# Exercise 2 : Exponential distribution and related distributions

### Question 1
```{r}
n <- 10000
u <- runif(n)
x <- -1/2 * log(1-u)
hist(x, breaks = 50, freq = FALSE)
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")
```