The order statistic x (1) x(2) x(n) of random variables i.i.d. xk ~ U (0, 1 1) for 1 k n partitions the interval (0,1) into n + 1 intervals with lengths {x (1),,x (k+1) = x(k), , 1 x (n)}. (a) With = min {x(1),, x(k+1) x(k), , 1 x (n)} * show that the random variable (n + 1) l has distribution Beta (1, n). (b) Show that - t t P (1 1+1) = n - L ( 1 w) n 1 dw = wn-1 dw "L -t = 1 (1 t)". (4) (5) (6) Interpret this result in case you want to sample uniform order statistics conditioned on 1 > 1/2 by rejection from unconditioned samples. n+1