If X is a Poisson variable such that $P(X=1)=2P(X=2),$ then $P(X=3)$ is :

$\frac{1}{6e}$

The correct answer is Option (4) → $\frac{1}{6e}$

In a Poisson Distribution, the PMF is,

$P(X=k)=\frac{e^{-λ}λ^k}{k!}$

and,

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

$⇒\frac{e^{-λ}λ^1}{1!}=2×\frac{e^{-λ}λ^2}{2!}$

$⇒λ=λ^2$

$⇒λ^2-λ=0$

$⇒λ(λ-1)=0⇒λ=1$

$∴P(X=3)=\frac{e^{-1}×1}{6}=\frac{1}{6e}$