Question

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

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

Answer 1

The correct option is B $\frac{\alpha }{3}$

$P(X=K)=\frac{{\lambda }^K{e}^{-\lambda }}{K!}$

$P(X=1)=P(X=2)$

$\Rightarrow \frac{\lambda e^{-\lambda }}{1!}=\frac{{\lambda }^2{e}^{-\lambda }}{2!}$

$\Rightarrow \lambda =2$ So, $\alpha =2e^{-2}$

Now,

$P(X=4)=\frac{{\lambda }^4{e}^{-\lambda }}{4!}$

$=\frac{2^4{e}^{-2}}{4!}$

$=\frac{\alpha }{3}$