Question

If $X$ is a random 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 $\frac{\alpha }{3}$

In Poisson distribution such that $\alpha =p(X=1)=p(X=2)$
$p(x;\mu )=\frac{e^{-\mu }{\mu }^x}{x!}$
=> $\alpha =\frac{e^{-\mu }{\mu }^1}{1!}=\frac{e^{-\mu }{\mu }^2}{2!}$ => $\alpha =e^{-\mu }\mu$
=> $e^{-\mu }\mu =\frac{e^{-\mu }{\mu }^2}{2}$
=> $\mu =2$
$p(4;\mu )=\frac{e^{-\mu }{\mu }^4}{4!}=\frac{e^{-\mu }\mu \times {\mu }^3}{24}=\frac{\alpha \times 2^3}{24}=\frac{\alpha }{3}$