Let Y and X be two random variables such that Y given X is exponential with mean 1/X, and X has the gamma density function with
where λ > −1 and α > 0.
a. Determine the posterior distribution of X given Y .
b. Show that Y has a Pareto distribution p(y) = (λ + 1)αλ+1(y + α)−λ−2, y>0.
c. Find the mean of variance of Y . Under what conditions on α and λ does the latter exist?
d. Verify the calculation of p (yt+1|y(t))and E Yt+1|y(t))