Question

Suppose there are n independent Gaussian r.v.s, X j ∼ N ( μ j , σ 2 j ) for j = 1 , 2 , . . . , n , with possibly different means and variances

(a)  For any constants a j’s, find the MGF of a linear combination of these n independent Gaussian r.v.s, i.e., the MGF ofY=a1X1+a2X2+· · ·anXn(=Pnj=1ajXj)

Hint: Since MX(t) =E(etX), it is the case that E(eatX) =MX(at), i.e., the MGF evaluated at at.

(b) Recall that the MGF uniquely defines the distribution of a random variable, i.e., if two random variables have the same MGF, then their distributions must be the same. Based on the functional form of the MGF of Y in (a), specify the name of Y’s distribution, including the parameter

Answer 1

For reference below is the MGF of Normal,