106 Multivariate Distributionshence the joint pdf ofY 1 andY 2 isfY 1 ,Y 2 (y 1 ,y 2 )={ | 2 |
4 e−y 1 −y (^2) (y 1 ,y 2 )∈T
0elsewhere.
Thus the pdf ofY 1 is given by
fY 1 (y 1 )=
⎧
⎨
⎩
∫∞
− 2 y 1
1
2 e
−y 1 −y (^2) dy 2 =^1
2 e
y (^1) −∞<y 1 < 0
∫∞
0
1
2 e
−y 1 −y (^2) dy 2 = 1
2 e
−y (^10) ≤y 1 <∞,
or
fY 1 (y 1 )=^12 e−|y^1 |, −∞<y 1 <∞. (2.2.4)
Recall from expression (1.9.20) of Chapter 1 thatY 1 has the Laplace distribution.
This pdf is also frequently called thedouble exponentialpdf.
Example 2.2.5.LetX 1 andX 2 have the joint pdf
fX 1 ,X 2 (x 1 ,x 2 )=
{
10 x 1 x^220 <x 1 <x 2 < 1
0elsewhere.
SupposeY 1 =X 1 /X 2 andY 2 =X 2. Hence, the inverse transformation isx 1 =y 1 y 2
andx 2 =y 2 , which has the Jacobian
J=
∣
∣
∣
∣
y 2 y 1
01
∣
∣
∣
∣=y^2.
The inequalities defining the supportSof (X 1 ,X 2 ) become
0 <y 1 y 2 ,y 1 y 2 <y 2 ,andy 2 < 1.
These inequalities are equivalent to
0 <y 1 <1and0<y 2 < 1 ,
which defines the support setT of (Y 1 ,Y 2 ). Hence, the joint pdf of (Y 1 ,Y 2 )is
fY 1 ,Y 2 (y 1 ,y 2 )=10y 1 y 2 y 22 |y 2 |=10y 1 y 24 , (y 1 ,y 2 )∈T.
The marginal pdfs are
fY 1 (y 1 )=
∫ 1
0
10 y 1 y^42 dy 2 =2y 1 , 0 <y 1 < 1 ,
zero elsewhere, and
fY 2 (y 2 )=
∫ 1
0
10 y 1 y^42 dy 1 =5y^42 , 0 <y 1 < 1 ,
zero elsewhere.