A random variable X follows a Poisson distribution with \(P(X = 0) = 0.0498\). The variance of X is (Given \(e^{-3} = 0.0498\))
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3
For a Poisson distribution with mean m,
\( \displaystyle P(X = 0) = \frac{e^{-m}\,m^{0}}{0!} = e^{-m} \)
So
\(e^{-m} = 0.0498 = e^{-3}\)
\(m = 3\)
In a Poisson distribution the variance equals the mean, so the variance of X is 3.
Why the other options are incorrect
Option (1) 9 — Incorrect. 9 is the square of the mean; the variance equals the mean itself.
Option (3) \(\sqrt{3}\) — Incorrect. \(\sqrt{3}\) is the standard deviation, not the variance.
Option (4) \(\displaystyle\frac{1}{3}\) — Incorrect. The variance equals the mean 3, not its reciprocal.