Let X be an absolutely continuous random variable with the following: Range(X) = (0, 1) I'(a + b) Γ(α)Γ(b) fx(x) = x-1(1

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Let X be an absolutely continuous random variable with the following: Range(X) = (0, 1) I'(a + b) Γ(α)Γ(b) fx(x) = x-1(1-x), 0<x< 1. This means X has the beta distribution, and we write X Beta(a, b). Now let Y~ Gamma(a + b, c) be independent of X, and define a third random variable Z = XY. (a) (5 pts) Write the joint distribution of X and Z in hierarchical form: X~ ??? Z|X = x ~ ???. (b) (10 pts) Based on this hierarchy, compute the marginal density of Z. Is it familiar?