How do I prove that the expectation of sample variance is equal to the population variance for any distribution?

Question

In other words, is $S^2$ unbiased for $\sigma^2$ for any distribution? I know how to prove this for the normal distribution, but is it possible for me to prove this generally?

Thank you

Answer 1

I don't know how you managed to prove it via properties of normal RVs, but you can follow the steps for a general derivation here. Note that Bessel corrected formula is the unbiased estimator. Of course, for the distributions we're referring to, the moments should be defined.