From 3d4a6d62b91ebc17b76470868090b48198ae3d47 Mon Sep 17 00:00:00 2001
From: 3582521b868d7c148df1de482ec5a734
 <3582521b868d7c148df1de482ec5a734@app-learninglab.inria.fr>
Date: Wed, 25 Oct 2023 20:48:44 +0000
Subject: [PATCH] Update toy_document_en.Rmd

---
 module2/exo1/toy_document_en.Rmd | 55 ++++++++++++++++++++------------
 1 file changed, 34 insertions(+), 21 deletions(-)

diff --git a/module2/exo1/toy_document_en.Rmd b/module2/exo1/toy_document_en.Rmd
index 13b258d..0fd1c3c 100644
--- a/module2/exo1/toy_document_en.Rmd
+++ b/module2/exo1/toy_document_en.Rmd
@@ -1,33 +1,46 @@
 ---
-title: "Your title"
-author: "Your name"
-date: "Today's date"
+title: "On the computation of pi"
+author: "Arnaud Legrand"
+date: "25 juin 2018"
 output: html_document
 ---
 
-
+	
 ```{r setup, include=FALSE}
 knitr::opts_chunk$set(echo = TRUE)
 ```
+## Asking the maths library
 
-## 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>.
-
-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:
+My computer tells me that $\pi$ is *approximatively*
 
-```{r cars}
-summary(cars)
+```{r}
+pi
 ```
-
-It is also straightforward to include figures. For example:
-
-```{r pressure, echo=FALSE}
-plot(pressure)
+## 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)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
+```
+## 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)	
+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 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)	
 ```
 
-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.
-
-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.
-- 
2.18.1