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5-9

The moments of a random variable can often be determined directly from the definition
in Equation S5-7, but there is an alternative procedure that is frequently useful that makes use
of a special function.

Definition
The moment generating function of the random variable X is the expected value of
tX
e and is denoted by M (t). That is,
X
e tx f 1x2, X discrete
a
x
tX
M X 1t2 E1e 2 µ (S5-8)
tx
e f 1x2 dx, X continuous

The moment generating function M (t) will exist only if the sum or integral in the above def-
X
inition converges. If the moment generating function of a random variable does exist, it can be
used to obtain all the origin moments of the random variable.

Let X be a random variable with moment generating function M (t). Then
X
r
d M X 1t2
` (S5-9)
¿ r r
dt t 0

Assuming that we can differentiate inside the summation and integral signs,
r tx
x e f 1x2, X discrete
a
r
d M X 1t2 x
dt r µ r tx
x e f 1x2 dx, X continuous

Now if we set t 0 in this expression, we find that
r
d M X 1t2 r
dt r ` E1X 2
t 0
EXAMPLE S5-5 Suppose that X has a binomial distribution, that is

n x n x
f 1x2 a b p 11 p2 , x 0, 1, p , n
x
Determine the moment generating function and use it to verify that the mean and variance of
2
the binomial random variable are np and np(1 p).
From the definition of a moment generating function, we have
n n n n
t x
tx
x
M 1t2 a e a b p 11 p2 n x a a b 1pe 2 11 p2 n x
X
x 0 x x 0 x