Monotonicity and aging properties of random sums

In this paper, we discuss the distributional properties of random sums. We first derive conditions under which the distribution of a binomial sum is PF2 and then show under the same conditions the distribution of a Poisson sum is PF2 by approximating a Poisson sum by a sequence of binomial sums. The PF2 property reveals the monotonicity property of the reversed failure rates of certain compound Poisson distributions. Further, we discuss a class of random sums and derive the NWUE aging property of random sums in the class. The result, together with existing results, shows that the aging properties of many random sums can be characterized uniquely by the aging properties of the primary distributions of the random sums whatever the underlying distributions of the random sums are.