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Given n events A 1, A 2, · ··, 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 S 1 and S 2 where S 1 = Σi P(Ai ) and S2 = Σ i<j Ρ (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 S 1 and S 2.

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In this article we study the estimation of the average excess life f3 in a two-parameter exponential distribution with a known linear relationship between a (the minimum life) and 8 of the form OL = a8, where a is known and positive. A comparison of the efficiencies of estimators which are linear combinations of the smallest sample value and the sample sum of deviations from the smallest sample value and the maximum likelihood estimators is made for various sample sizes and different values of a. It is shown that these estimators are dominated in the risk by the minimum-risk scale equivariant estimator based on sufficient statistics. A class of Bayes estimators for inverted gamma priors is constructed and shown to include a minimum-risk scale equivariant estimator in it. All the members of this class can be computed easily.