设随机变量 $X_{1}, X_{2}, \cdots, X_{n}(n>1)$ 独立同分布, 且其方差为 $\sigma^{2}>0$. 令 $Y=\frac{1}{n} \sum_{i=1}^{n} X_{i}$, 则 ( )
$\text{A.}$ $\operatorname{Cov}\left(X_{1}, Y\right)=\frac{\sigma^{2}}{n}$.
$\text{B.}$ $\operatorname{Cov}\left(X_{1}, Y\right)=\sigma^{2}$.
$\text{C.}$ $D\left(X_{1}+Y\right)=\frac{n+2}{n} \sigma^{2}$.
$\text{D.}$ $D\left(X_{1}-Y\right)=\frac{n+1}{n} \sigma^{2}$.