df$Accept) -#+end_src +#+end_src #+RESULTS: : [1] 3.156 -- 2.18.1