Update toy_document_en.Rmd

module2/exo1/toy_document_en.Rmd

 ---
-title: "Your title"
-author: "Your name"
-date: "Today's date"
+title: "On the computation of pi"
+author: "El hassane Nour"
+date: "November 1st, 2022"
 output: html_document
 ---
 
@@ -10,16 +10,41 @@ output: html_document
 knitr::opts_chunk$set(echo = TRUE)
 ```
 
-## Some explanations
 
-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>.
+## Asking the maths library
+My computer tells me that $\pi$ is approximatively
+```{r pi}
+pi
+```
+
+
+##Buffon’s needle
+Applying the method of [Buffon’s needle](https://en.wikipedia.org/wiki/Buffon%27s_needle_problem), we get the approssimazione
+```{r Buffon}
+set.seed(42)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
+```
 
-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:
 
-```{r cars}
-summary(cars)
+##Using a surface fraction argument
+A method that is easier to understand and does not make use of the $\sin$ function is based on the fact that if $X \sim U(0,1)$ and $Y \sim U(0,1)$, then $P[X^2+Y^2 \le 1]=\pi / 4 (see [“Monte Carlo method” on Wikipedia](https://en.wikipedia.org/wiki/Monte_Carlo_method)). The following code uses this approach:
+```{r surface-frac}
+set.seed(42)
+N = 1000
+df = data.frame(X = runif(N), Y = runif(N))
+df$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 therefore straightforward to obtain a (not really good) approximation to \pi by counting how many times, on average, $X^2 + Y^2$ is smaller than 1 :
+```{r surface-frac2}
+4*mean(df$Accept)
 ```
 
+
 It is also straightforward to include figures. For example:
 
 ```{r pressure, echo=FALSE}