Discussion
A random sample X1, X2, ....., Xn is drawn from a Poisson distribution with parameter \(\lambda\). Which one of the following is an unbiased estimator of \(\lambda^{2}\) ?
Answer: C
The unbiased estimator is (1/n) Σ Xi(Xi − 1).
For a Poisson variable, the second factorial moment is
\[ \begin{aligned} E[X(X - 1)] &= E(X^{2}) - E(X) \\ &= (\lambda + \lambda^{2}) - \lambda \\ &= \lambda^{2} \end{aligned} \]
so the average of \(X_{i}(X_{i} - 1)\) has expectation \(\lambda^{2}\).
The others are biased. \(E(\bar{X}^{2}) = \lambda^{2} + \lambda/n\), \(E(X_{i}^{2}) = \lambda^{2} + \lambda\), and \(E[X_{i}(X_{i} + 1)] = \lambda^{2} + 2\lambda\).
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