Question

If X follows a binomial distribution with parameters $n=100$ and $p=1/3$, find r for which $P(X=r)$ is maximum

• A. $33$
• B. $22$
• C. $11$
• D. $15$

Answer 1

The correct option is A $33$

$P(X=r)$ will be maximum when r is the mode. There are $2$ cases for mode of binomial distribution.

Case $1$: If $(n+1)p$ is an integer the binomial distribution is bimodal and the two modal values are $(n+1)p$ and $(n+1)p-1$

Case $2$: If $(n+1)p$ is not an integer there exist unique modal value and its the integral part of $(n+1)p$

The given binomial distribution has parameters $n=100,p=\frac{1}{3}$

We have $(n+1)p=\frac{101}{3}=33.67$

Hence unique mode is $33$ the integral part of $(n+1)p$