2.22 Let X(1)

FX(X(i))

FX(X(iþ1))

for 1  i  n  1 and Vn ¼ FX(X(n))

(a) Find the marginal distribution of Vr, 1  r  n.

(b) Find the joint distribution of Vr and FX(X(r1)), 1  r  n  1, and show that they are independent.

(c) Find the joint distribution of V1, V2, . . . , Vn.

(d) Show that V1, V2, . . . , Vn are independent.

(e) Show that V1,V2 2 ,V3 3 , . . . ,Vn n are independent and identically distributed with the uniform (0, 1) distribution.