Let X be Poisson Distributed with expectation λt. Show that the estimator is unbiased and has a variance that goes towards 0 when the times goes to infinity.
λ^=X/t
I know that for these questions there is a good thing to have some kind of expectation, of how to progress with the question in relevance. However, I don't, I just don't understand at all. I have had 0 similar examples, and no tuition either.
E(X) is λt, and that's everything.
We have $E(X/t)=\lambda t/t=\lambda$, unbiased for $\lambda$. Now, for a Poisson r.v. $E(X)=Var(X)$. So $Var(X/t)=(1/t^2)Var(X)=(1/t^2)E(X)=\lambda/t$, goes to 0 when $t$ goes to infinity.