Answer to Question #334027 in Statistics and Probability for Dinor123

Question

Suppose that the continuous random variable has a Gamma distribution with shape parameter α and scale parameter β, that is X ∼ Γ(α, β) .

(a) Show that the moment generating function of the random variable X is Mx(t) = (1 −t/β)^−α ,

where t < β and use the result to derive the expectation and variance of X.

(b) Hence show that if (x1, · · · , xn) are independent random variable such that each has a Gamma distribution with parameter P αi and β where i = 1, 2, 3, · · · , n , then the random variable Y =sum n, i=1 Xi , has a Gamma distribution with parameter Sum n, i=1 αi and β.

(c) Deduce from your results in (ii) that if (x1, x2, · · · , xn) are iid exponential distributed with

parameter β, then Sn = Sum Xi ∼ Γ(n, β).

Answer 1

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