Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 186

Using ( 3.159 ), ( 3.158 ), and ( 3.175 ) we get:

¼x 1 ¼z;

x 2 x 1
10

J ¼ (3.176)

and

f X 1 X 2 ð x 1 ; x 2 Þ
jj

f X 1 X 2 z; y=z ð Þ

f Yr ðy; zÞ¼ ¼ : (3.177)

jj

Finally, the desired PDF is obtained from ( 3.171 ) and ( 3.174 ):

1

1
jj

f Y ðyÞ¼ f X 1 X 2 ðz; y=zÞ d z: (3.178)

1

3.4.2 Nonunique Transformation

In this case, the infinitesimal area (d y 1 d y 2 ) corresponds to two or more infinitesi-
mal areas (d x 1 d x 2 ), resulting in:

P i

x i 1 ¼ g 1

f X 1 X 2 ðx i 1 ; x i 2 Þ

f Y 1 Y 2 ðy 1 ; y 2 Þ¼ : (3.179)

Jðx i 1 ; x i 2 Þ

ðy 1 ; y 2 Þ

1

x i 2 ¼ g 1 ðy 1 ; y 2 Þ

2

Example 3.4.3 Find the joint PDF of the random variables Y 1 and Y 2 ,

q

X 1 þ X 2

X 1
X 2 ;

Y 1 ¼ Y 2 ¼

; (3.180)

if the given joint PDF of the random variables X 1 and X 2 is given as:

x 1 þ x 2

1

2 ps 2 e

2 s 2

f X 1 X 2 ðx 1 ; x 2 Þ¼ : (3.181)

Solution From ( 3.180 ), we have:

q

x 1 þ x 2

x 1
x 2 :

y 1 ¼ y 2 ¼ (3.182)

;