Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 183

the elementary area (d x 1 d x 2 ) is mapped one-to-one onto a corresponding infinites-
imal area (d y 1 d y 2 ). As a result, the corresponding probabilities are equal:

Pfx 1 < X 1 x 1 þ d x 1 ; x 2 < X 2 x 2 þ d x 2 g¼Pfy 1 < Y 1
y 1 þ d y 1 ; y 2 < Y 2 y 2 þ d y 2 g: (3.154)

The probabilities in ( 3.154 ) can be expressed in terms of their corresponding
joint densities:

Pfx 1 < X 1 x 1 þ d x 1 ; x 2 < X 2 x 2 þ d x 2 g¼f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 ;
Pfy 1 < Y 1 y 1 þ d y 1 ; y 2 < Y 2 y 2 þ d y 2 g¼f Y 1 Y 2 ðy 1 ; y 2 Þ d y 1 d y 2 : (3.155)

From ( 3.154 ) and ( 3.155 ), we have:

f Y 1 Y 2 ðy 1 ; y 2 Þ d y 1 d y 2 ¼ f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 : (3.156)

The infinitesimal areas (d y 1 d y 2 ) and (d x 1 d x 2 ) are related as,

d y 1 d y 2
Jðx 1 x 2 Þ ;

d x 1 d x 2 ¼ (3.157)

where J ( x 1 , x 2 ) is the Jacobian of the transformation ( 3.153 ) [PAP65, p. 201]:

@g 1
@x 1

@g 1
@x 2

Jðx 1 ; x 2 Þ¼ : (3.158)

@g 2
@x 1

@g 2
@x 2

Finally, from ( 3.156 ) and ( 3.157 ) we get:

x 1 ¼g 1 ðy 1 ;y 2 Þ
x 2 ¼g 1

f X 1 X 2 ð x 1 ; x 2 Þ
Jðx 1 ; x 2 Þ

f Y 1 Y 2 ðy 1 ; y 2 Þ¼ ; (3.159)

j j

2 ðy 1 ;y 2 Þ

where g 1
i

, i ¼ 1, 2 is the inverse transformation of ( 3.153 ),

x 1 ¼ g 1 ðy 1 ; y 2 Þ

1

(3.160)

x 2 ¼ g 1
2

ðy 1 ; y 2 Þ:

Note that in ( 3.159 ) the absolute value of the Jacobian must be used because the
joint density cannot be negative as opposite to the Jacobian which can be either
positive or negative.
If for certain values y 1 , y 2 , there is no real solution ( 3.160 ), then

f Y 1 Y 2 ðy 1 ; y 2 Þ¼ 0 : (3.161)