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88 2. Expectations of Functions of Random Variables
claim that X and Y have different probability distributions because their two
pdfs are obviously different when c ≠ 0. In the Figure 2.4.1, these two pdfs
have been plotted when c = 1/2. Here all the moments of X are finite, but in the
Exercise 2.4.5 one would verify that the mgf for the random variable X is not
finite. In another related Exercise 2.4.6, examples of other pairs of random
variables with analogous characteristics can be found. !
Finite moments alone may not determine a distribution
uniquely. The Example 2.4.4 highlights this point.
Depending on the particular situation, however, an infinite sequence of all
finite moments may or may not characterize a probability distribution uniquely.
Any elaborate discussion of such issues would be out of scope of this book. The
readers may additionally consult the Section 3, Chapter 7 in Feller (1971) on
Moment Problems. Chung (1974) also includes relevant details.
2.5 The Probability Generating Function
We have seen that a mgf generates the moments. Analogously, there is another
function which generates the probabilities. Suppose that X is a non-negative
random variable. A probability generating function (pgf) of X is defined by
The explicit form of a pgf is often found when the random variable X is
integer-valued. From this point onward, let us include only non-negative inte-
ger-valued random variables in this discussion.
Why this function P (t) is called a pgf should become clear once we write
it out fully as follows: X
where p = P(X = i) for i = 1, 2, ... . We see immediately that the coefficient
i
of t is p , the probability that X = i for i = 1, 2, ... .
i
i
In the light of the Theorem 2.3.1 one can justify the following result:
