If a random variable X follows Poisson distribution such that $3P(X=2)=2P(X=1)$, then the variance of X is :

$\frac{4}{3}$

The correct answer is Option (1) → $\frac{4}{3}$

A Poisson-distributed random variable X is,

$P(X=k)=\frac{e^{-λ}λ^k}{k!}$  [λ = Variance]

and,

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

$3×\frac{e^{-λ}λ^2}{2!}=2×\frac{e^{-λ}λ^1}{1!}$

$⇒\frac{3λ^2}{2}=2λ$

$⇒λ(3λ-4)=0$

$⇒λ=\frac{4}{3}$