In this paper, the joint distribution of the sum and maximum of independent, not necessarily identically distributed, nonnegative random variables is studied for two cases: i) continuous and ii) discrete random variables. First, a recursive formula of the joint cumulative distribution function (CDF) is derived in both cases. Then, recurrence relations of the joint probability density function (PDF) and the joint probability mass function (PMF) are given in the former and the latter case, respectively. Interestingly, there is a fundamental difference between the joint PDF and PMF. The proofs are simple and mainly based on the following tools from calculus and discrete mathematics: differentiation under the integral sign (also known as Leibniz's integral rule), the law of total probability, and mathematical induction. Finally, this work generalizes previous results in the literature.

翻译：本文研究独立（不必同分布）非负随机变量之和与最大值的联合分布在两种情形下的性质：i) 连续型随机变量；ii) 离散型随机变量。首先，针对两种情形分别推导出联合累积分布函数（CDF）的递推公式。随后，在前一情形中给出联合概率密度函数（PDF）的递推关系，在后一情形中给出联合概率质量函数（PMF）的递推关系。值得注意的是，联合PDF与联合PMF之间存在根本性差异。证明过程简洁且主要基于微积分与离散数学中的以下工具：积分号下求导（亦称莱布尼茨积分法则）、全概率公式以及数学归纳法。最后，本研究推广了文献中的既有结果。