Question

The maximum value of the variance of Binomial distribution with parameters \(n\) and \(p\) is:

• A. \(\frac{1}{2}\)
• B. \(\frac{n}{4}\)
• C. \(np(1-p)\)
• D. \(2n\)

Hint:

For a Binomial distribution, \[ \mathrm{Var}(X)=np(1-p). \] The expression \(p(1-p)\) attains its maximum value \(\frac14\) at \(p=\frac12\).

Answer 1

The Correct Option is B

Step 1: Recall variance of Binomial distribution.
For \(X\sim B(n,p)\), \(\text{Var}(X)=np(1-p)\).

Step 2: Maximise p(1-p) over p in [0,1].
\(p(1-p)\) is a downward parabola maximised at \(p=\frac{1}{2}\), giving maximum value \(\frac{1}{4}\). So maximum variance \(=\dfrac{n}{4}\). \[ \boxed{\dfrac{n}{4}} \]