7.3.4 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 subintervals [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 on [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

Determine the mean of L.

Transcribed Image Text:

X if Y X, L= 1-X if Y>X.