Question

If $X$ is a poisson variate such that $\alpha =P(X=1)=P(X=2)$, then $P(X=4)$ is equal to

• A. $2\alpha$
• B. $\frac{\alpha }{3}$
• C. $\alpha e^{-2}$
• D. $\alpha e^2$

Answer 1

Answer: (B)

Given $X$ is a poisson variate such that
$\alpha =P(X=1)=P(X=2)$
$\therefore \frac{e^{-\lambda }\lambda }{1!}=\frac{e^{-\lambda }{\lambda }^2}{2!}$
$\Rightarrow \lambda =2$
$\therefore \alpha =P(X=1)$
$=e^{-2}\times 2=\frac{2}{e^2}$ .....(i)
$\therefore P(X=4)=\frac{e^{-\lambda }{\lambda }^4}{4!}=\frac{e^{-2}(2{)}^4}{24}$
$=\frac{e^{-2}\times 16}{24}=\frac{2}{3}{e}^{-2}=\frac{1}{3}\alpha$ [from Eq. (i)]