9. Let X ~ Bin(n,p) be a binomial r.v. Recall the derivation of Var(X ) in the lecture. We used Var(X) = E(X")-(EX)', and used the binomial theorem to evaluate E(X2). Alternatively, one could use the representation of X = )_ Y of X as sum of Bernoulli r.v.'s, Yi ~ Bern(p), independently. Use this representation and Corollary 3.4 from the lecture to find Var(X). Which of the following arguments proves the claim? (if multiple choices are correct, mark any of those) (a) E(X') = E E(Y?) = mp3 and EX = EE(Y) = np = Var(X) = E(X?) - (EX)' = np- - (np)? = mp3(1 - n) (b) By Corollary 3.4 E(X') = _ _E()?) = ap, and EX = CE(Y) = np = Var(X) = E(X? ) - (EX)? = np - (np)? = np-(1 -n) 2 (c) Var(X) = _ Var(Y,), by Corollary 3.4, since the Y, are independent. Var(Y) = E(Y?) - (Evi)' = p- p= = p(1 - p), and therefore Var(X) = np(1 - p) (d) Var(X) = EL_, Var(Y) + 2 );; Cov(Yi, Y;) = ap(1 -p) +(n - 1)p. (e) none of these