If two independent random variables X and Y are binomially... | Agrigyan

Mode of sum of independent binomial variates:

• $3, 2$ would imply bimodal distribution, which occurs only if $(n+1)p$ is an integer.
• By the additive property of the binomial distribution, since $p$ is identical, $Z = X + Y \sim \operatorname{Binomial}(n_1 + n_2, p) = \operatorname{Binomial}(4 + 5, 1/3) = \operatorname{Binomial}(9, 1/3)$. The mode is given by $\lfloor (n + 1)p \rfloor$. Here $(9 + 1) \times (1/3) = 10/3 = 3.333$. Since $10/3$ is non-integral, there is a unique mode equal to the integer part: $\lfloor 3.333 \rfloor = 3$.
• $3, 4$ is incorrect.
• $4$ is the ceiling, not the mode.