On the maximum of the weighted binomial sum $(1+a{)}^{-r}{\sum }_{i=0}^r\left(\genfrac{}{}{0px}{}{m}{i}\right)a^i$

Recently, Glasby and Paseman considered the following sequence of binomial sums $\{2^{-r}{\sum }_{i=0}^r\left(\genfrac{}{}{0px}{}{m}{i}\right){\}}_{r=0}^m$ and showed that this sequence is unimodal and attains its maximum value at $r=\lfloor \frac{m}{3}\rfloor +1$ for $m\in {\mathbb{Z}}_{\ge 0}\setminus \{0,3,6,9,12\}$. They also analyzed the asymptotic behavior of the maximum value of the sequence as $m$ approaches infinity. In the present work, we generalize their results by considering the sequence $\{(1+a{)}^{-r}{\sum }_{i=0}^r\left(\genfrac{}{}{0px}{}{m}{i}\right)a^i{\}}_{r=0}^m$ for integers $a\ge 1$. We also consider a family of discrete probability distributions that naturally arises from this sequence.