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Recently, Glasby and Paseman considered the following sequence of binomialsums $\{2^{-r}\sum_{i=0}^{r}\binom{m}{i}\}_{r=0}^{m}$ and showed that thissequence is unimodal and attains its maximum value at$r=\lfloor\frac{m}{3}\rfloor+1$ for$m\in\mathbb{Z}_{\geq0}\setminus\{0,3,6,9,12\}$. They also analyzed theasymptotic behavior of the maximum value of the sequence as $m$ approachesinfinity. In the present work, we generalize their results by considering thesequence $\{(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}\}_{r=0}^{m}$ for integers$a \geq 1$. We also consider a family of discrete probability distributionsthat naturally arises from this sequence.

On the maximum of the weighted binomial sum $(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}$