Theoretical analysis reveals exact modes and antimodes for weighted sums of discrete uniform random variables, highlighting lattice point geometric properties.
This paper studies the weighted sum Z=aX+bY, where X and Y are independent discrete uniform random variables. The modes and antimodes of Z are determined by counting lattice points. When N=b, the antimodes are exactly the values with a unique representation. Their number is also obtained and eventually becomes 2ab.
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Po Hu (2026) studied this question.
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