Applied Statistics and Probability for Engineers

5-11

Let. Then dx du, and this last expression above becomes

Now the integral is just the total area under a standard normal density, which is 1, so the mo-

ment generating function of a normal random variable is

Differentiating this function twice with respect to tand setting t0 in the result, we find

Therefore, the variance of the normal random variable is

Moment generating functions have many important and useful properties. One of the

most important of these is the uniqueness property.That is, the moment generating function

of a random variable is unique when it exists, so if we have two random variables Xand Y, say,

with moment generating functions MX(t) and MY(t), then if MX(t)MY(t) for all values of t,

both Xand Yhave the same probability distribution. Some of the other useful properties of the

moment generating function are summarized as follows.

^2 ¿ 2 ^2 ^2 ^2 ^2 ^2

dMX 1 t 2

dt

`

t 0

¿ 1  and

d^2 MX 1 t 2

dt^2

`

t 0

¿ 2 ^2 ^2

MX 1 t 2 et

(^2) t (^2
2)
MX 1 t 2 et
(^2) t (^2
2)
(^) 


1
12 
eu
(^2
2)
du
u 3 x 1 t^224 
If Xis a random variable and ais a constant, then
(1)
(2)
If X 1 , X 2 ,, Xnare independent random variables with moment generating functions
(t), (t),... , (t), respectively, and if YX 1 X 2 Xn, then the mo-
ment generating function of Yis
(3) MY 1 t 2 MX 11 t 2 MX 21 t 2 pMXn 1 t 2 (S5-10)
MX p
MX 1 MX 2 n
p
MaX 1 t 2 MX 1 at 2
MXa 1 t 2 eatMX 1 t 2
Properties of
Moment
Generating
Functions
Property (1) follows from. Property (2)
follows from. Consider property (3) for the case
where the X’s are continuous random variables:

 

   



   

p



   

e

t 1 x 1 x 2 pxn (^2) f 1 x
1 , x 2 ,p, xn^2 dx 1 dx 2 pdxn
MY 1 t 2 E 1 etY 2 E 3 et^1 X^1 X^2 
pXn 2
4
MaX 1 t 2 E 3 et^1 aX^24 E 3 e^1 at^2 X 4 MX 1 at 2
MXa 1 t 2 E 3 et^1 Xa^24 eatE 1 etX 2 eatMX 1 t 2
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