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

Note that, if X_{1}≤ y, X_{1}≤ y, X_{3}≤ y and X_{4}> y, then Y ≤ y.

(i) Write down three similar arrangements of the X_{i}that also imply that Y ≤ y.

(ii) Write down one more, slightly different arrangement of the X_{i}that implies that Y ≤ y.

(iii) Deduce that Y has cumulative distribution function given by

{0, y≤0

F(y)= {4y^{3}+ 3y^{3}, 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