Explicit results on conditional distributions of generalized exponential mixtures

For independent exponentially distributed random variables, with distinct rates we consider sums for which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum given that a subset sum exceeds a certain threshold value 0 ]]>, and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for. Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.