1.13

(a) If X is binomial b(p, n), show that E







x n − p







= 2 n − 1 k − 1



pk

(1 − p)

n−k+1 for k − 1 n ≤ p ≤

k n

.

(b) Graph the risk function of part (i) for n = 4 and n = 5.

[Hint: For (a), use the identity

n x



(x − np) = n

n − 1 x − 1



(1 − p) −

n − 1 x



p



, 1 ≤ x ≤ n.

(Johnson 1957–1958, and Blyth 1980).]