Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

192 Some Special Distributions

whereeis a random variable (often called random error) with aN(0,σ^2 ) distribu-

tion. Conversely, it follows immediately that ifXsatisfies expression (3.4.13) with

edistributedN(0,σ^2 )thenXhas aN(μ, σ^2 ) distribution.

We close this part of the section with three important results.

Example 3.4.4(All the Moments of a Normal Distribution).Recall that in Ex-
ample 1.9.7, we derived all the moments of a standard normal random variable by
using its moment generating function. We can use this to obtain all the moments
ofX,whereXhas aN(μ, σ^2 ) distribution. From expression (3.4.13), we can write
X=σZ+μ,whereZhas aN(0,1) distribution. Hence, for all nonnegative integers
ka simple application of the binomial theorem yields

E(Xk)=E[(σZ+μ)k]=

∑k

j=0

(

k

j

)

σjE(Zj)μk−j. (3.4.14)

Recall from Example 1.9.7 that all the odd moments ofZare 0, while all the even
moments are given by expression (1.9.3). These can be substituted into expression
(3.4.14) to derive the moments ofX.

Theorem 3.4.1.If the random variableXisN(μ, σ^2 ),σ^2 > 0 , then the random
variableV=(X−μ)^2 /σ^2 isχ^2 (1).
Proof.BecauseV=W^2 ,whereW=(X−μ)/σisN(0,1), the cdfG(v)forV
is, forv≥0,
G(v)=P(W^2 ≤v)=P(−

√

v≤W≤

√

v).

That is,

G(v)=2

∫√v

0

1

√

2 π

e−w

(^2) / 2
dw, 0 ≤v,
and
G(v)=0,v< 0.
If we change the variable of integration by writingw=
√
y,then
G(v)=
∫v
0
1
√
2 π
√
y
e−y/^2 dy, 0 ≤v.
Hence the pdfg(v)=G′(v) of the continuous-type random variableVis
g(v)=
{ 1
√π√ 2 v^1 /^2 −^1 e−v/^20 <v<∞
0elsewhere.
Sinceg(v)isapdf ∫
∞
0
g(v)dv=1;
hence, it must be that Γ(^12 )=
√
πand thusVisχ^2 (1).
One of the most important properties of the normal distribution is its additivity
under independence.