3. Let X1,X2, . . . ,Xn be a sample from an absolutely continuous DF F with PDF f .

Show that EX(r)

≈ F−1



r n+1



and var(X(r)) ≈ r(n−r+1)

(n+1)2(n+2)

1

{f [F−1(r/n+1)]}2 .

[Hint: Let Y be an RV with mean μ and φ be a Borel function such that Eφ(Y) exists.

Expand φ(Y) about the point μ by a Taylor series expansion, and use the fact that F(X(r)) = U(r).]