Order Statistics - Let X1, . . . , Xn be i.i.d. continuous r.v.s with PDF f and a strictly increasing CDF F. Suppose that we know that the jth order statistic of n i.i.d. Unif(0, 1) r.v.s is a Beta(j, n j + 1), but we have forgotten the formula andderivation for the distribution of the jth order statistic of X1, . . . , Xn. Show how we can recover the PDF of X(j) using a change of variables.

(Hint: Use the fact that F1(U) has c.d.f F if U Unif(0,1). Also note that since F is strictly increasing on its support so is F 1 on (0, 1). Therefore, we can represent Xi = F 1(Ui). Now note that because F1 is strictly increasing, X(j) = F1(U(j)). Therefore, we need to find the p.d.f of F1(U(j)) and we know the p.d.f of U(j). We are in good shape now to use the 1d change of variables theorem. Or you can use the CDF approach as well here.)