Question

For a Binomial variate $X$ with $n=6$, if$P(X=2)=9P(X=4)$, then its variance is

• A. $\frac{8}{9}$
• B. $\frac{1}{4}$
• C. $\frac{9}{8}$
• D. $4$

Answer 1

The correct option is C

$\frac{9}{8}$

Finding the variance:

Given, $P(X=2)=9P(X=4)$

Therefore,

$\begin{array}{rcl}{}^{6}C_2{p}^2{(1-p)}^4& =& 9{}^{6}C_4{p}^4{(1-p)}^2\\ 9p^2& =& {(1-p)}^{2}3p=(1-p)\\ p& =& \frac{1}{4}\end{array}$

And $q=\frac{3}{4}$

Now putting the value in the formula of variance.

We know that, Variance$\mathbf{=}\mathbf{}\mathit{n}\mathit{p}\mathit{q}$

Variance$=6\times \frac{1}{4}\times \frac{3}{4}$

Variance$=\frac{18}{36}$

Variance$=\frac{9}{8}$

Hence, the correct option is $\mathbf{\left(}\mathit{C}\mathbf{\right)}$