# 概率论 Cheat Sheet 19：随机变量函数的联合分布

设 $X_1, X_2$ 是联合连续的随机变量，具有联合密度函数 $F_{X_1, X_2}$，$Y_1, Y_2$ 为 $X_1, X_2$ 的函数，要计算 $Y_1, Y_2$ 的联合分布，设 $Y_1 = g_1(X_1, X_2)$，$Y_2 = g_2(X_1, X_2)$，函数 $g_1, g_2$ 满足以下两个条件：

（1）有下列方程组

y_1 = g_1(x_1, x_2) \\
y_2 = g_2(x_1, x_2)

（2）函数 $g_1, g_2$ 对一切 $(x_1, x_2)$ 有连续偏导数，并且下面的 $2 \times 2$ 行列式对一切 $(x_1, x_2)$ 有

J(x_1, x_2) = \begin{vmatrix}
\frac{\partial g_1}{\partial x_1} & \frac{\partial g_1}{\partial x_2} \\
\frac{\partial g_2}{\partial x_1} & \frac{\partial g_2}{\partial x_2} \end{vmatrix}
\equiv \frac{\partial g_1}{\partial x_1}\frac{\partial g_2}{\partial x_2} – \frac{\partial g_1}{\partial x_2}\frac{\partial g_2}{\partial x_1} \neq 0

f_{Y_1, Y_2}(y_1, y_2) = f_{X_1, X_2}(x_1, x_2) \vert J(x_1, x_2) \vert^{-1} \tag{1}

设已知 $n$ 个随机变量 $X_1, \cdots, X_n$ 的联合密度函数，$Y_1, \cdots, Y_n$ 为 $X_1, \cdots, X_n$ 的函数，令

Y_1 = g_1(X_1, \cdots, X_n), \; Y_2 = g_2(X_1, \cdots, X_n), \; \cdots, \; Y_n = g_n(X_1, \cdots, X_n)

J(x_1, \cdots, x_n) = \begin{vmatrix}
\frac{\partial g_1}{\partial x_1} & \frac{\partial g_1}{\partial x_2} & \cdots & \frac{\partial g_1}{\partial x_n} \\
\frac{\partial g_2}{\partial x_1} & \frac{\partial g_2}{\partial x_2} & \cdots & \frac{\partial g_2}{\partial x_n} \\
\vdots & \vdots & \vdots & \vdots \\
\frac{\partial g_n}{\partial x_1} & \frac{\partial g_n}{\partial x_2} & \cdots & \frac{\partial g_n}{\partial x_n}
\end{vmatrix} \neq 0

y_1 = g_1(x_1, \cdots, x_n) \\
y_2 = g_2(x_1, \cdots, x_n) \\
\vdots \\
y_n = g_n(x_1, \cdots, x_n)

x_1 = h_1(y_1, \cdots, y_n) \\
\vdots \\
x_n = h_n(y_1, \cdots, y_n)

f_{Y_1, \cdots, Y_n} = f_{X_1, \cdots, X_n}(x_1, \cdots, x_n) |J(x_1, \cdots, x_n)|^{-1} \tag{2}