Update toy_document_en.Rmd

module2/exo1/toy_document_en.Rmd

----
-title: "On the computation of pi"
-author: "Lijuan Ren"
-date: "20 Sept 2021"
-output: html_document
----
-
+---	---
+title: "On the computation of pi"	title: "On the computation of pi"
+author: "Arnaud Legrand"	author: "Lijuan Ren"
+date: "25 juin 2018"	date: "20 Sept 2021"
+output: html_document	output: html_document
+---	---
+```{r setup, include=FALSE}	## Asking the maths library
+knitr::opts_chunk$set(echo = TRUE)	pi
+```	## [1] 3.141593
 ## Asking the maths library	
+My computer tells me that $\pi$ is *approximatively*	
+```{r}	
 pi	
-## [1] 3.141593
-
-
-## Buffon’s needle
-set.seed(42)
-N = 100000
-x = runif(N)
-theta = pi/2*runif(N)
-2/(mean(x+sin(theta)>1))
-
-
-## Using a surface fraction argument
-set.seed(42)
-N = 1000
-df = data.frame(X = runif(N), Y = runif(N))
-df$Accept = (df$X**2 + df$Y**2 <=1)
-library(ggplot2)
-ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw()
-4*mean(df$Accept)
-
-
-
-
+```	
+## Buffon's needle	## Buffon’s needle
+Applying the method of [Buffon's needle](https://en.wikipedia.org/wiki/Buffon%27s_needle_problem), we get the __approximation__	
+```{r}	
+set.seed(42)	set.seed(42)
+N = 100000	N = 100000
+x = runif(N)	x = runif(N)
+theta = pi/2*runif(N)	theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))	2/(mean(x+sin(theta)>1))
+```	
+## Using a surface fraction argument	## 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 $P[X^2+Y^2\leq 1] = \pi/4$ (see ["Monte Carlo method" on Wikipedia](https://en.wikipedia.org/wiki/Monte_Carlo_method)). The following code uses this approach:	
+```{r}	
+set.seed(42)	set.seed(42)
+N = 1000	N = 1000
+df = data.frame(X = runif(N), Y = runif(N))	df = data.frame(X = runif(N), Y = runif(N))
+df$Accept = (df$X**2 + df$Y**2 <=1)	df$Accept = (df$X**2 + df$Y**2 <=1)
+library(ggplot2)	library(ggplot2)
+ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw()	ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw()
+```	
+It is then 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}	
+4*mean(df$Accept)	4*mean(df$Accept)
+```
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