Page 104 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 82

82 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

obtain r successes. 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.

If X is a negative binomial random variable with parameters p and r,

2
E1X2 r p and V1X2 r11 p2 p 2 (3-12)

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 X denote the number of requests until all three servers fail, and let X 1 , X 2 , and X 3
denote the number of requests before a failure of the first, second, and third servers used,
respectively. Now, X X X X 3 . Also, the requests are assumed to comprise independ-
1
2
ent trials with constant probability of failure p 0.0005. Furthermore, a spare server is not
affected by the number of requests before it is activated. Therefore, X has a negative binomial
distribution with p 0.0005 and r 3. Consequently,
E1X2 3 0.0005 6000 requests

What is the probability that all three servers fail within five requests? The probability is
P1X 52 and

P1X 52 P1X 32 P1X 42 P1X 52
3 4
3 3 3 2
0.0005 a b 0.0005 10.99952 a b 0.0005 10.99952
2 2
1.25
10 10 3.75
10 10 7.49
10 10
1.249
10 9

EXERCISES FOR SECTION 3-7
3-71. Suppose the random variable X has a geometric 3-72. Suppose the random variable X has a geometric
distribution with p 0.5. Determine the following proba- distribution with a mean of 2.5. Determine the following
bilities: probabilities:
(a) P1X 12 (b) P1X 42 (a) P1X 12 (b) P1X 42
(c) P1X 82 (d) P1X 22 (c) P1X 52 (d) P1X 32
(e) P1X 22 (e) P1X 32