Technology developed in a predecessor paper (Chen and Seneta (1996)) is applied to provide, in a unified manner, a sharpening of bivariate Bonferroni-type bounds on P(v1≥r, v2≥u) obtained by Galambos and Lee (1992; upper bound) and Chen and Seneta (1986; lower bound).

In the paper we first show how to convert a generalized Bonferroni-type inequality into an estimation for the generating function of the number of occurring events, then we give estimates for the deviation of two discrete probability distributions in terms of the maximum distance between their generating functions over the interval [0, 1].

Let (A1A2, · ··, An) be a set of n events on a probability space. Let be the sum of the probabilities of all intersections of r events, and Mn the number of events in the set which occur. The classical Bonferroni inequalities provide upper and lower bounds for the probabilities P(Mn = m), and equal to partial sums of series of the form which give the exact probabilities. These inequalities have recently been extended by J. Galambos to give sharper bounds.

Here we present straightforward proofs of the Bonferroni inequalities, using indicator functions, and show how they lead naturally to new simple proofs of the Galambos inequalities.