I am in a new class and I do not understand my instructor's proofs. I have a question about a specific part of the proof. The expression is this: $$E\left[\sum_{i=0}^n Y_i\overline{Y_n}\right]$$ For random variables that are I.I.D. I concluded that you could treat $\overline{Y_n}$ like a constant. It is defined in the question as $\dfrac{1}{n}\left(\displaystyle\sum_{i=0}^n Y_i\right)$. It is also given that $E[Y_i]=\mu$.
So then I got: $$=\overline{Y_n}E\left[\sum_{i=0}^n Y_i\right]$$ $$=\mu nE[Y_i]$$ Finally, $E\left[\displaystyle\sum_{i=0}^n Y_i\overline{Y_n}\right] = n\mu^2$.
My instructor got a final answer of $\sigma^2 + n\mu^2$ with the following justification:
What am I missing? Thank you so much!