Page 154 - Applied Statistics And Probability For Engineers

c04.qxd 5/10/02 5:20 PM Page 132 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

132 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

2.0
r λ
1 1
1.6 8.3 2
7.5 3.75

1.2
f (x)

0.8

0.4

0.0
0 2 4 6 8 10 12
x
Figure 4-26 Gamma probability density functions

for selected values of and r.

the gamma distribution in which 1 2 and r equals one of the values 1 2, 1, 3 2, 2, p . This
distribution is used extensively in interval estimation and tests of hypotheses that are discussed
in subsequent chapters.

EXERCISES FOR SECTION 4-10
4-96. Calls to a telephone system follow a Poisson distribu- (b) What is the probability that the time until the third failure
tion with a mean of five calls per minute. exceeds 50,000 hours?
(a) What is the name applied to the distribution and parame- 4-100. In a data communication system, several messages
ter values of the time until the tenth call? that arrive at a node are bundled into a packet before they
(b) What is the mean time until the tenth call? are transmitted over the network. Assume the messages ar-
(c) What is the mean time between the ninth and tenth calls? rive at the node according to a Poisson process with 30
4-97. Continuation of Exercise 4-96. messages per minute. Five messages are used to form a
(a) What is the probability that exactly four calls occur within packet.
one minute? (a) What is the mean time until a packet is formed, that is, un-
(b) If 10 separate one-minute intervals are chosen, what is the til five messages arrived at the node?
probability that all intervals contain more than two calls? (b) What is the standard deviation of the time until a packet is
4-98. Raw materials are studied for contamination. Suppose formed?
that the number of particles of contamination per pound of (c) What is the probability that a packet is formed in less than
material is a Poisson random variable with a mean of 0.01 par- 10 seconds?
ticle per pound. (d) What is the probability that a packet is formed in less than
(a) What is the expected number of pounds of material re- 5 seconds?
quired to obtain 15 particles of contamination? 4-101. Errors caused by contamination on optical disks oc-
(b) What is the standard deviation of the pounds of materials cur at the rate of one error every 10 5 bits. Assume the errors
required to obtain 15 particles of contamination? follow a Poisson distribution.
4-99. The time between failures of a laser in a cytogenics ma- (a) What is the mean number of bits until five errors occur?
chine is exponentially distributed with a mean of 25,000 hours. (b) What is the standard deviation of the number of bits until
(a) What is the expected time until the second failure? five errors occur?