Unbiased estimator from a random sample

Question

Problem

I really don't understand the answer to the below question. It's only 2 marks, so it should be reasonably simple.

Attempt

I have expanded out the brackets and got as far as $E(Y) = E(X^2) - E(X)^2$ but it feels very lengthy.

Question

Answer 1

Notice that $$\begin{eqnarray*}S^2&=&\frac{1}{7}\sum_{k=1}^8(X_k-\overline{X})^2\\&=&\frac{1}{7}\left[\sum_{k=1}^8X_k^2-2\overline{X}\sum_{k=1}^8X_k+8\overline{X}^2\right] \\ &=& \frac{1}{7}\left[\sum_{k=1}^8X_k^2-2\overline{X}\cdot 8\overline{X}+8\overline{X}^2\right]\\&=&\frac{1}{7}\left[\sum_{k=1}^8X_k^2-8\overline{X}^2\right] \\ &=& \frac{8}{7}Y \end{eqnarray*}$$ So, $$\sigma^2=\mathbb{E}(S^2)=\frac{8}{7}\mathbb{E}(Y) \implies \mathbb{E}(Y)={7\sigma^2 \over 8} $$