(The independent sampler MCMC). We wish to sample a target distribution Γ(α,β) using the independent MCMC algorithm. We will use as the proposal distribution γ(bαc,b). (a) Derive the corresponding...

(The independent sampler MCMC). We wish to sample a target distribution Γ(α,β) using the independent MCMC algorithm. We will use as the proposal distribution γ(bαc,b).

(a) Derive the corresponding Accept-Reject method. Explain why, when β = 1, the choice of b minimizing the average number of simulations per sample is b = bαc/α.

(b) Generate 10000 Γ(5,5/5.25) random variables to derive a Γ(5.25,1) sample.

(c) Use the same sample in the corresponding Metropolis-Hastings algorithm to generate 10000 Γ(5.25,1) random variables.

(d) Compare the algorithms using (i) their acceptance rates and (ii) the estimates of the mean and variance of the Γ(5.25,1) along with their errors.