设 $\left(X_1, Y_1\right),\left(X_2, Y_2\right), \cdots,\left(X_n, Y_n\right)$ 为来自总体 $N\left(\mu_1, \mu_2\right.$; $\left.\sigma_1^2, \sigma_2^2 ; \rho\right)$ 简单随机样本，令 $\theta=\mu_1-\mu_2, \bar{X}=\frac{1}{n} \sum_{i=1}^n X_i$, $\overline{\boldsymbol{Y}}=\frac{1}{n} \sum_{i=1}^n Y_i, \hat{\boldsymbol{\theta}}=\overline{\boldsymbol{X}}-\overline{\boldsymbol{Y}}$ ，则 $(\quad)$
A
$E(\hat{\theta})=\theta, D(\hat{\theta})=\frac{\sigma_1^2+\sigma_2^2}{n}$
B
$E(\hat{\theta})=\theta, D(\hat{\theta})=\frac{\sigma_1^2+\sigma_2^2-2 \rho \sigma_1 \sigma_2}{n}$
C
$E(\hat{\theta}) \neq \theta, D(\hat{\theta})=\frac{\sigma_1^2+\sigma_2^2}{n}$
D
$E(\hat{\theta}) \neq \theta, \quad D(\hat{\theta})=\frac{\sigma_1^2+\sigma_2^2-2 \rho \sigma_1 \sigma_2}{n}$
E
F