voici rendu exercice 1 module 2

module2/exo1/toy_document_en.Rmd

 ---
-title: "Your title"
-author: "Your name"
-date: "Today's date"
+title: "À propos du calcul de pi"
+author: "Louis Hognon"
+date: "17/04/2020"
 output: html_document
 ---
 
 
-```{r setup, include=FALSE}
-knitr::opts_chunk$set(echo = TRUE)
-```
+# En demandant à la lib maths ###################
 
-## Some explanations
+Mon ordinateur m’indique que $\pi$ vaut approximativement
 
-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>.
+```{r}
+pi
+```
 
-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:
+# En utilisant la méthode des aiguilles de Buffon ###############
 
-```{r cars}
-summary(cars)
+Mais calculé avec la **méthode** des [aiguilles de Buffon](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), on obtiendrait comme approximation :
+```{r}
+set.seed(42)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
 ```
 
-It is also straightforward to include figures. For example:
+# Avec un argument “fréquentiel” de surface #############
 
-```{r pressure, echo=FALSE}
-plot(pressure)
+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]= $\pi$ /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}
+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()
 ```
 
-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.
+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:
 
-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.
+```{r}
+4*mean(df$Accept)
+```
 
-Now it's your turn! You can delete all this information and replace it by your computational document.