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 $10$

In a binomial distribution, mean, $np=10$ and variance, $npq=5$

[Since, variance=square of standard deviation]

$\therefore 10\times q=5\Rightarrow q=\frac{1}{2}$
$\Rightarrow p=1-q=1-\frac{1}{2}=\frac{1}{2}$
Let $x$ be the mode. Then,
$np+q>x>np-q$
$\Rightarrow 10+\frac{1}{2}>x>10-\frac{1}{2}$
$\Rightarrow \frac{21}{2}>x>\frac{19}{2}\Rightarrow 9.5<x<10.5$
Hence, option (d) is correct.