Applied Statistics and Probability for Engineers

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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