1. Let X(1),X(2), . . .X(n) be the set of order statistics of independent RVs X1,X2, . . . ,Xn with common PDF f (x) =



βe−xβ if x ≥ 0, 0 otherwise.

(a) Show that X(r) and X(s)

−X(r) are independent for any s > r.

(b) Find the PDF of X(r+1)

−X(r).

(c) Let Z1 = nX(1),Z2 = (n−1)(X(2)

−X(1)), Z3 = (n−2)(X(3)

−X(2)), . . . ,Zn =

(X(n)

− X(n−1). Show that (Z1,Z2, . . . ,Zn) and (X1,X2, . . . ,Xn) are identically distributed.