From bd3743ae94b6bd1d3fad6b961e5e5663d026a761 Mon Sep 17 00:00:00 2001 From: af867c30e46f4b444ec768838323b91d Date: Sat, 6 Jun 2020 10:39:57 +0000 Subject: [PATCH] Update toy_document_orgmode_R_en.org --- module2/exo1/toy_document_orgmode_R_en.org | 11 ++++++++++- 1 file changed, 10 insertions(+), 1 deletion(-) diff --git a/module2/exo1/toy_document_orgmode_R_en.org b/module2/exo1/toy_document_orgmode_R_en.org index a782a5f..24ee120 100644 --- a/module2/exo1/toy_document_orgmode_R_en.org +++ b/module2/exo1/toy_document_orgmode_R_en.org @@ -20,8 +20,10 @@ pi #+RESULTS: : [1] 3.141593 + * Buffon's needle Applying the method of [[https://en.wikipedia.org/wiki/Buffon%2527s_needle_problem][Buffon's needle]], we get the *approximation* + #+begin_src R :results output :session *R* :exports both set.seed(42) N = 100000 @@ -29,10 +31,13 @@ x = runif(N) theta = pi/2*runif(N) 2/(mean(x+sin(theta)>1)) #+end_src + #+RESULTS: : [1] 3.14327 + * 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 [[https://en.wikipedia.org/wiki/Monte_Carlo_method]["Monte Carlo method" on Wikipedia]]). The following code uses this approach: + #+begin_src R :results output graphics :file figure_pi_mc1.png :exports both :width 600 :height 400 :session *R* set.seed(42) N = 1000 @@ -41,11 +46,15 @@ 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() #+end_src + #+RESULTS: [[file:figure_pi_mc1.png]] + 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: + #+begin_src R :results output :session *R* :exports both 4*mean(df$Accept) #+end_src + #+RESULTS: -: [1] 3.156 \ No newline at end of file +: [1] 3.156 -- 2.18.1