Question

Problem 3 (m.g.f. of Binomial random variables). We proved that the m.g.f. ψX(t) “generates” the moments of the random variable X by differentiation, and computation at t = 0 (rather than by integration or summation, which is typically harder). For example, ψ 0 X(0) = E(X), ψ00 X(0) = E(X 2 ), . . . , ψ(n) X (0) = E(X n ), where the superscript (n) indicates the n th derivative. (a) Assume that X ∼ Binomial(n, p), i.e. with p.f. fX(k) = n k p k (1 − p) n−k , for k = 0, 1, . . . , n. Find the function ψX(t), in terms of the parameters n and p. Hint: You will need to use the binomial theorem, which states thatX n k=0 n k a k b n−k = (a + b) n . (b) Find E(X) and Var(X) using the method described above. Note that this is much simpler than computing, for example, E(X) = X n k=0 k n k p k (1− p) n−k .

Answer 1