1. (a) Let X ∈ R be a random variable and f : R → R a function. Introducing any notation you use, give the formula for the Monte Carlo estimate ZN for the expectation E (f(X)) .

(b) Prove that the Monte Carlo estimate from part (a) has mean squared error

MSE(ZN ) = σ 2 /N

and explain what σ 2 represents in this equation

(c) How should we choose f, if we want to compute an estimate for the probability p = P(X ∈ A)? What is the mean squared error in this case, as a function of N and p?

(d) Assume that X ∼ Exp(λ) and that we want to estimate H = R ∞ 0 h(x) dx for a given function h.

2. (a) In lectures we have introduced the importance sampling estimate

3. (a) The Raleigh distribution with scale parameter σ > 0 has density 

4. (a) Give the definition of a Markov Chain with discrete state space S

(b) Introducing any notation you use, state the Random Walk Metropolis algorithm for the state space S = Z = {. . . , −2, −1, 0, 1, 2, . . .}. What is the purpose of this algorithm?

(c) Given p ∈ (0, 1), use the Random Walk Metropolis algorithm to construct a Markov chain whose stationary distribution is the geometric distribution, with weights πk = (1 − p) k−1p for all k ∈ N.

(d) i. Using the Markov chain from part (c), state the Markov Chain Monte Carlo (MCMC) estimator for the expectation E (f(X)) where f is a given function, and X is geometrically distributed with parameter p.

ii. Give an (approximate) formula for the mean squared error of your estimator.

iii. Explain, why sometimes a burn-in period is used for MCMC estimators.

(e) Is the use of the Random Walk Metropolis algorithm in part (d)i appropriate? Justify your answer.

5. Assume we are given data x1, . . . , xn ∈ R, which were obtained by taking independent random samples from a random variable X.