32.

a. Let X1, . . . , Xn be a random sample from a uniform distribution on [0, ]. Then the mle of  is ˆ


Y
max(Xi

).

Use the fact that Y  y iff each Xi  y to derive the cdf of Y. Then show that the pdf of Y
max(Xi

) is fY(y)
{

ny



n n

1

 0  y  

0 otherwise

b. Use the result of part

(a) to show that the mle is biased but that (n 1)max(Xi

)/n is unbiased.