  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.
