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Reliability for some bivariate gamma distributions. (English) Zbl 1094.62132

Summary: In the area of stress-strength models, there has been a large amount of work as regards estimation of the reliability \(R=\text{Pr} (X<Y)\). The algebraic form for \(R=\text{Pr}(X<Y)\) has been worked out for the vast majority of the well-known distributions when \(X\) and \(Y\) are independent random variables belonging to the same univariate family. We consider forms of \(R\) when \((X,Y)\) follows a bivariate distribution with dependence between \(X\) and \(Y\). In particular, we derive explicit expressions for \(R\) when the joint distribution is bivariate gamma. The calculations involve the use of special functions.

MSC:

62N05 Reliability and life testing
62H10 Multivariate distribution of statistics
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