Applied Statistics and Probability for Engineers

132 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

the gamma distribution in which and requals one of the values 12, 1, 32, 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

 1 2

Figure 4-26 Gamma probability density functions

for selected values of and  r.

0

0.0

0.4

0.8

1.2

1.6

2.0

2 4 6 8 10 12

x

1

8.3

7.5

1

2

3.75

r λ

f (x)

4-96. Calls to a telephone system follow a Poisson distribu-

tion with a mean of five calls per minute.

(a) What is the name applied to the distribution and parame-

ter values of the time until the tenth call?

(b) What is the mean time until the tenth call?

(c) What is the mean time between the ninth and tenth calls?

4-97. Continuation of Exercise 4-96.

(a) What is the probability that exactly four calls occur within

one minute?

(b) If 10 separate one-minute intervals are chosen, what is the

probability that all intervals contain more than two calls?

4-98. Raw materials are studied for contamination. Suppose

that the number of particles of contamination per pound of

material is a Poisson random variable with a mean of 0.01 par-

ticle per pound.

(a) What is the expected number of pounds of material re-

quired to obtain 15 particles of contamination?

(b) What is the standard deviation of the pounds of materials

required to obtain 15 particles of contamination?

4-99.The time between failures of a laser in a cytogenics ma-

chine is exponentially distributed with a mean of 25,000 hours.

(a) What is the expected time until the second failure?

(b) What is the probability that the time until the third failure

exceeds 50,000 hours?

4-100. In a data communication system, several messages

that arrive at a node are bundled into a packet before they

are transmitted over the network. Assume the messages ar-

rive at the node according to a Poisson process with

messages per minute. Five messages are used to form a

packet.

(a) What is the mean time until a packet is formed, that is, un-

til five messages arrived at the node?

(b) What is the standard deviation of the time until a packet is

formed?

(c) What is the probability that a packet is formed in less than

10 seconds?

(d) What is the probability that a packet is formed in less than

5 seconds?

4-101. Errors caused by contamination on optical disks oc-

cur at the rate of one error every bits. Assume the errors

follow a Poisson distribution.

(a) What is the mean number of bits until five errors occur?

(b) What is the standard deviation of the number of bits until

five errors occur?

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