1. Balls and bins revisited. There are 10 bins numbered 1,2,...,10. n balls are thrown into the 10 bins. For each ball, the probability that it falls into bin i is 1 for i = 1,2,...,10.

10

Different balls are thrown independently of each other. Let Y be the number of balls in bin 1. Let Z be the total number of balls in bins 6, 7, 8, 9, 10.

(a) Find P(Y = y|Z = z). Please specify the range of y, z.

(b) Given Z = z, find the estimator y?(z) that minimizes conditional MSE E[(y?(z)?Y )2|Z =

z].

(c) Find the conditional MSE E[(y?(z) ? Y )2|Z = z] for the estimator in part (b).

(d) Find the linear LMS estimator of Y given Z = z.

(e) Find E[Z] and Var[Z].

(f) Find E[Y ] and Var[Y ].

(g) Find Cov(Y, Z).

Hint: Use the law of total expectation. Try to first determine the conditional pmf.

(h) Find the linear LMS estimator of Z given Y .

(i) Find the corresponding (overall) MSE for the estimator in part (h).

1. Balls and bins revisited. There are 10 bins numbered 1,2, . . . , 10. n balls are thrown into the 10 bins. For each ball, the probability that it falls into bin i is % for i = 1,2, . . . , 10. Dierent balls are thrown independently of each other. Let Y be the number of balls in bin 1. Let Z be the total number of balls in bins 6, 7,8, 9, 10. (a) Find P(Y = y|Z = 2). Please specify the range of y, z. (b) Given Z = 2, nd the estimator 33(2) that minimizes conditional MSE E[(g(z) Y)2|Z = (c) Find the conditional MSE E[(3Q(z) Y)2|Z = z] for the estimator in part (b). (d) Find the linear LMS estimator of Y given Z = z. (e) Find E[Z] and Var[Z]. (f) Find E[Y] and Var[Y]. (g) Find Cov(Y, Z). Hint: Use the law of total expectation. Try to rst determine the conditional pmf. (h) Find the linear LMS estimator of Z given Y. (1) Find the corresponding (overall) MSE for the estimator in part (h)