(a) Let Y1, , Yn be independent identically distributed

Chapter , Problem 13

(choose chapter or problem)

(a) Let Y1, , Yn be independent identically distributed random variables and let Y = E + ... + Yn. Show that Y E[Yl l Y] = -. n (b) Let a and W be independent zero-mean normal random variables, with positive integer variances k and m, respectively. Use the result of part (a) to find E[a I a + W] , and verify that this agrees with the conditional expectation formula in Example 8.3. Hint: Think of 8 and W as sums of independent random variables. (c) Repeat part (b) for the case where e and W are independent Poisson random variables with integer means ..x and IL, respectively. Solution. (a) By symmetry, we see that E[Yi I Y] is the same for all i. Furthermore, E[Y} + ... + Yn I Y] E[Y I Y] = Y. Therefore, E[Yll Y] = Yin. (b) We can think of a and W as sums of independent standard normal random variables: W=Wl + " ' + Wm, We identify Y with a + W and use the result from part (a), to obtain Thus, 8+W E[8d e + W] = k . +m k E[e I e + W] = E[e} + ... + ek Ie + W] = k (8 + W). +m Sec. The formula for the conditional mean derived in Example 8.3, specialized to the current context (zero prior mean and a single measurement) shows that the conditional expectation is of the form consistent with the answer obtained here. (c) We recall that the sum of independent Poisson random variables is Poisson. Thus the argument in part (b) goes through, by thinking of 8 and W as sums of A (respectively, J.t) independent Poisson random variables with mean one. We then obtain E[818 + W] =