Page 103 - Probability and Statistical Inference

80 2. Expectations of Functions of Random Variables

A finite mgf M (t) determines a unique infinite sequence
X
of moments of a random variable X.

r
Remark 2.3.1 Incidentally, the sequence of moments {η = E(X ): r = 1,
r
2, ...} is sometimes referred to as the sequence of positive integral moments
of X. The Theorem 2.3.1 provides an explicit tool to restore all the positive
integral moments of X by successively differentiating its mgf and then letting
t = 0 in the expression. It is interesting to note that the negative integral mo-
ments of X, that is when r = –1, –2, ... in the Definition 2.3.1, are also hidden
inside the same mgf. These negative moments of X can be restored by imple-
menting a process of successive integration of the mgf, an operation which is
viewed as the opposite of differentiation. Precise statements of the regularity
conditions under which the negative moments of X can be derived with this
approach are found in Cressie et al. (1981).

The positive integral moments of X can be found by
successively differentiating its mgf with respect to t
and letting t = 0 in the final derivative.

The following result is very useful and simple. We leave its proof as an exer-
cise. There will be ample opportunities to use this theorem in the sequel.
Theorem 2.3.2 Suppose that a random variable X has the mgf M (t), for |
X
t | < a with some a > 0. Let Y = cX + d be another random variable where c, d
are fixed real numbers. Then, the mgf M (t) of Y is related to the mgf M (t) as
Y
X
follows:

In the following subsections, we show the derivations of the mgf in the
case of a few specific distributions. We also exhibit some immediate applica-
tions of the Theorem 2.3.1.

2.3.1 The Binomial Distribution
x
Suppose that X has the Binomial (n, p) distribution with its pmf f(x) = p (1
– p) for x = 0, 1, ..., n and 0 < p < 1, given by (1.7.2). Here, for all fixed t ∈
n–x
ℜ we can express M (t) as
X