Here is the proof of biased and unbiased sample variance. Since they are random sample they are i.i.d.
From the process of
$$\sum^n_{i=1}\mathbb{E}(Y^2_i)-n\mathbb{E}(\bar{Y}^2)$$
I was thinking about
$$\mathbb{E}(Y_i^2)=\mathbb{E}(Y_i)\mathbb{E}(Y_i)$$,similarly for $\mathbb{E}(\bar{Y}^2)$, why this is not the case?
Thank you for any comments
Book： Mathematical Statistics with Applications, Seventh Edition. Dennis D. Wackerly, William Mendenhall III, Richard L. Scheaffer.