# Exercise 9 : Estimation of Pi

## Methode 1

```{r}
n <- 15e4

pi_1 <- function(n) {
  U <- runif(n, 0, 1)
  return(4 / n * sum(sqrt(1 - U^2)))
}
pi_1(n)
```

## Methode 2
```{r}
n <- 15e4

pi_2 <- function(n) {
  U1 <- runif(n, 0, 1)
  U2 <- runif(n, 0, 1)
  return(4 / n * sum(U1^2 + U2^2 <= 1))
}
pi_2(n)
```

## Best Estimator of pi
```{r}
n <- 1000
m <- 15e4

sample_1 <- replicate(n, pi_1(m))
sample_2 <- replicate(n, pi_2(m))

cat(sprintf("[Methode 1] Mean: %s. Variance: %s \n", mean(sample_1), var(sample_1)))
cat(sprintf("[Methode 2] Mean: %s. Variance: %s", mean(sample_2), var(sample_2)))
```