If $X_1,X_2$ constitute a random sample of size n=2 from a Poisson Population show that the mean of the sample is a sufficient estimator of the parameter $\lambda$ .

Since the sum of Poissons is also Poisson with parameter $\lambda+\lambda=2\lambda$
Then $E[Y]=2\lambda$. where Y constitute of two Poisson distributions each with parameter $\lambda$.

Do I have to show that $2\lambda$ is sufficient for $\lambda$.
I don't understand what the estimator is here?


Probably you mean that the sample average is sufficient statistics for $\lambda$. So, if $X_1, X_2$ are independent draws from Poisson distribution with parameter $\lambda$ then the answer is "yes".

Check here: http://en.wikipedia.org/wiki/Sufficient_statistic

Using Fisher–Neyman factorization theorem:


where $h(x)=(x_1!x_2!)^{-1}$ and $g_\lambda(T(x))=\lambda^{(x_1+x_2)/2}e^{2-2\lambda}$


You're not showing sufficiency at all. The only thing your calculation shows is that $\bar X = (X_1 + X_2)/2$ is an unbiased estimator of $\lambda$.

To show that $\bar X$ is a sufficient statistic, you have to do something else. The most straightforward approach is to use the Factorization Theorem, which seeks the expression of the joint distribution for $X_1, X_2$ as $$f_{X_1, X_2}(x_1, x_2) = g(T(x_1, x_2) \mid \lambda) h(x_1, x_2),$$ for some function $T$ that depends only on the sample; and in your case, $T(x_1, x_2) = \bar x = (x_1 + x_2)/2$.

  • $\begingroup$ I didn't knew that the estimator that I should show that is sufficient is $(x_1+x_2)/2$. $\endgroup$ – clarkson Apr 9 '14 at 5:41

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