Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.4. The Normal Distribution 193

Theorem 3.4.2.LetX 1 ,...,Xnbe independent random variables such that, for

i=1,...,n,Xihas aN(μi,σi^2 )distribution. LetY=

∑n

i=1aiXi,wherea^1 ,...,an

are constants. Then the distribution ofYisN(

∑n

i=1aiμi,

∑n

i=1a

2

iσ

2

i).

Proof:By Theorem 2.6.1, fort∈R,themgfofYis

MY(t)=

∏n

i=1

exp

{

taiμi+(1/2)t^2 a^2 iσ^2 i

}

=exp

{

t

∑n

i=1

aiμi+(1/2)t^2

∑n

i=1

a^2 iσ^2 i

}

,

which is the mgf of aN(

∑n

i=1aiμi,

∑n

i=1a

2

iσ

2

i) distribution.

A simple corollary to this result gives the distribution of the sample meanX=
n−^1

∑n

i=1XiwhenX^1 ,X^2 ,...Xnrepresents a random sample from aN(μ, σ

(^2) ).
Corollary 3.4.1.LetX 1 ,...,Xnbe iid random variables with a commonN(μ, σ^2 )
distribution. LetX=n−^1
∑n
i=1Xi.ThenXhas aN(μ, σ
(^2) /n)distribution.
To prove this corollary, simply takeai=(1/n),μi=μ,andσi^2 =σ^2 ,for
i=1, 2 ,...,n, in Theorem 3.4.2.

3.4.1 ∗ContaminatedNormals.....................

We next discuss a random variable whose distribution is a mixture of normals. As
with the normal, we begin with a standardized random variable.
Suppose we are observing a random variable that most of the time follows a
standard normal distribution but occasionally follows a normal distribution with
a larger variance. In applications, we might say that most of the data are “good”
but that there are occasional outliers. To make this precise letZhave aN(0,1)
distribution; letI 1 − be a discrete random variable defined by

I 1 − =

{

1 with probability 1−

0 with probability,

and assume thatZandI 1 − are independent. LetW=ZI 1 − +σcZ(1−I 1 − ).

ThenWis the random variable of interest.

The independence ofZandI 1 − imply that the cdf ofWis

FW(w)=P[W≤w]=P[W≤w, I 1 − =1]+P[W≤w, I 1 − =0]

= P[W≤w|I 1 − =1]P[I 1 − =1]

+P[W≤w|I 1 − =0]P[I 1 − =0]

= P[Z≤w](1− )+P[Z≤w/σc].

=Φ(w)(1− )+Φ(w/σc) (3.4.15)

Therefore, we have shown that the distribution ofW is a mixture of normals.
Further, becauseW=ZI 1 − +σcZ(1−I 1 − ), we have

E(W)=0andVar(W)=1+ (σ^2 c−1); (3.4.16)