If X is a binomial variate with the range {0,1,2,3,4,5,6} and P(X=2) = 4P(X=4), then the parameter p of X is 
 

Given n=6

and $\mathrm{p}(\mathrm{x}=2)=4\mathrm{p}(\mathrm{x}=4)$

$$\begin{gathered} \Rightarrow {}^6{\mathrm{c}}_2 {\mathrm{p}}^2 {\mathrm{q}}^4=4\left({}^6{\mathrm{c}}_4 {\mathrm{p}}^4 {\mathrm{q}}^2\right) \\ \Rightarrow {\mathrm{q}}^2=4{\mathrm{p}}^2 \left(\because {}^6{\mathrm{c}}_4={}^6{\mathrm{c}}_2\right) \\ \Rightarrow {(1-\mathrm{p})}^2=4{\mathrm{p}}^2 \\ \Rightarrow 3{\mathrm{p}}^2+2\mathrm{p}-1=0 \\ \Rightarrow (\mathrm{p}+1) (3\mathrm{p}-1)=0 \end{gathered}$$

$\Rightarrow \mathrm{p}=\frac{1}{3}$ ( $\because 0\le p\le 1$ )