Let Y and X be two random variables such that Y given X is exponential with mean 1 /X , and X has…

Let and be two random variables such that given is exponential with mean 1/X, and has the gamma density function with

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where λ > −1 and α > 0.

a. Determine the posterior distribution of given .

b. Show that has a Pareto distribution p(y) (λ + 1λ+1(y α)λ−2, y>0.

c. Find the mean of variance of . 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))

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