diff --git a/module2/exo1/toy_document_en.Rmd b/module2/exo1/toy_document_en.Rmd index 6a788fe1a17b36974017dad410da13c21267aaea..d54119e8d8094d7e82b3627646a718d25f3ffe23 100644 --- a/module2/exo1/toy_document_en.Rmd +++ b/module2/exo1/toy_document_en.Rmd @@ -4,13 +4,20 @@ author: "Arnaud Legrand" date: "25 juin 2018" output: html_document --- + +```{r setup, include=FALSE} +knitr::opts_chunk$set(echo = TRUE) +``` + ## Asking the maths library My computer tells me that $\pi$ is *approximatively* + ```{r} pi ``` ## 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 @@ -20,6 +27,7 @@ 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: + ```{r} set.seed(42) N = 1000 @@ -29,6 +37,7 @@ 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) ```