Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.8. Linear Combinations of Random Variables 151

2.7.5.LetX 1 ,X 2 ,X 3 be iid with common pdff(x)=e−x,x>0, 0 elsewhere.
Find the joint pdf ofY 1 =X 1 /X 2 ,Y 2 =X 3 /(X 1 +X 2 ), andY 3 =X 1 +X 2 .Are
Y 1 ,Y 2 ,Y 3 mutually independent?

2.7.6.LetX 1 ,X 2 have the joint pdff(x 1 ,x 2 )=1/π, 0 <x^21 +x^22 < 1 .Let
Y 1 =X 12 +X 22 andY 2 =X 2. Find the joint pdf ofY 1 andY 2.

2.7.7.LetX 1 ,X 2 ,X 3 ,X 4 have the joint pdff(x 1 ,x 2 ,x 3 ,x 4 ) = 24, 0<x 1 <x 2 <
x 3 <x 4 < 1 ,0 elsewhere. Find the joint pdf ofY 1 =X 1 /X 2 ,Y 2 =X 2 /X 3 ,Y 3 =
X 3 /X 4 ,Y 4 =X 4 and show that they are mutually independent.

2.7.8.LetX 1 ,X 2 ,X 3 be iid with common mgfM(t) = ((3/4) + (1/4)et)^2 , for all
t∈R.

(a)Determine the probabilities,P(X 1 =k),k=0, 1 ,2.

(b)Find the mgf ofY =X 1 +X 2 +X 3 and then determine the probabilities,

P(Y=k),k=0, 1 , 2 ,...,6.

2.8 LinearCombinationsofRandomVariables...............

In this section, we summarize some results on linear combinations of random vari-

ables that follow from Section 2.6. These results will prove to be quite useful in

Chapter 3 as well as in succeeding chapters.

Let (X 1 ,...,Xn)′denote a random vector. In this section, we consider linear

combinations of these variables, writing them , generally, as

T=

∑n

i=1

aiXi, (2.8.1)

for specified constantsa 1 ,...,an. We obtain expressions for the mean and variance
ofT.
The mean ofTfollows immediately from linearity of expectation. For reference,
we state it formally as a theorem.

Theorem 2.8.1. SupposeTis given by expression (2.8.1). SupposeE(Xi)−μi,
fori=1,...,n.Then

E(T)=

∑n

i=1

aiμi. (2.8.2)

In order to obtain the variance ofT, we first state a general result on covariances.

Theorem 2.8.2.SupposeTis the linear combination (2.8.1) and thatWis another

linear combination given byW =

∑m

i=1biYi, for random variablesY^1 ,...,Ymand

specified constantsb 1 ,...,bm.LetT =

∑n

i=1aiXi and letW =

∑m

i=1biYi.If

E[Xi^2 ]<∞,andE[Yj^2 ]<∞fori=1,...,nandj=1,...,m,then

Cov(T,W)=

∑n

i=1

∑m

j=1

aibjCov(Xi,Yj). (2.8.3)