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. In addition, this work generalizes previous results in the literature, and finally presents several extensions of the methodology.

翻译：本文研究了独立但不一定同分布的非负随机变量和与最大值的联合分布，涵盖两种情况：i) 连续随机变量；ii) 离散随机变量。首先，针对两种情况分别推导了联合累积分布函数（CDF）的递归公式。随后，针对连续情形给出了联合概率密度函数（PDF）的递推关系，针对离散情形给出了联合概率质量函数（PMF）的递推关系。值得注意的是，联合PDF与PMF之间存在本质性差异。证明过程简洁，主要基于微积分与离散数学中的以下工具：积分号下求导（即莱布尼茨积分法则）、全概率公式及数学归纳法。此外，本研究推广了文献中的已有结果，并最终提出了该方法的若干扩展形式。