Question

If $X$ is a poisson variable such that $P(X=2)=\frac{2}{3}P(X=1)$, then $P(x=3)$ is

• A. $e^{\frac{-4}{3}}$
• B. $\frac{64}{162}{e}^{\frac{-4}{3}}$
• C. $e^{\frac{-3}{4}}$
• D. $e^{\frac{3}{4}}$

Answer 1

The correct option is B $\frac{64}{162}{e}^{\frac{-4}{3}}$

$P(X=2)=\frac{2}{3}P(X=1)$

$P$$(2;\mu )=\frac{e^{-\mu }{\mu }^2}{2!}=\frac{2}{3}P(1;\mu )$=$\frac{2}{3}\frac{e^{-\mu }{\mu }^1}{1!}$

$e^{-\mu }\mu (\frac{\mu }{2}-\frac{2}{3})=0$

either $\mu =0or\mu =\frac{4}{3}$

but here $\mu =\frac{4}{3}$$P(X=3)=$ $\frac{e^{-\mu }{\mu }^3}{3!}=\frac{e^{\frac{-4}{3}}(\frac{4}{3}{)}^3}{6}=\frac{64}{162}{e}^{\frac{-4}{3}}$