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5-11

2
Let u 3x 1 t 24
. Then dx du, and this last expression above becomes

2 2
2
M 1t2 e t t
2 1 e u
2 du
X 12

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

2 2
M 1t2 e t t
2
X
Differentiating this function twice with respect to t and setting t 0 in the result, we find

2
dM X 1t2 d M X 1t2 2 2
` ¿ and 2 ` ¿
2
1
dt t 0 dt t 0
Therefore, the variance of the normal random variable is
2
2
2
2
2
¿ 2
2
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 X and Y, say,
with moment generating functions M X (t) and M Y (t), then if M X (t) M Y (t) for all values of t,
both X and Y have the same probability distribution. Some of the other useful properties of the
moment generating function are summarized as follows.

Properties of
Moment If X is a random variable and a is a constant, then
Generating
at
Functions (1) M X a 1t2 e M 1t2
X
(2) M aX 1t2 M 1at2
X
If X , X 2 , p , X n are independent random variables with moment generating functions
1
(t), respectively, and if Y X 1 X 2 p X n , then the mo-
M X 1 (t), M X 2 (t), . . . , M X n
ment generating function of Y is
(3) M Y 1t2 M X 1 1t2 M X 2 1t2 p M X n 1t2 (S5-10)

tX
at
at
Property (1) follows from M X a 1t2 E3e t 1X a2 4 e E1e 2 e M 1t2 . Property (2)
X
follows from M 1t2 E3e t 1aX2 4 E3e 1at2X 4 M 1at2 . Consider property (3) for the case
aX X
where the X’s are continuous random variables:
tY
M 1t2 E1e 2 E3e t 1X 1 X 2 p X n 2 4
Y

p e t 1x 1 x 2 p x n 2 f 1x , x , p , x 2 dx dx p dx
1 2 n 1 2 n