1. Let X be a. strictly positive random variable with cdf F1112) := lP'(X S 3:). (a) Prove that the following three conditions are equivalent: I F(:r) is a concave function of in E [0, oo); o the distribution of X has a. right continuous, weakly decreasing probability density function at) for :1: E (0, oo); o X g U Z for some distribution of Z and U uniform [0, 1] independent of Z; (b) Assuming these conditions hold, derive a. simple formula. for NZ 5 .T) in terms of F(s:) and f (1r)