8. Let X1, ..., Xn be a sample from a continuous distribution with distribution function F. In this exercise, we want to show that EF(X(k)) = k/(n + 1). Define U; = F(X;) for i = 1, ..., n. (i) Show that the random variables U1, ..., Un form a sample from the uniform distribution on [0, 1]. (ii) Show that the distribution function F(k) of Uck) is given by n F( k(x) = > n x' (1 - x) "- j. j=k (iii) Show that the density f(k) of U(k) is given by n! f ( 1) (x) = x*-1 (1 - x ) "- k (k - 1)!(n - k)! (iv) Show that EU ( k) = k/(n + 1)