If X has a Poisson distribution such that $P(X=1)=P(X=2)$ then $P(X=3)$ is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $\frac{4}{3}e^{-2}$
for a Poisson distribution,
$P(X=k)=\frac{e^{-λ}λ^k}{k!},k=0,1,2$
$P(X=1)=P(X=2)$
$\frac{e^{-λ}λ^1}{1!}=\frac{e^{-λ}λ^2}{2!}$
$λ=\frac{λ^2}{2}⇒λ=2$
$∴P(X=3)=\frac{e^{-2}.2^3}{3!}=\frac{4}{3}e^{-2}$