Finding $E\Bigl(\overline{Y^2}\Bigm|\overline{Y\vphantom{Y^2}}\Bigr)$ by Basu's theorem? Suppose $Y_1,\ldots,Y_n$ are a random sample of normal distribution $\mathcal{N}(\mu,1)$. If $\overline{Y^2}=\displaystyle\frac{1}{n}\sum_{i=1}^n Y_i^2$, how can I find $E\Bigl(\overline{Y^2}\Bigm|\overline{Y\vphantom{Y^2}}\Bigr)$ by Basu's theorem?

By Basu's theorem $\frac{1}{n-1}\sum_{i=1}^n (Y_i- \bar{Y})^2$ is independent of $\bar{Y}$. Hence
$$ \mathbb{E} \left(\frac{1}{n-1}\sum_{i=1}^n (Y_i- \bar{Y})^2 \mid \bar{Y} \right) = \mathbb{E} \left(\frac{1}{n-1}\sum_{i=1}^n (Y_i- \bar{Y})^2 \right)$$
Develop both sides. You expression will appear on the left-hand side.
