diff --git a/module2/exo1/toy_document_en.Rmd b/module2/exo1/toy_document_en.Rmd index 30151911087f72a4653759488edb2c8135de2d69..8ca3190ba6918bc0676bc2187b0cfa12ba9fb62b 100644 --- a/module2/exo1/toy_document_en.Rmd +++ b/module2/exo1/toy_document_en.Rmd @@ -17,7 +17,7 @@ My computer tells me that $\pi$ is *approximatively* pi ``` -## 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__ @@ -34,16 +34,17 @@ theta = pi/2*runif(N) 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$Y**2 <=1) -library(ggplot2) +set.seed(42) +N = 1000 +df = data.frame(X = runif(N), Y = runif(N)) +df$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$Accept) -``` +```{r} +4*mean(df$Accept) +``` \ No newline at end of file