1. Write a general Metropolis or Metropolis-Hastings algorithm to generate a sample from a target distribution π, where π(x) ∝ x34(1 - x)38(2 + x)125. Use the proposal density as q(x, y) = 1 on the interval [0, 1].

2. For the bivariate density given in Example 13.5.5, starting with three different values of y0, say, 1/3, 1/2, and 2/3 n = 15, and α = 1,β = 2, obtain the ?rst three realizations of the Gibbs sequence. Comment on the in?uence of the initial values.

3. Consider a problem of sampling bivariate random variables with joint density given by rce-(x+y+4xy), x ≥ 0,y ≥ 0 f(x, y) = 0, otherwise.

(a) Find f (x |y ) and f (y |x ).

(b) Write a Gibbs procedure to generate samples from this distribution. Discuss why it is easier to use the Gibbs sampler for this case.

(c) Starting from an arbitrary point, obtain the ?rst three sample points.