19. Consider the following two complete conditional distributions, originally analyzed by Casella and George (1992):

f(x|y) ∝ ye−yx, 0

f(y|x) ∝ xe−xy, 0

(a) Obtain an estimate of the marginal distribution of X when B = 10 using the Gibbs sampler.

(b) Now suppose B = ∞, so that the complete conditional distributions are ordinary (untruncated) exponential distributions. Show analytically that fx(t)=1/t is a solution to the integral equation fx(x) =  

fx|y(x|y)fy|t(y|t)dy

fx(t)dt in this case. Would a Gibbs sampler converge to this solution? Why or why not?