Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

152 Multivariate Distributions

Proof:Using the definition of the covariance and Theorem 2.8.1, we have the first

equality below, while the second equality follows from the linearity ofE:

Cov(T,W)=E

⎡

⎣

∑n

i=1

∑m

j=1

(aiXi−aiE(Xi))(bjYj−bjE(Yj))

⎤

⎦

=

∑n

i=1

∑m

j=1

aibjE[(Xi−E(Xi))(Yj−E(Yj))],

which is the desired result.

To obtain the variance ofT, simply replaceW byTin expression (2.8.3). We
state the result as a corollary:

Corollary 2.8.1.LetT=

∑n

i=1aiXi.ProvidedE[X

2

i]<∞,fori=1,...,n,

Var(T)=Cov(T,T)=

∑n

i=1

a^2 iVar(Xi)+2

∑

i<j

aiajCov(Xi,Xj). (2.8.4)

Note that ifX 1 ,...,Xnare independent random variables, then by Theorem
2.5.2 all the pairwise covariances are 0; i.e., Cov(Xi,Xj) = 0 for alli =j.This
leads to a simplification of (2.8.4), which we record in the following corollary.

Corollary 2.8.2.IfX 1 ,...,Xnare independent random variables and Var(Xi)=
σi^2 ,fori=1,...,n,then

Var(T)=

∑n

i=1

a^2 iσi^2. (2.8.5)

Note that we need onlyXiandXjto be uncorrelated for alli =jto obtain this
result.
Next, in addition to independence, we assume that the random variables have
the same distribution. We call such a collection of random variables arandom
samplewhich we now state in a formal definition.
Definition 2.8.1.If the random variables X 1 ,X 2 ,...,Xn are independent and
identically distributed, i.e. eachXihas the same distribution, then we say that
these random variables constitute arandom sampleof sizenfrom that common
distribution. We abbreviate independent and identically distributed byiid.
In the next two examples, we find some properties of two functions of a random
sample, namely the sample mean and variance.

Example 2.8.1(Sample Mean). LetX 1 ,...,Xnbe independent and identically
distributed random variables with common meanμand varianceσ^2 .Thesample
meanis defined byX=n−^1

∑n
i=1Xi. This is a linear combination of the sample
observations withai≡n−^1 ; hence, by Theorem 2.8.1 and Corollary 2.8.2, we have

E(X)=μand Var(X)=σ

2

n. (2.8.6)

BecauseE(X)=μ,weoftensaythatXisunbiasedforμ.