---
title: "Exercise 2 R-studio"
author: "Husnain Arshad"
date: "8/14/2022"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

# À propos de pi

```{r}
pi 
```

## Buffon’s needle
Applying the method of Buffon’s needle:[Buffon's needle](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon)

```{r}
set.seed(42)
N = 100000
x = runif(N)
theta = pi/2*runif(N)
2/(mean(x+sin(theta)>1))
```

## Using Fraction arguement
A method that is easier to understand and does not make use of the sin function is based on the fact that if  $X\sim U(0,1)$ et $Y\sim U(0,1)$ then $P[X^2+Y^2\leq 1] = \pi/4$ [Mont carlo method](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 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 :

```{r}
4*mean(df$Accept)
```