Question

Suppose $X$ follows binomial distribution with parameters $n=100$ and $p=\frac{1}{3}$ then $P(x=r)$ is maximum when $r=$

• A. $49$
• B. $50$
• C. $12$
• D. $33$

Answer 1

The correct option is D $33$

P(X=r) will be maximum when r is the mode. There are two cases for mode of binomial distribution:
Case 1: If (n+1)/p is an integer the Binomial distribution is bi modal 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 exists unique modal value and it's the integral part of (n+1)/p.
The Given Binomial distribution has parameters, n= 100 and p= 1/3.
We have (n+1)/p = 101/3 = 33.67, which is not an integer.
Hence the unique mode is 33, the integral part of (n+1)/p.