\begin{aligned} &E(Y)=E\left[\sum_{i=1}^{n}\left(X_{i}+X_{n+i}-2 \bar{X}\right)^{2}\right]=E\left\{\sum_{i=1}^{n}\left[\left(X_{i}-\overline{X_{1}}\right)+\left(X_{n+i}-\overline{X_{2}}\right)\right]^{2}\right\} \\ &=E\left\{\sum_{i=1}^{n}\left[\left(X_{i}-\overline{X_{1}}\right)^{2}+2\left(X_{i}-\overline{X_{1}}\right)\left(X_{n+i}-\overline{X_{2}}\right)+\left(X_{n+i}-\overline{X_{2}}\right)^{2}\right]\right\} \\ &=E\left[\sum_{i=1}^{n}\left(X_{i}-\overline{X_{1}}\right)^{2}\right]+E\left\{\sum_{i=1}^{n}\left[2\left(X_{i}-\overline{X_{1}}\right)\left(X_{n+i}-\overline{X_{2}}\right)\right]\right\}+E\left[\sum_{i=1}^{n}\left(X_{n+i}-\overline{X_{2}}\right)^{2}\right] \end{aligned}

$$E S^{2}=E\left[\frac{1}{n-1} \sum_{i=1}^{n}\left(X_{i}-\overline{X_{1}}\right)^{2}\right]=\sigma^{2}$$

\begin{aligned} &=2 \sum_{i=1}^{n} E\left[\left(X_{i}-\overline{X_{1}}\right)\left(X_{n+i}-\overline{X_{2}}\right)\right]=\sum_{i=1}^{n} E\left(X_{i} X_{n+i}-X_{i} \overline{X_{2}}-\overline{X_{1}} X_{n+i}+\overline{X_{1}} \overline{X_{2}}\right) \\ &=\sum_{i=1}^{n}\left(E X_{i} X_{n+i}-E X_{i} \overline{X_{2}}-E \overline{X_{1}} X_{n+i}+E \overline{X_{1}} \overline{X_{2}}\right) \end{aligned}

\begin{aligned} &E X_{i} X_{n+i}=E X_{i} E X_{n+i}=u^{2}, E X_{i} \overline{X_{2}}=E X_{i} E \overline{X_{2}}=u^{2} \\ &E \overline{X_{1}} X_{n+i}=E \overline{X_{1}} E X_{n+i}=u^{2}, E \overline{X_{1}} \overline{X_{2}}=E \overline{X_{1}} E \overline{X_{2}}=u^{2} \end{aligned}

$$=\sum_{i=1}^{n}\left(E X_{i} X_{n+i}-E X_{i} \overline{X_{2}}-E \overline{X_{1}} X_{n+i}+E \overline{X_{1}} \overline{X_{2}}\right)=\sum_{i=1}^{n}\left(u^{2}-u^{2}-u^{2}+u^{2}\right)=0$$

\begin{aligned} E(Y) &=E\left[\sum_{i=1}^{n}\left(X_{i}-\overline{X_{1}}\right)^{2}\right]+E\left\{\sum_{i=1}^{n}\left[2\left(X_{i}-\overline{X_{1}}\right)\left(X_{n+i}-\overline{X_{2}}\right)\right]\right\}+E\left[\sum_{i=1}^{n}\left(X_{n+i}-\overline{X_{2}}\right)^{2}\right] \\ &=(n-1) \sigma^{2}+(n-1) \sigma^{2}=2(n-1) \sigma^{2} \end{aligned}