72 Probability and Distributions
us to interchange the order of differentiation and integration (or summation in the
discrete case). That is, ifXis continuous,
M′(t)=
dM(t)
dt
=
d
dt
∫∞
−∞
etxf(x)dx=
∫∞
−∞
d
dt
etxf(x)dx=
∫∞
−∞
xetxf(x)dx.
Likewise, ifXis a discrete random variable,
M′(t)=
dM(t)
dt
=
∑
x
xetxp(x).
Upon settingt=0,wehaveineithercase
M′(0) =E(X)=μ.
The second derivative ofM(t)is
M′′(t)=
∫∞
−∞
x^2 etxf(x)dx or
∑
x
x^2 etxp(x),
so thatM′′(0) =E(X^2 ). Accordingly, Var(X)equals
σ^2 =E(X^2 )−μ^2 =M′′(0)−[M′(0)]^2.
For example, ifM(t)=(1−t)−^1 ,t<1, as in the illustration above, then
M′(t)=(1−t)−^2 and M′′(t)=2(1−t)−^3.
Hence
μ=M′(0) = 1
and
σ^2 =M′′(0)−μ^2 =2−1=1.
Of course, we could have computedμandσ^2 from the pdf by
μ=
∫∞
−∞
xf(x)dx and σ^2 =
∫∞
−∞
x^2 f(x)dx−μ^2 ,
respectively. Sometimes one way is easier than the other.
In general, ifmis a positive integer and ifM(m)(t)meansthemth derivative of
M(t), we have, by repeated differentiation with respect tot,
M(m)(0) =E(Xm).
Now
E(Xm)=
∫∞
−∞
xmf(x)dx or
∑
x
xmp(x),
and the integrals (or sums) of this sort are, in mechanics, calledmoments.Since
M(t) generates the values ofE(Xm),m=1, 2 , 3 ,...,it is called the moment-
generating function (mgf). In fact, we sometimes callE(Xm)themth momentof
the distribution, or themth moment ofX.
The next two examples concern random variables whose distributions do not
have mgfs.