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190 4. Functions of Random Variables and Sampling Distribution

4.3 Using the Moment Generating Function

The moment generating function (mgf) of a random variable was introduced
in the Section 2.3. In the Section 2.4 we had emphasized that a finite mgf
indeed pinpoints a unique distribution (Theorem 2.4.1). The implication we
wish to reinforce is that a finite mgf corresponds to the probability distribu-
tion of a uniquely determined random variable. For this reason alone, the finite
mgf’s can play important roles in deriving the distribution of functions of
random variables.
One may ponder over what kinds of functions of random variables we should
be looking at in the first place. We may point out that in statistics one fre-
quently faces the problem of finding the distribution of a very special type of
function, namely a linear function of independent and identically distributed
(iid) random variables. The sample mean and its distribution, for example, fall
in this category. In this vein, the following result captures the basic idea. It
also provides an important tool for future use.
Theorem 4.3.1 Consider a real valued random variable X having its mgf
i
M (t) = E(e ) for i = 1, ..., n. Suppose that X , ..., X are independent. Then,
tX
n
1
Xi
i
the mgf of is given by where a , ..., a are any
n
1
arbitrary but otherwise fixed real numbers.
Proof The proof merely uses the Theorem 3.5.1 which allows us to split
the expected value of a product of n independent random variables as the
product of the n individual expected values. We write

which completes the proof. ¢
First, find the mgf M (t). Visually match this mgf with that of one
U
of the standard distributions. Then, U has that same distribution.
A simple looking result such as the Theorem 4.3.1 has deep implications.
First, the mgf M (t) of U is determined by invoking Theorem 4.3.1. In view
U
of the Theorem 2.4.1, since the mgf of U and the distribution of U corre-
spond uniquely to each other, all we have to do then is to match the form of
this mgf M (t) with that of one of the standard distributions. This way we
U
will identify the distribution of U. There are many situations where this simple
approach works just fine.