This problem is designed to aid in the understanding of length-biased sampling. Let X be a uniformly distributed random variable on [0,1]. Then X divides [0,1] into the subinterval [0, X) and (X,1]. By symmetry, each subinterval has mean length 1/2. Now pick one of these subintervals at random in the following way:
Let Y be independent of X and uniformly distributed in [0,1], and pick the subinterval [0, X] or (X,1] that Y falls in. Let L be the length of the subinterval so chosen. Formally,
L = { X if Y<_X
1 - X if Y > X}
A) Determine the mean of L.