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