Update toy_document_en.Rmd

module2/exo1/toy_document_en.Rmd

@@ -5,7 +5,6 @@ date: "25 juin 2018"
 output: html_document
 ---
 
-
 ```{r setup, include=FALSE}
 knitr::opts_chunk$set(echo = TRUE)
 ```
@@ -29,7 +28,7 @@ theta = pi/2*runif(N)
 ```
 
 ## 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:
+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)
@@ -38,9 +37,12 @@ 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 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 :
+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)
 ```
+