设随机变量 $X_{1}, X_{2}, \dots, X_{n} (n > 1)$ 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( ) $\left(\mathrm{A}\right)\operatorname {Cov}\left(X_{1},Y\right) = \frac{\sigma^{2}}{n}.$ (B) $\operatorname{Cov}(X_1, Y) = \sigma^2$ . (C) $D(X_{1} + Y) = \frac{n + 2}{n}\sigma^{2}.$ $\left(\mathrm{D}\right) D\left(X_{1} - Y\right) = \frac{n + 1}{n} \sigma^{2}$ . # 三、解答题(本题共9小题，满分94分．解答应写出文字说明、证明过程或演算步骤)