Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

108 Multivariate Distributions

So the mgf ofY=(1/2)(X 1 −X 2 )isgivenby

E(etY)=

∫∞

0

∫∞

0

et(x^1 −x^2 )/^2

1

4

e−(x^1 +x^2 )/^2 dx 1 dx 2

=

[∫∞

0

1

2

e−x^1 (1−t)/^2 dx 1

][∫∞

0

1

2

e−x^2 (1+t)/^2 dx 2

]

=

[

1

1 −t

][

1

1+t

]

=

1

1 −t^2

provided that 1−t>0and1+t>0; i.e.,− 1 <t<1. However, the mgf of a

Laplace distribution with pdf (1.9.20) is

∫∞

−∞

etx

e−|x|

2

dx =

∫ 0

−∞

e(1+t)x

2

dx+

∫∞

0

e(t−1)x

2

dx

=

1

2(1 +t)

+

1

2(1−t)

=

1

1 −t^2

,

provided− 1 <t<1. Thus, by the uniqueness of mgfs,Yhas a Laplace distribution
with pdf (1.9.20).

EXERCISES
2.2.1.Ifp(x 1 ,x 2 )=(^23 )x^1 +x^2 (^13 )^2 −x^1 −x^2 ,(x 1 ,x 2 )=(0,0),(0,1),(1,0),(1,1), zero
elsewhere, is the joint pmf ofX 1 andX 2 , find the joint pmf ofY 1 =X 1 −X 2 and
Y 2 =X 1 +X 2.

2.2.2. LetX 1 andX 2 have the joint pmfp(x 1 ,x 2 )=x 1 x 2 / 36 ,x 1 =1, 2 ,3and
x 2 =1, 2 ,3, zero elsewhere. Find first the joint pmf ofY 1 =X 1 X 2 andY 2 =X 2 ,
and then find the marginal pmf ofY 1.

2.2.3.LetX 1 andX 2 have the joint pdfh(x 1 ,x 2 )=2e−x^1 −x^2 , 0 <x 1 <x 2 <∞,
zero elsewhere. Find the joint pdf ofY 1 =2X 1 andY 2 =X 2 −X 1.

2.2.4.LetX 1 andX 2 have the joint pdfh(x 1 ,x 2 )=8x 1 x 2 , 0 <x 1 <x 2 <1, zero
elsewhere. Find the joint pdf ofY 1 =X 1 /X 2 andY 2 =X 2.
Hint: Use the inequalities 0<y 1 y 2 <y 2 <1 in considering the mapping fromS
ontoT.

2.2.5.LetX 1 andX 2 be continuous random variables with the joint probability
density functionfX 1 ,X 2 (x 1 ,x 2 ),−∞<xi<∞,i=1,2. LetY 1 =X 1 +X 2 and
Y 2 =X 2.

(a)Find the joint pdffY 1 ,Y 2.

(b)Show that

fY 1 (y 1 )=

∫∞

−∞

fX 1 ,X 2 (y 1 −y 2 ,y 2 )dy 2 , (2.2.5)

which is sometimes called theconvolution formula.