Suppose that Xi ~ U(0, 1), i = 1, 2, 3, 4, and that the Xi are mutually independent. Let Y denote the second largest of the Xi (two of the Xi are smaller than Y and one is larger).

Note that, if X1 ≤ y, X1 ≤ y, X3 ≤ y and X4 > y, then Y ≤ y.

(i) Write down three similar arrangements of the Xi that also imply that Y ≤ y.
(ii) Write down one more, slightly different arrangement of the Xi that implies that Y ≤ y.
(iii) Deduce that Y has cumulative distribution function given by


{0, y≤0
F(y)= {4y3 + 3y3 , y E (0,1).
{1, y >/ 1


(iv) Deduce the form of the density function of Y and find E(Y) and var(Y).

Any help would be much appreciated. I cant think of how to do any of this questions