Exercise 2 (Monte Carlo methods) (a) Describe a Monte Carlo approximation of R 0 e 5x14x2 dx. Hint: Use the exponential distribution with parameter 5. (b) Bonus task: We aim at approximating the integral I[f] := R 1 0 f(x) dx with a Monte Carlo approximation IN [f]. Let us denote by the square root of the variance of f, i.e. := Z 1 0 (f(x) I[f])2 dx1/2 . Show that the root mean square error (RMSE) {E(I[f] IN [f])2}1/2 satisfies E(I[f] IN [f])2 1/2 = N (1) and reflect on this result. Hint: To prove (1) consider Yi := 1 (f(Xi) I[f]) with Xi uniformly distributed and SN := 1 N PN i=1 Yi