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.

中文翻译：

---

广义指数混合条件分布的显式结果

对于独立的指数分布随机变量$X_i$,$i\in {\mathcal{N}}$, 具有不同的速率${\lambda}_i$我们考虑总和$\sum_{i\in\mathcal{A}} X_i$为了$\mathcal{A}\subseteq {\mathcal{N}}$遵循广义指数混合分布。我们提供了关于总和的条件分布的新的明确结果$\sum_{i\in {\mathcal{N}}}X_i$给定子集总和$\sum_{j\in \mathcal{A}}X_j$超过某个阈值$t>0$，反之亦然。此外，我们研究了这些条件分布的特征尾行为$t\to\infty$. 最后，我们通过提供可靠性理论和风险管理的例子来说明我们的概率结果如何在实践中应用。