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