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

Recurrence relations for the joint distribution of the sum and maximum of independent random variables