设二维随机变量(X,Y)的概率密度为f(x,y)=2-x-y ,0

题目

设二维随机变量(X,Y)的概率密度为f(x,y)=2-x-y ,0

答案

设T=X-Y则X=(Z+T)/2Y=(Z-T)/2f(z,t)=f(x(z,t),y(z,t))*|det(jacobian)|jacobian=[(dx/dz,dx/dt),(dy/dz,dy/dt)]=[(1/2,1/2)(1/2,-1/2)]|det(jacobian)|=|-1/4-1/4|=1/2f(x(z,t),y(z,t))=2-(x+y)=2-zf(z,t)=(2-z)/2T=...