Statistics

    

Exact Global Modes of Two-Term Weighted Sums of I.i.d. Geometric Random Variables

Authors: Po Hu

Let X and Y be independent geometric random variables on Z≥0 with P(X = k) = p(1 − p)k. For positive integers a, b, we determine the global modes of Z = aX + bY . The probability mass function may have gaps and local rises and falls. We prove that every global mode is a multiple of lcm (a, b) and that there are at most two global modes. A logarithmic threshold gives their exact locations and determines whether one or two modes occur. We also give an exact formula for every probability and a small-p asymptotic formula for the mode. The proof parametrizes the solutions of ax + by = n by residue classes and then compares the maxima of consecutive blocks.

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[v1] 2026-08-17 22:34:33

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