多维随机变量函数分布

 设二维随机变量$(X,Y)$的联合密度函数为$p(x,y)$，如果函数$$\{\begin{array}{rl}u& =g_1(x,y)\\ v& =g_2(x,y)\end{array}$$有连续偏导数，且存在唯一的反函数$$\{\begin{array}{rl}x& =x(u,v)\\ y& =y(u,v)\end{array}$$其变换的雅可行列式$$J=\frac{\partial (x,y)}{\partial (u,v)}=\left|\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial y}{\partial u}\\ \frac{\partial x}{\partial v}& \frac{\partial y}{\partial v}\end{array}\right|={\left(\frac{\partial (u,v)}{\partial (x,y)}\right)}^{-1}={\left(\left|\begin{array}{cc}\frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}\end{array}\right|\right)}^{-1}\ne 0$$若$$\{\begin{array}{rl}U& =g_1(X,Y)\\ V& =g_2(X,Y)\end{array}$$则$(U,V)$的联合密度函数为$$p(u,v)=p(x(u,v),y(u,v))|J|$$