Suppose that $X_1$ and $X_2$ are two random variables sampled from a Poisson distribution with parameter $\mu$. Let $T_1=\bar{X}$ be the sample mean and let $T_2=(1/3)X_1 +(2/3)X_2$.

Are T1 and T2 both unbiased estimators?

  • $\begingroup$ This looks remarkably like homework. $\endgroup$ – Nick Sabbe Mar 15 '13 at 7:53
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    $\begingroup$ @NickSabbe Homework is okay on CrossValidated if the self-study tag (or the equivalent, homework, which now maps to it) is used. If the OP adds such a tag, and follows the guidelines in the tag wiki and the FAQ relating to homework, then it's explicitly okay. $\endgroup$ – Glen_b -Reinstate Monica Mar 15 '13 at 9:50
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    $\begingroup$ The question should appear in the text. And the answer is yes as the OP should realise by computing the expectation and using the linearity of this operator. $\endgroup$ – Xi'an Mar 15 '13 at 10:57
  • $\begingroup$ @Xi'an Fair comment. I changed it and made a few minor corrections. Maretha, please check the edits leave your question asking the right question, and please, if this is for self-study or in any way related to some subject, add the self-study tag. $\endgroup$ – Glen_b -Reinstate Monica Mar 15 '13 at 12:42

$E(T_1) = E[\frac{1}{2}(X_1+X_2)] = \frac{1}{2}E[X_1+X_2] = \frac{1}{2}(E[X_1]+E[X_2]) = \frac{1}{2}(\mu+\mu)=\mu$

$E(T_2) = E[\frac{1}{3}X_1+\frac{2}{3}X_2] = \frac{1}{3}E[X_1]+\frac{2}{3}E[X_2] = \frac{1}{3}\mu+\frac{2}{3}\mu=\mu$

Consequently, both $T_1$ and $T_2$ are unbiased estimators of $\mu$.


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