If computation facilities are available, write a program for the Gibbs sampler of Exercise 11.4.7. Run your program for α = 10, β = 4, m = 3000, and n = 6000. Obtain estimates (and confidence intervals) of E(X) and E(Y) and compare them with the true parameters.

Exercise 11.4.7

Consider the following mixed discrete-continuous pdf for a random vector (X, Y), (discussed in Casella and George, 1992):

for α > 0 and β > 0.
(a) Show that this function is indeed a joint, mixed discrete-continuous pdf by finding the proper constant of proportionality.
(b) Determine the conditional pdfs f(x|y) and f(y|x).
(c) Write the Gibbs sampler algorithm to generate random samples on X and Y.
(d) Determine the marginal distributions of X and Y.

Transcribed Image Text:

f(x, y) x { (2)yz+a-1 (1-y)n-x+8-1 0 x = 0, 1,..., n, 0 < y < 1 elsewhere,