X follows a Poisson distribution with mean 3. Then \(E(X^{2})\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 12
For a Poisson distribution, the variance equals the mean, so \(\mathrm{Var}(X) = 3\).
Since \(\mathrm{Var}(X) = E(X^{2}) - [E(X)]^{2}\),
\(E(X^{2}) = \mathrm{Var}(X) + [E(X)]^{2}\)
\(= 3 + 3^{2} = 12\)
Why the other options are incorrect
Option (2) 9 — Incorrect. This is \([E(X)]^{2}\) only; the variance 3 has to be added.
Option (3) 3 — Incorrect. This is the mean (and the variance), not \(E(X^{2})\).
Option (4) 6 — Incorrect. This is mean + variance; the mean has to be squared before adding.