Question

Suppose X is a Gamma random variable with shape parameter α and scale parameter θ > 0, i.e., the pdf is given as, f(x|α, θ) = 1 Γ(α)θ α x α−1 e −x/θ , 0 < x < ∞, (1) where α > 0, θ > 0 and Γ(a) = Z ∞ 0 x a−1 e −x dx. HINT: see section 3.2 of the textbook. (a) What is the support of X? That is, X = ? (b) Show that the function given in (1) is a true pdf, i.e., that f(x) > 0 for all x ∈ X and Z ∞ −∞ f(x)dx = 1. (c) Show that the moment generating function M(t) for the density defined in (1) exists and is given by M(t) = 1 (1 − θt) α , t < 1 θ (2) (d) Using the moment generating function in (2), show that µ = E(X) = αθ, EX2 = αθ2 + α 2 θ 2 , and σ 2 = αθ2

Answer 1