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Explicit results on conditional distributions of generalized exponential mixtures. (English) Zbl 1454.62066

Summary: For independent exponentially distributed random variables \(X_i\), \(i\in\mathcal{N}\), with distinct rates \(\lambda_i\) we consider sums \(\sum_{i\in\mathcal{A}}X_i\) for \(\mathcal{A}\subseteq\mathcal{N}\) which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum \(\sum_{i\in\mathcal{N}}X_i\) given that a subset sum \(\sum_{j\in\mathcal{A}}X_j\) exceeds a certain threshold value \(t>0\), and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for \(t\to\infty\). Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.

MSC:

62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
60K10 Applications of renewal theory (reliability, demand theory, etc.)
90B25 Reliability, availability, maintenance, inspection in operations research

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