If a random variable X follows Poisson distribution such that $3P(X=1) =P(X+2),$ then $P(X=4)$ is ,

$54e^{-6}$

The correct answer is Option (3) → $54e^{-6}$

Random variable X follows Poisson distributions,

$P(X=k)=\frac{e^{-λ}λ^k}{k!}$

$3P(X=1)=P(X=2)$

$3λe^{-λ}=\frac{λ^2e^{-λ}}{2!}$

$⇒λ(λ-6)=0⇒λ=6$

$∴P(X=4)=\frac{e^{-6}6^4}{4!}=54e^{-6}$