Lehmann and Scheffé have shown how to construct uniformly most powerful unbiased tests of certain hypotheses when the assumed distributions belong to an “exponential family.” The present paper is concerned with particular cases which arise from independent gamma variates with scale parameters θ1and θ2. Conditional distributions are given which are appropriate for testing γ = γ0, where γ =c1/θ1+c2/θ2. As an application, suppose that a series system has two dissimilar components with expected lives θ1and θ2. When component failures are exponentially distributed, so are system failures, the mean being 1/(1/θ1+1/θ2). From independent estimates of θ1and θ2confidence limits can be found for this mean, or for the probability of successful system operation up to any fixed “mission time.” With appropriate restrictions, more general distributions including the Weibull can also be treated.