Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 182

Therefore, if random variables are uncorrelated, they can be either dependent or
independent. However, there are some exceptions such as the case of uncorrelated
normal random variables. If normal random variables are uncorrelated it follows
that they are independent (see Chap. 4 ) .

3.4 Transformation of Random Variables

3.4.1 One-to-One Transformation

Consider two random variables X 1 and X 2 with a known joint PDF f X 1 X 2 ðx 1 ; x 2 Þ .Asa
result of the transformation

Y 1 ¼ g 1 ðX 1 ; X 2 Þ
Y 2 ¼ g 2 ðX 1 ; X 2 Þ

(3.153)

new random variables Y 1 and Y 2 are obtained, as shown in Fig. 3.13 .
We are looking for the joint density of the transformed random variables ( 3.153 ).
The result depends on the type of transformation. Here we consider a simple case in
which the infinitesimal area in the ( x 1 , x 2 ) system has a one-to-one correspondence
to the infinitesimal area in the ( y 1 , y 2 ) system, as shown in Fig. 3.14 . In other words,

Fig. 3.13 Transformation of two random variables

Fig. 3.14 Mapping from ( x 1 , x 2 ) space onto ( y 1 , y 2 ) space