Could somebody please quickly clear up a confusion of mine regarding the following?
Let there be a random sample from the Poisson distribution with parameter $\lambda$ as well as an an unbiased estimator of $\lambda$, $\tilde{\lambda}=S^2$. How is $\tilde{\lambda}$ unbiased?
More specifically, knowing that $\lambda=\mathbb{E}X=var X$, how is the expectation of the variance equivalent to the variance?
Thanks a bunch and apologies in advance if this seems trivial.
Not exactly. The expectation of the sample variance is equal to the population variance
$$\mathbb{E}[S^2]=\mathbb{E}\left[\frac{1}{n-1} \Sigma_i(X_i-\overline{X}_n)^2 \right]=\frac{1}{n-1}\mathbb{E}\left[ \Sigma_i(X_i-\mu)^2-n(\overline{X}_n-\mu)^2 \right]=$$
$$=\frac{1}{n-1}\left[ \Sigma_i\mathbb{E}(X_i-\mu)^2-n\mathbb{E}(\overline{X}_n-\mu)^2 \right]=\frac{1}{n-1}\left[n\sigma^2-n\mathbb{V}[\overline{X}_n]\right]=$$
$$=\frac{1}{n-1}\left[n\sigma^2-n\frac{\sigma^2}{n}\right]=\sigma^2=\lambda$$