14 - Change of Variables Formula for Joint Densities

Change of Variables Formula for Joint Densities

Suppose $T:(x,y)\mapsto (u,v)$ is a one-to-one mapping from $D\subseteq {\mathbb{R}}^2$ onto range $R\subseteq {\mathbb{R}}^2$

Jacobian $J$ as function of $(u,v)$ is defined by

$$J=\det \left(\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}\right)=\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}$$

assuming that partial derivatives exist and are continuous

If $X,Y$ have joint probability density function $f_{X,Y}$ then

Random variables $U,V$ defined by

$$(U,V)=T(X,Y)$$

are jointly continuous with joint probability density function $f_{U,V}$ defined by

$$f_{U,V}(u,v)=\{\begin{array}{ll}{f}_{X,Y}(x(u,v),y(u,v))|J(u,v)|& \text{ if }(u,v)\in R\\ & \\ 0& \text{ otherwise}\end{array}$$

Proof

Suppose $A\subseteq D$ and $T(A)=B$ then as $T$ is one-to-one

$$\begin{array}{rl}\mathbb{P}((U,V)\in B)& =\mathbb{P}((X,Y)\in A)\\ & ={\iint }_A{f}_{X,Y}(x,y)dxdy\\ & ={\iint }_B{f}_{X,Y}(x(u,v),y(u,v))|J(u,v)|dudv\end{array}$$

Hence final integrand is joint pdf of $(U,V)$