If a random variable X follows Poisson distribution such that $3P(X=1) =P(X+2),$ then $P(X=4)$ is ,
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $54e^{-6}$
Random variable X follows Poisson distributions,
$P(X=k)=\frac{e^{-λ}λ^k}{k!}$
$3P(X=1)=P(X=2)$
$3λe^{-λ}=\frac{λ^2e^{-λ}}{2!}$
$⇒λ(λ-6)=0⇒λ=6$
$∴P(X=4)=\frac{e^{-6}6^4}{4!}=54e^{-6}$