设 $\left(X_i, Y_i\right), i=1,2, \cdots, n$ 是来自二维正态分布 $N\left(\mu_1, \mu_2\right.$ ， $\left.\sigma_1^2, \sigma_2^2, \rho\right)$ 的样本．
又设

$$
\bar{X}=\frac{1}{n} \sum_{i=1}^n X_i, \bar{Y}=\frac{1}{n} \sum_{i=1}^n Y_i
$$
$$
\begin{gathered}
S_x^2=\frac{1}{n} \sum_{i=1}^n\left(X_i-\bar{X}\right)^2, S_y^2=\frac{1}{n} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)^2 \\
r=\frac{\sum_{i=1}^n\left(X_i-\bar{X}\right)\left(Y_i-\bar{Y}\right)}{\sqrt{\sum_{i=1}^n\left(X_i-\bar{X}\right)^2} \cdot \sqrt{\sum_{i=1}^n\left(Y_i-\bar{Y}\right)^2}}
\end{gathered}
$$


求

$$
\frac{\bar{X}-\bar{Y}-\left(\mu_1-\mu_2\right)}{\sqrt{S_x^2+S_y^2-2 r S_x S_y}} \sqrt{n-1}
$$


的分布．