8.4 Key definitions and Relationships

Let $(X,Y)$ be a continuous bivariate rv that is mapped by a 1-1 transformation to another continuous rv, $(S,T)$.

1. 1.

   $f_{S,T}(s,t)=f_{X,Y}(x,y)|det(J)|$, where $J=\partial (x,y)/\partial (s,t)$. Relabelling and re-arranging $f_{S,T}(s,t)=f_{X,Y}(x,y){|det(J)|}^{-1}$, where $J=\partial (s,t)/\partial (x,y)$.

2. 2.

   Be careful to define the joint range of $S$ and $T$.

3. 3.

   If interest lies in $S=g_1(X,Y)$ then define a dummy variable $T=g_2(X,Y)$ such that the transformation $(X,Y)\to (S,T)$ is 1-1.