This problem is a follow up on the problem described in HW1.
In this problem we are asked to derive analytically the PDF of the random variable \(Y\) by conditioning on the box selected.
\(Y\) is the random variable which is the observation from the following experiment: Generate random variable \(X\) from uniform \(\left [ 0,1\right ] \). Put this number is box labeled \(S\) and put twice this number in a box labeled \(\bar {S}\). Next, we pick one of these 2 boxes by random. If the number inside the box selected is found to be greater than 1, then we switch the boxes and pick the number inside the second box. The random variable \(Y\) is the final number selected.