Update toy_document_en.Rmd

module2/exo1/toy_document_en.Rmd

+# On the computation of pi
+
+title: "Mr"
+author: "TEGEGNE"
+
+date: "11 June 2020"
+
+output: ## Asking the maths library
+
+My computer tells me that π is approximatively
+
 ---
-title: "Your title"
-author: "Your name"
-date: "Today's date"
-output: html_document
----
+pi
+
+
+```
+## [1] 3.141593
+```
+
+## Buffon’s needle
 
+Applying the method of (https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), we get the (**approximation**)
 
-```{r setup, include=FALSE}
-knitr::opts_chunk$set(echo = TRUE)
+```
+set.seed(42)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
 ```
 
-## Some explanations
+```
+## [1] 3.14327
+```
 
-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>.
+## Using a surface fraction argument
 
-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:
+A method that is easier to understand and does not make use of the (**sin**) function is based on the fact that if (**X∼U(0,1)**) and (**Y∼U(0,1),**) then P(**[X2+Y2≤1]=π/4 **)(see “Mhttps://en.wikipedia.org/wiki/Monte_Carlo_method") The following code uses this approach:
 
-```{r cars}
-summary(cars)
+```
+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:
+It is therefore straightforward to obtain a (not really good) approximation to π by counting how many times, on average, (**X2**)+(**Y2**) is smaller than
+1 :
 
-```{r pressure, echo=FALSE}
-plot(pressure)
+```
+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.
+```
+## [1] 3.156
+```
 
-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.