Hi, please answer each part of the problem using the given information. Please also show all steps/any work involved to better my understanding. Please don't repost the work of other posts. Please do not repost the work of other posts. Thanks! This is a STATS problem NOT Artificial Intelligence. Please stop putting it under AI.

Random interval length. Let X ~ Unif [0, 1]. Then X divides the unit interval into two subintervals [0, X] and (X, 1]. One of these subintervals is picked in the following way. Let the random variable Y ~ Unif [0, 1] be independent of X and pick the subinterval that Y falls into. Define the length of the subinterval chosen as L. Thus L = X if Y X. What is the minimum MSE estimate of L given that you observed Y? Your answer should be a function only of Y. (Hint: You can solve this problem without first finding the conditional pdf by using the fact that L is a function of the independent r.v.s X, Y. Of course it's fine if you decide to find the conditional pdf first.)