Add exo 2

M1/Monte Carlo Methods/Exercise2.rmd

@@ -16,12 +16,32 @@ qqplot(x, rexp(n, rate = 2))
 ```{r}
 lambda <- 1.5
 n <- 10000
+d <- 10
 S <- numeric(n)
 for (i in 1:n) {
-    u <- runif(10)
+    u <- runif(d)
     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")
+curve(dgamma(x, shape = d, rate = lambda), add = TRUE, col = "red")
+```
+
+### Question 3
+```{r}
+lambda <- 4
+n <- 10000
+X <- numeric(n)
+p <- 1
+for (i in 1:n) {
+    S <- -1/lambda * log(1-runif(1))
+    while (S <= 1) {
+      S <- x - 1/lambda * log(1-runif(1))
+      p <- p+1
+    }
+    X[i] <- p
+    p <- 1
+}
+hist(X, freq = FALSE, breaks = 50)
+curve(dpois(x, lambda), add = TRUE, col = "red")
 ```