Explicit results on conditional distributions of generalized exponential mixtures

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Date

2020

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Abstract

For independent exponentially distributed random variables Xi, i ∈ N with distinct rates λi we consider sums ∑i∈AXi for A⊆N which follow generalized exponential mixture (GEM) distributions. We provide novel explicit results on the conditional distribution of the total sum ∑i∈NXi giventhat a subset sum ∑j∈NXj exceeds a certain threshold value t > 0, and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for t → ∞,. Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.

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conditional distribution, tail behavior, risk management, reliability theory, phase-type distributions, generalized exponential mixture distribution, exponential distribution, convolution

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