Quiz question 111 a psychology experiment, a random number of people, X , volunteer to be studied. Each of them is randomly assigned to either the control group (with probability 0.2) or the treatment group (with probability 0.8), all independently. Let Y be the number of people assigned to the treatment group. The distribution of X is unknown. However, we know that ]E(X) = 40 and Var(X) = 25. (a) What IS the conditional distribution of Y given X. ? (1)) Find [E ( Y |X) (c) Find lE(Y ). (d)l: Find COV(X Y). Hint: use Theorem 8.10 on page 165': Cov(X, Y) = Cov(X,lE ( Y | X)). (e) Find Var(Y). Hint: use the Variance Partition Fomnuia: Var(Y) : lE(Var (Y | X)) + Var(lE(Y | X)), see Theorem 8.11 on page 163. What is the conditional variance for the conditional dis- tribution you found in part (a)? (f) [Rind pxnx