Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.2. Transformations: Bivariate Random Variables 105

y 2

(0, 0) y 1

y 2 = y 1 y 2 = 2 – y 1

y 2 = – y 1 y 2 = y 1 – 2

T

Figure 2.2.3:The support of (Y 1 ,Y 2 ) of Example 2.2.3.

The marginal pdf ofY 1 is given by

fY 1 (y 1 )=

∫∞

−∞

fY 1 ,Y 2 (y 1 ,y 2 )dy 2.

If we refer to Figure 2.2.3, we can see that

fY 1 (y 1 )=

⎧

⎪⎨

⎪⎩

∫y 1

−y 1

1

∫^2 dy^2 =y^10 <y^1 ≤^1

2 −y 1

y 1 − 2

1

2 dy^2 =2−y^11 <y^1 <^2

0elsewhere,

which agrees with expression (2.2.2) of Example 2.2.2. In a similar manner, the
marginal pdffY 2 (y 2 )isgivenby

fY 2 (y 2 )=

⎧

⎪⎨

⎪⎩

∫y 2 +2

−y 2

1

∫^2 dy^1 =y^2 +1 −^1 <y^2 ≤^0

2 −y 2

y 2

1

2 dy^1 =1−y^20 <y^2 <^1

0elsewhere.

Example 2.2.4.LetY 1 =^12 (X 1 −X 2 ), whereX 1 andX 2 have the joint pdf

fX 1 ,X 2 (x 1 ,x 2 )=

{ 1

4 exp

(

−x^1 + 2 x^2

)

0 <x 1 <∞, 0 <x 2 <∞

0elsewhere.

LetY 2 =X 2 so thaty 1 =^12 (x 1 −x 2 ),y 2 =x 2 or, equivalently,x 1 =2y 1 +y 2 ,x 2 =
y 2 , define a one-to-one transformation fromS={(x 1 ,x 2 ):0<x 1 <∞, 0 <x 2 <
∞}ontoT ={(y 1 ,y 2 ):− 2 y 1 <y 2 and 0<y 2 <∞, −∞<y 1 <∞}.The
Jacobian of the transformation is

J=

∣

∣

∣

∣

21

01

∣

∣

∣

∣=2;