74 Probability and Distributionsconsider this alternative method. The functionM(t) is represented by the following
Maclaurin’s series:^8et(^2) / 2
=1+
1
1!
(
t^2
2
)
- 1
2!
(
t^2
2
) 2
+···+
1
k!
(
t^2
2
)k
+···
=1+
1
2!
t^2 +
(3)(1)
4!
t^4 +···+
(2k−1)···(3)(1)
(2k)!
t^2 k+···.
In general, though, from calculus the Maclaurin’s series forM(t)is
M(t)=M(0) +
M′(0)
1!
t+
M′′(0)
2!
t^2 +···+
M(m)(0)
m!
tm+···
=1+
E(X)
1!
t+
E(X^2 )
2!
t^2 +···+
E(Xm)
m!
tm+···.
Thus the coefficient of (tm/m!) in the Maclaurin’s series representation ofM(t)is
E(Xm). So, for our particularM(t), we have
E(X^2 k)=(2k−1)(2k−3)···(3)(1) =
(2k)!
2 kk!
,k=1, 2 , 3 ,...(1.9.3)
E(X^2 k−^1 )=0,k=1, 2 , 3 ,.... (1.9.4)
We make use of this result in Section 3.4.
Remark 1.9.1.As Examples 1.9.5 and 1.9.6 show, distributions may not have
moment-generating functions. In a more advanced course, we would letidenote
the imaginary unit,tan arbitrary real, and we would defineφ(t)=E(eitX). This
expectation exists foreverydistribution and it is called thecharacteristic func-
tionof the distribution. To see whyφ(t) exists for all realt,wenote,inthe
continuous case, that its absolute value
|φ(t)|=
∣
∣
∣
∣
∫∞
−∞
eitxf(x)dx
∣
∣
∣
∣≤
∫∞
−∞
|eitxf(x)|dx.
However,|f(x)|=f(x)sincef(x) is nonnegative and
|eitx|=|costx+isintx|=
√
cos^2 tx+sin^2 tx=1.
Thus
|φ(t)|≤
∫∞
−∞
f(x)dx=1.
Accordingly, the integral forφ(t) exists for all real values oft. In the discrete
case, a summation would replace the integral. In reference to Example 1.9.6, it can
be shown that the characteristic function of the Cauchy distribution is given by
φ(t)=exp{−|t|},−∞<t<∞.
(^8) See Chapter 2 ofMathematical Comments.