From 5d242a381b877959f1ca64246045e875caaf0cee Mon Sep 17 00:00:00 2001
From: Arthur DANJOU <arthurdanjou@outlook.fr>
Date: Wed, 18 Sep 2024 17:01:12 +0200
Subject: [PATCH] Add question 2

---
 M1/Monte Carlo Methods/Exercise2.rmd | 16 ++++++++++++++++
 1 file changed, 16 insertions(+)

diff --git a/M1/Monte Carlo Methods/Exercise2.rmd b/M1/Monte Carlo Methods/Exercise2.rmd
index f3a27d1..29f1ece 100644
--- a/M1/Monte Carlo Methods/Exercise2.rmd	
+++ b/M1/Monte Carlo Methods/Exercise2.rmd	
@@ -1,5 +1,6 @@
 # Exercise 2 : Exponential distribution and related distributions
 
+### Question 1
 ```{r}
 n <- 10000
 u <- runif(n)
@@ -9,3 +10,18 @@ curve(dexp(x, rate = 2), add = TRUE, col = "red")
 
 qqplot(x, rexp(n, rate = 2))
 ```
+
+### Question 2
+
+```{r}
+lambda <- 1.5
+n <- 10000
+S <- numeric(n)
+for (i in 1:n) {
+    u <- runif(10)
+    x <- -1/lambda * log(1-u)
+    S[i] <- sum(x)
+}
+hist(S, freq = FALSE, breaks = 50)
+curve(dgamma(x, shape = 10, rate = lambda), add = TRUE, col = "red")
+```