4. (40 pts) A stick of length 1 is broken at a uniformly random point, yielding two pieces. Let X and Y be the lengths of the shorter and longer pieces, respectively, and let R : X / Y be the ratio of the lengths X and Y. (a Find the CDF and PDF of X. ( ) 1)) Find the CDF and PDF of Y. (c) Find the CDF and PDF of R. ) (d Find the expected values of X , Y, and R. (Hint: Let U denote the breaking point. Express X and Y each as a flmction of U. Express R as a function of X.)