Statistics

6. (3 pts) Suppose there are n independent Gaussian r.v.s, X; ~ N(u;, ; ) for j = 1, 2, ..., n, with possibly different means and variances (i.e., may not be identically distributed). For any constants a;'s, find the MGF of a linear function of these n independent Gaussian r.v.s, i.e., the MGF of Y = alX1 + a2X2+ . . . anXn (= Cha; X;). Hint: Since Mx (t) = E(et* ), it is the case that E(eatx ) = Mx(at), i.e., the MGF of X evaluated at at