Question

The mode of the Binomial distribution for which mean and standard deviation are $10$ and $\sqrt{5}$ respectively, is

• A. $7$
• B. $8$
• C. $9$
• D. $10$

Answer 1

The correct option is D

$10$

Explanation for the correct option:

Find the mode of the given binomial distribution

In the question, a Binomial distribution for which mean and standard deviation are $10$ and $\sqrt{5}$ respectively is given.

According to the binomial distribution, the mean can be given by $np$. Where $n$ is the number of trials and $p$ is the probability of success.

And the variance can be given by $npq$ where $q$ is the probability of failure.

We know that variance is equal to the square of standard deviation.

Therefore, variance is $5$.

So, $np=10...\left(1\right)$

and $npq=5...\left(2\right)$

From equation $\left(1\right)$ and $\left(2\right)$, we get,

$$\begin{gathered} \Rightarrow \frac{npq}{np}=\frac{5}{10} \\ \Rightarrow q=\frac{1}{2} \end{gathered}$$.

Assume that, $m$ be the mode of the given binomial distribution.

So,

$$\begin{gathered} np-q<m<np+q \\ \Rightarrow 10-\frac{1}{2}<m<10+\frac{1}{2} \\ \Rightarrow 9.5<m<10.5 \end{gathered}$$

Therefore, the mode of the Binomial distribution for which mean and standard deviation are $10$ and $\sqrt{5}$ respectively, is $10$.

Hence, correct option is $D$,