PDF Method for Deriving Distribution for Multivariable Transformation

Method

• For Continuous RV $X,Y$
• With Joint PDF $f_{X,Y}(x,y)$
• With differentiable Injective Transformation $(Z,W)=h(X,Y)$ The Joint PDF of $(Z,W)$ is given by: $f_{Z,W}(z,w)=\frac{f_{X,Y}(h^{-1}(z,w))}{|J(h^{-1}(z,w))|}$
• Where $J(x,y)$ is Jacobian Matrix

Inverting

$(X,Y)=h^{-1}(Z,W)$