# Exercise 7 : Rejection

```{r}
n <- 5000

f <- function(x, y) {
  return(
    1 / pi * (x^2 + y^2 <= 1)
  )
}

M <- 4 / pi
g <- function(x, y) {
  return(
    1 / 4 * (x >= -1 & x <= 1 & y >= -1 & y <= 1)
  )
}

x <- NULL
y <- NULL
while (length(x) < n) {
  U <- runif(1)
  X <- runif(1, -1, 1)
  Y <- runif(1, -1, 1)
  x <- append(x, X[U <= (f(X, Y) / (M * g(X, Y)))])
  y <- append(y, Y[U <= (f(X, Y) / (M * g(X, Y)))])
}

t <- seq(-1, 1, 0.01)
plot(x, y)
contour(t, t, outer(t, t, Vectorize(f)), add = TRUE, col = "red", lwd = 2)
```