9.3. Noncentralχ^2 andF-Distributions 523
The integral exists ift<^12. To evaluate the integral, note that
tx^2 i
σ^2
−
(xi−μi)^2
2 σ^2
=−
x^2 i(1− 2 t)
2 σ^2
+
2 μixi
2 σ^2
−
μ^2 i
2 σ^2
=
tμ^2 i
σ^2 (1− 2 t)
−
1 − 2 t
2 σ^2
(
xi−
μi
1 − 2 t
) 2
.
Accordingly, witht<^12 ,wehave
E
[
exp
(
tX^2 i
σ^2
)]
=exp
[
tμ^2 i
σ^2 (1− 2 t)
]∫∞
−∞
1
σ
√
2 π
exp
[
−
1 − 2 t
2 σ^2
(
xi−
μi
1 − 2 t
) 2 ]
dxi.
If we multiply the integrand by
√
1 − 2 t,t<^12 , we have the integral of a normal
pdf with meanμi/(1− 2 t)andvarianceσ^2 /(1− 2 t). Thus
E
[
exp
(
tXi^2
σ^2
)]
=
1
√
1 − 2 t
exp
[
tμ^2 i
σ^2 (1− 2 t)
]
,
and the mgf ofY=
∑n
1 X
2
i/σ
(^2) is given by
M(t)=
1
(1− 2 t)n/^2
exp
[
t
∑n
1 μ
2
i
σ^2 (1− 2 t)
]
,t<
1
2
. (9.3.1)
A random variable that has the mgf
M(t)=
1
(1− 2 t)r/^2
etθ/(1−^2 t), (9.3.2)
wheret<^12 ,0<θ,andris a positive integer, is said to have anoncentral
chi-square distributionwithrdegrees of freedom and noncentrality parameter
θ. If one sets the noncentrality parameterθ= 0, one hasM(t)=(1− 2 t)−r/^2 ,
which is the mgf of a random variable that isχ^2 (r). Such a random variable can
appropriately be called acentral chi-square variable. We shall use the symbol
χ^2 (r, θ) to denote a noncentral chi-square distribution that has the parametersr
andθ; and we shall say that a random variable isχ^2 (r, θ) when that random variable
has this kind of distribution. The symbolχ^2 (r,0) is equivalent toχ^2 (r). Thus our
random variableY=
∑n
1 X
2
i/σ
(^2) of this section isχ 2 (n,∑n
1 μ
2
i/σ
2 ). The mean of
Yis given by
E(Y)=
1
σ^2
∑n
i=1
E(X^2 i)=
1
σ^2
∑n
i=1
(σ^2 +μ^2 i)=n+θ, (9.3.3)
i.e., the mean of the centralχ^2 plus the noncentrality parameter. If eachμiis equal
to zero, thenYisχ^2 (n,0) or, more simply,Y isχ^2 (n)withmeann.
The noncentralχ^2 -variables, in which we have interest, are certain quadratic
forms in normally distributed variables divided by a varianceσ^2 .Inourexam-
ple it is worth noting that the noncentrality parameter of
∑n
1 X
2
i/σ
(^2) ,whichis