(Continuation of Problem 6.1.5.) Let X(n) = maxi(X1 , X2 , • X ), the maximum of a sample of size n from the distribution n described in Problem 6.1.5. (a) Show that P(X(n) ~ x) = (x/6)n for x [O, 6)....

(Continuation of Problem 6.1.5.) Let X(n) = maxi(X1 , X2 , •

X ), the maximum of a sample of size n from the distribution n

described in Problem 6.1.5.

(a) Show that P(X(n) ~ x) = (x/6)n for x [O, 6).

( ) n-1 n (b) Show that fX(n) x = nx /6 on [O, 6) and 0 elsewhere.

(c) Find Var(X(n)) and Bias (X(n)).

(d) Use Problem 6.1.2 to demonstrate that X(n) is a consistent

estimator of e.

(e) Find e(2X, X(n)) and the limit of this efficiency as n ~ oo.