Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

124 Multivariate Distributions

Example 2.4.5(Example 2.1.10, Continued). Let (X, Y) be a pair of random

variables with the joint pdf

f(x, y)=

{

e−y 0 <x<y<∞

0elsewhere.

In Example 2.1.10, we showed that the mgf of (X, Y)is

M(t 1 ,t 2 )=

∫∞

0

∫∞

x

exp(t 1 x+t 2 y−y)dydx

=

1

(1−t 1 −t 2 )(1−t 2 )

,

provided thatt 1 +t 2 <1andt 2 <1. BecauseM(t 1 ,t 2 ) =M(t 1 ,0)M(t 1 ,0), the

random variables are dependent.

Example 2.4.6(Exercise 2.1.15, Continued).For the random variableX 1 andX 2

defined in Exercise 2.1.15, we showed that the joint mgf is

M(t 1 ,t 2 )=

[

exp{t 1 }

2 −exp{t 1 }

][

exp{t 2 }

2 −exp{t 2 }

]

,ti<log 2,i=1, 2.

We showed further thatM(t 1 ,t 2 )=M(t 1 ,0)M(0,t 2 ). Hence,X 1 andX 2 are inde-
pendent random variables.

EXERCISES

2.4.1.Show that the random variablesX 1 andX 2 with joint pdf

f(x 1 ,x 2 )=

{

12 x 1 x 2 (1−x 2 )0<x 1 < 1 , 0 <x 2 < 1

0elsewhere

are independent.

2.4.2.If the random variablesX 1 andX 2 have the joint pdff(x 1 ,x 2 )=2e−x^1 −x^2 , 0 <
x 1 <x 2 , 0 <x 2 <∞, zero elsewhere, show thatX 1 andX 2 are dependent.

2.4.3.Letp(x 1 ,x 2 )= 161 ,x 1 =1, 2 , 3 ,4, andx 2 =1, 2 , 3 ,4, zero elsewhere, be the

joint pmf ofX 1 andX 2. Show thatX 1 andX 2 are independent.

2.4.4.FindP(0<X 1 <^13 , 0 <X 2 <^13 ) if the random variablesX 1 andX 2 have
the joint pdff(x 1 ,x 2 )=4x 1 (1−x 2 ), 0 <x 1 < 1 , 0 <x 2 <1, zero elsewhere.

2.4.5.Find the probability of the union of the eventsa<X 1 <b,−∞<X 2 <∞,
and−∞<X 1 <∞,c<X 2 <difX 1 andX 2 are two independent variables with
P(a<X 1 <b)=^23 andP(c<X 2 <d)=^58.