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.
The moment generating function MX(t) will exist only if the sum or integral in the above def-
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.
Assuming that we can differentiate inside the summation and integral signs,
Now if we set t0 in this expression, we find that
EXAMPLE S5-5 Suppose that Xhas a binomial distribution, that is
Determine the moment generating function and use it to verify that the mean and variance of
the binomial random variable are npand ^2 np(1p).
From the definition of a moment generating function, we have
MX 1 t 2 a
n
x 0
etx a
n
x
b px 11 p 2 nxa
n
x 0
a
n
x
b 1 pet 2 x 11 p 2 nx
f 1 x 2 a
n
x
b px 11 p 2 nx, x0, 1,p, n
drMX 1 t 2
dtr
`
t 0
E 1 Xr 2
dr MX 1 t 2
dtr
μ
a
x
xretxf 1 x 2 , X discrete
xretxf 1 x 2 dx, X continuous
The moment generating functionof the random variable Xis the expected value of
etXand is denoted by MX(t). That is,
MX 1 t 2 E 1 etX 2 μ (S5-8)
a
x
etx f 1 x 2 , X discrete
etxf 1 x 2 dx, X continuous
Definition
Let Xbe a random variable with moment generating function MX(t). Then
¿r (S5-9)
dr MX 1 t 2
dtr
`
t 0
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