done

module2/exo2/exercice_en.Rmd

 ---
-title: "Your title"
-author: "Your name"
-date: "Today's date"
-output: html_document
+title: "À propos du calcul de pi"
+author: "Agnieszka Duniec"
+date: "4 juin 2020"
+output:
+  html_document: default
+  pdf_document: default
 ---
 
+## En demandant à la lib maths
+
+Mon ordinateur m’indique que _π_ vaut _approximativement_
 
 ```{r setup, include=FALSE}
 knitr::opts_chunk$set(echo = TRUE)
 ```
+```{r}
+pi
+```
+
+## En utilisant la méthode des aiguilles de Buffon
+
+Mais calculé avec la __méthode__ des [aiguilles de Buffon](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), on obtiendrait comme __approximation__ :
 
-## Some explanations
+```{r}
+set.seed(42)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
+```
 
-This is an R Markdown document that you can easily export to HTML, PDF, and MS Word formats. For more information on R Markdown, see <http://rmarkdown.rstudio.com>.
+## Avec un argument “fréquentiel” de surface
 
-When you click on the button **Knit**, the document will be compiled in order to re-execute the R code and to include the results into the final document. As we have shown in the video, R code is inserted as follows:
+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[X2+Y2≤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:
 
-```{r cars}
-summary(cars)
+```{r}
+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()
 ```
 
-It is also straightforward to include figures. For example:
+Il est alors aisé d’obtenir une approximation (pas terrible) de π en comptant combien de fois, en moyenne, X2+Y2 est inférieur à 1:
 
-```{r pressure, echo=FALSE}
-plot(pressure)
+```{r}
+4*mean(df$Accept)
 ```
 
-Note the parameter `echo = FALSE` that indicates that the code will not appear in the final version of the document. We recommend not to use this parameter in the context of this MOOC, because we want your data analyses to be perfectly transparent and reproducible.
 
-Since the results are not stored in Rmd files, you should generate an HTML or PDF version of your exercises and commit them. Otherwise reading and checking your analysis will be difficult for anyone else but you.
 
-Now it's your turn! You can delete all this information and replace it by your computational document.
+
+
+