# 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
d <- 10
S <- numeric(n)
for (i in 1:n) {
  u <- runif(d)
  x <- -1 / lambda * log(1 - u)
  S[i] <- sum(x)
}
hist(S, freq = FALSE, breaks = 50)
curve(dgamma(x, shape = d, rate = lambda), add = TRUE, col = "red")
```

### Question 3
```{r}
n <- 10000
S <- numeric(n)
i <- 1
for (j in (1:n)) {
  x <- -1 / 4 * log(1 - runif(1))
  while (x <= 1) {
    i <- i + 1
    x <- x - 1 / 4 * log(1 - runif(1))
  }
  S[j] <- i
  i <- 1
}
hist(S, freq = FALSE)
curve(dpois(S, lambda = 4), add = TRUE, col = "red")
```