Applied Statistics and Probability for Engineers

5-12

Since the X’s are independent,

and one may write

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 1 and X 2 are two independent Poisson random variables with parameters  1 and

 2 , respectively. Find the probability distribution of YX 1 X 2.

The moment generating function of a Poisson random variable with parameter is

so the moment generating functions of X 1 and X 2 are and

respectively. Using Equation S5-10, we find that the moment generating function of

YX 1 X 2 is

which is recognized as the moment generating function of a Poisson random variable with pa-

rameter  1  2. Therefore, we have shown that the sum of two independent Poisson random

variables with parameters  1 and  2 is a Poisson random variable with parameters equal to the

sum of the two parameters  1  2.

EXERCISES FOR SECTION 5-9

MY 1 t 2 MX 11 t 2 MX 21 t 2 e^11 e

t 12

e^21 e

t 12

e 1 ^1 ^221 e

t 12

MX 21 t 2 e^21 e

t 12

MX 11 t 2 e^11 e ,

t 12

MX 1 t 2 e^1 e

t 12

MX 1 1 t 2  MX 21 t 2 pMXn 1 t 2

MY 1 t 2  





etx^1 fX 11 x 12 dx (^1) 


e
tx (^2) f

X 21 x 22 dx 2 p^ 





e

txn (^) f
Xn^1 xn^2 dxn
f 1 x 1 , x 2 ,p, xn 2 fX 11 x 12 fX 21 x 22 pfXn 1 xn 2
S5-13. A random variable Xhas the discrete uniform distri-
bution
(a) Show that the moment generating function is
(b) Use MX(t) to find the mean and variance of X.
S5-14. A random variable Xhas the Poisson distribution
MX 1 t 2 
et^11 e tm 2
m 11 et 2
f 1 x 2 
1
m,^ x1,^ 2,p, m
(a) Show that the moment generating function is
(b) Use MX(t) to find the mean and variance of the Poisson
random variable.
S5-15. The geometric random variable Xhas probability
distribution
f 1 x 2  11 p 2 x^1 p, x1, 2,p
MX 1 t 2 e^1 e
t 12
f 1 x 2 
ex
x!
, x0, 1,p
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