If a random variable X follows Poisson distribution such that \(3P(X=2)=2P(X=1)\), then the variance of X is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{4}{3}\)
For a Poisson distribution with parameter λ:
\( \displaystyle P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!} \)
The given condition becomes:
\( \displaystyle 3 \times \frac{e^{-\lambda}\lambda^2}{2} = 2 \times e^{-\lambda}\lambda \)
Dividing both sides by \(e^{-\lambda}\lambda\) (which is not zero):
\( \displaystyle \frac{3\lambda}{2} = 2 \)
\( \displaystyle \lambda = \frac{4}{3} \)
In a Poisson distribution, variance = mean = λ, so the variance is \(\frac{4}{3}\).