From c2bba253a5844f0a28165c19fdba18763dc01cd0 Mon Sep 17 00:00:00 2001 From: c319822a5d066547a0de4251b9819297 Date: Thu, 3 Oct 2024 14:25:47 +0000 Subject: [PATCH] Update toy_document_en.Rmd --- module2/exo1/toy_document_en.Rmd | 19 ++++++++++++------- 1 file changed, 12 insertions(+), 7 deletions(-) diff --git a/module2/exo1/toy_document_en.Rmd b/module2/exo1/toy_document_en.Rmd index 06139c5..f032a56 100644 --- a/module2/exo1/toy_document_en.Rmd +++ b/module2/exo1/toy_document_en.Rmd @@ -4,15 +4,19 @@ author: "Arnaud Legrand" date: "25 juin 2018" output: html_document --- -##Asking the maths library -My computer tells me that π is *approximatively* + +```{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** +## 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 @@ -20,8 +24,8 @@ 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*∼*U*(0,1)** and ***Y*∼*U*(0,1)**, then **P[X^2+Y^2≤1]=π/4 **(see [“Monte Carlo method” on Wikipedia](https://en.wikipedia.org/wiki/Monte_Carlo_method)). The following code uses this approach: +## 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 @@ -30,7 +34,8 @@ 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 therefore straightforward to obtain a (not really good) approximation to π 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) ``` -- 2.18.1