1. Compute Pr ( Y3 < >0.5 <>Y7) if Y, <>···<>Y9 are the order statistics of a random sample of size 9 from a distribution of the continuous type.
2. Find the smallest value of n for which Pr ( Y, <>0.5 <>Yn) > 0.99, where Y, <>··· Yn are the order statistics of a random sample of size n from a distribution of the continuous type.
3. Let Y, Y2 denote the order statistics of a random sample of size 2 from a distribution which is N(µ, u2), where u2 is known.
(a) Show that Pr ( Y1 µ Y2 ) = !and compute the expected value of the random length Y2 - Y, .
(b) If X is the mean of this sample, find the constant c such that Pr (X - cu µ X + cu) = !,and compare the length of this random interval with the expected value of that of part (a).