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5-12

Since the X’s are independent,

1x 2 2 p 1x n 2
f 1x 1 , x 2 , p , x n 2 f X 1 1x 1 2 f X 2 f X n
and one may write

tx 1 tx 2 p tx n
M 1t2 e f 1x 2 dx e f 1x 2 dx e f 1x 2 dx n
2
n
1
Y X 1 1 X 2 2 X n

M 1t2 M 1t2 p M 1t2
X 1 X 2 X n
For the case when the X’s are discrete, we would use the same approach replacing integrals
with summations.
Equation S5-10 is particularly useful. In many situations we need to find the distribution
of the sum of two or more independent random variables, and often this result makes the prob-
lem very easy. This is illustrated in the following example.

EXAMPLE S5-7 Suppose that X and X are two independent Poisson random variables with parameters and
1
1
2
, respectively. Find the probability distribution of Y X X .
1
2
2
The moment generating function of a Poisson random variable with parameter is
t
1e 12
M 1t2 e
X
t
t
so the moment generating functions of X 1 and X 2 are M 1t2 e 1 1e 12 and M 1t2 e 2 1e 12 ,
X 1 X 2
respectively. Using Equation S5-10, we find that the moment generating function of
Y X 1 X 2 is
t
t
t
e
M 1t2 M 1t2 M 1t2 e 1 1e 12 2 1e 12 e 1 1 2 21e 12
Y
X 2
X 1
which is recognized as the moment generating function of a Poisson random variable with pa-
. Therefore, we have shown that the sum of two independent Poisson random
rameter 1 2
variables with parameters and is a Poisson random variable with parameters equal to the
1
2
sum of the two parameters 2 .
1
EXERCISES FOR SECTION 5-9
x
S5-13. A random variable X has the discrete uniform distri- e
f 1x2 , x 0, 1, p
bution x !
1 (a) Show that the moment generating function is
f 1x2 , x 1, 2, p , m
m
t
M X 1t 2 e 1e 12
(a) Show that the moment generating function is
(b) Use M X (t) to find the mean and variance of the Poisson
tm
e 11 e 2 random variable.
t
M X 1t 2 t
m11 e 2 S5-15. The geometric random variable X has probability
distribution
(b) Use M X (t) to find the mean and variance of X.
f 1x2 11 p2 x 1 p, x 1, 2, p
S5-14. A random variable X has the Poisson distribution