Question

A random variable $X$ follows Binomial distribution with mean $\alpha$ and variance $\beta$. Then,

• A. $0<\alpha <\beta$
• B. $0<\beta <\alpha$
• C. $\alpha <0<\beta$
• D. $\beta <0<\alpha$

Answer 1

The correct option is B

$0<\beta <\alpha$

Explanation for the correct option:

Step 1: Find the ratio of mean and variance:

Write the equation of mean and variance.

mean, $\alpha =np$

Variance, $\beta =npq$

Find the ratio, $\frac{\beta }{\alpha }=q$

Step 2: Find the relation of $\alpha$ and $\beta$.

Use the value of $q$ to find the relation.

$$\begin{gathered} \because 0<q<1 \\ \therefore \beta <\alpha \end{gathered}$$

Also, note that $n,p$ and $q$ are positive values.

So both mean and variance need to be positive.

$\therefore 0<\beta <\alpha$

Hence, option (B) is the correct answer.