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 $\frac{64}{162}{e}^{\frac{-4}{3}}$

Poisson probability formula :

$P(x,\mu )=\frac{\left(e^{-\mu }\right)\left({\mu }^x\right)}{x!}$

$x\to$ actual number of occuring sucesses

$\mu \to$ average of the number of success with in

a specified region

$P(x=2)=\frac{e^{-\mu }{\mu }^2}{2!}$

$P(x=1)=\frac{e^{-\mu }\mu }{11}$

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

$\frac{e^{-\mu }{\mu }^2}{2}=\frac{2}{3}\times \frac{e^{-\mu }\mu }{1}$

$\mu =\frac{4}{3}$ ____ (1)

Now, $P(x=3)=\frac{e^{-\mu }{\mu }^3}{3!}$

$=\frac{e^{-\mu }{\mu }^3}{6}$

substituting the value of $\mu$ from (1)

$=\frac{e^{-4/3}}{6}{\left(\frac{4}{3}\right)}^3$

$=\frac{e^{-4/3}}{6}\times \frac{64}{27}$

$P(x=3)=\frac{64e^{-4/3}}{162}$