Applied Statistics and Probability for Engineers

82 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

obtain rsuccesses. That is, the number of successes is predetermined, and the number of trials

is random. In this sense, a negative binomial random variable can be considered the opposite,

or negative, of a binomial random variable.

The description of a negative binomial random variable as a sum of geometric random

variables leads to the following results for the mean and variance. Sums of random variables

are studied in Chapter 5.

EXAMPLE 3-25 A Web site contains three identical computer servers. Only one is used to operate the site, and

the other two are spares that can be activated in case the primary system fails. The probability

of a failure in the primary computer (or any activated spare system) from a request for service

is 0.0005. Assuming that each request represents an independent trial, what is the mean num-

ber of requests until failure of all three servers?

Let Xdenote the number of requests until all three servers fail, and let , , and

denote the number of requests before a failure of the first, second, and third servers used,

respectively. Now,. Also, the requests are assumed to comprise independ-

ent trials with constant probability of failure p0.0005. Furthermore, a spare server is not

affected by the number of requests before it is activated. Therefore, Xhas a negative binomial

distribution with p0.0005 and r3. Consequently,

What is the probability that all three servers fail within five requests? The probability is

and

EXERCISES FOR SECTION 3-7

1.249 

10 ^9

1.25 

10 ^10 3.75 

10 ^10 7.49 

10 ^10

0.0005^3 a

3

2

b 0.0005^31 0.9995 2     a

4

2

b 0.0005^31 0.9995 22

P 1 X 52 P 1 X 32     P 1 X 42   P 1 X 52

P 1 X 52

E 1 X 2  (^3) 0.0005 6000 requests
XX 1 X 2 X 3
X 1 X 2 X 3
If Xis a negative binomial random variable with parameters pand r,
E 1 X 2 rp and 2 V 1 X 2 r 11 p (^2) p^2 (3-12)
3-71. Suppose the random variable Xhas a geometric
distribution with p0.5. Determine the following proba-
bilities:
(a) (b)
(c) (d)
(e)P 1 X 22
P 1 X 82 P 1 X 22
P 1 X 12 P 1 X 42
3-72. Suppose the random variable Xhas a geometric
distribution with a mean of 2.5. Determine the following
probabilities:
(a) (b)
(c) (d)
(e)P 1 X 32
P 1 X 52 P 1 X 32
P 1 X 12 P 1 X 42
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