Assume that X1 and X2 are poisson r v

3. Assume that X] and X2 are Poisson random variables with rates 01 and 012 where 0 represents a random environment modulating the rates , and 12. We assume that given the random environment 0, X] and X2 are conditionally independent. Suppose that uncertainty about the environment is described by a gamma distribution, that is, A ~ G(a, B). (a) Obtain the marginal distribution p(X;), j = 1, 2. Do you recognize this distribution ? (b) Obtain the bivariate distribution of X1 and X2, that is, p(X1, X2). (c) Obtain the conditional distribution of X] given X2 and the conditional mean of X1 given X2. Is the conditional mean linear in X2 ? (d) Obtain the correlation between X1 and X2. (e) If we generalize the above distribution from 2 to k dimension, what can you say about correlation between any specific components X; and X