Let pk, k ≧ 0, be a probability distribution having finite binomial moments Br, r ≧ 0, and the probability generating function U(z) with a radius of convergence α (≧ 1). In this note explicit and recursive formulae are derived allowing computation of the pk in terms of the Br if α > 1. Ch. Jordan's formula, which holds if α > 2, turns out to be a special case.

We derive the Sobel–Uppuluri and Galambos-type extensions of the Bonferroni bounds, and further extensions of the same nature, as consequences of a single non-probabilistic inequality. The methodology follows that of Galambos.

Let A1, A2, · ··, An be events on a probability space. Denote by Sk the kth binomial moment of the number Mn of those A 's which occur. Sharp lower and upper bounds of Sm will be given in terms of Sk and Sl. The same method can be applied for proving Bonferroni–Galambos-type inequalities.

Given n events A1, A2, · ··, An, bounds are obtained for the probability of the occurrence of at least m out of the n events. The bounds are linear in terms of S1 and S2 where S1 = Σi P(Ai) and S2 = Σi<jΡ (Α i,Α j). The bounds are illustrated with an example. For m = 1 the bounds are the same as the best known linear bounds in terms of S1 and S2.

Let A1, A2, ···, An be events on a given probability space and let Br, n be the event that exactly r of the A's occur. Let further Sk (n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n), as n→ + ∞, in terms of limits of the Sk(n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X1, X2, ···, Xn by choosing Aj = {Xj ≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.