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)]