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128 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

(b) Determine x such that the probability that you wait less 4-89. Continuation of Exercise 4-88.
than x minutes is 0.90. (a) If 30 separate one-hour intervals are chosen, what is the
(c) Determine x such that the probability that you wait less probability that no interval contains more than three arrivals?
than x minutes is 0.50. (b) Determine the length of an interval of time (in hours) such
4-83. The distance between major cracks in a highway fol- that the probability that no arrivals occur during the inter-
lows an exponential distribution with a mean of 5 miles. val is 0.10.
(a) What is the probability that there are no major cracks in a 4-90. The time between calls to a corporate office is expo-
10-mile stretch of the highway? nentially distributed with a mean of 10 minutes.
(b) What is the probability that there are two major cracks in (a) What is the probability that there are more than three calls
a 10-mile stretch of the highway? in one-half hour?
(c) What is the standard deviation of the distance between (b) What is the probability that there are no calls within one-
major cracks? half hour?
4-84. Continuation of Exercise 4-83. (c) Determine x such that the probability that there are no
(a) What is the probability that the first major crack occurs calls within x hours is 0.01.
between 12 and 15 miles of the start of inspection? 4-91. Continuation of Exercise 4-90.
(b) What is the probability that there are no major cracks in (a) What is the probability that there are no calls within a two-
two separate 5-mile stretches of the highway? hour interval?
(c) Given that there are no cracks in the first 5 miles in- (b) If four nonoverlapping one-half hour intervals are se-
spected, what is the probability that there are no major lected, what is the probability that none of these intervals
cracks in the next 10 miles inspected? contains any call?
4-85. The lifetime of a mechanical assembly in a vibration (c) Explain the relationship between the results in part (a)
test is exponentially distributed with a mean of 400 hours. and (b).
(a) What is the probability that an assembly on test fails in 4-92. If the random variable X has an exponential distribu-

less than 100 hours? tion with mean , determine the following:
(b) What is the probability that an assembly operates for more (a) P1X 2 (b) P1X 2 2
than 500 hours before failure? (c) P1X 3 2
(c) If an assembly has been on test for 400 hours without a fail- (d) How do the results depend on ?
ure, what is the probability of a failure in the next 100 hours?
4-93. Assume that the flaws along a magnetic tape follow a
4-86. Continuation of Exercise 4-85. Poisson distribution with a mean of 0.2 flaw per meter. Let X
(a) If 10 assemblies are tested, what is the probability that at denote the distance between two successive flaws.
least one fails in less than 100 hours? Assume that the as- (a) What is the mean of X?
semblies fail independently. (b) What is the probability that there are no flaws in 10 con-
(b) If 10 assemblies are tested, what is the probability that all secutive meters of tape?
have failed by 800 hours? Assume the assemblies fail (c) Does your answer to part (b) change if the 10 meters are
independently. not consecutive?
4-87. When a bus service reduces fares, a particular trip (d) How many meters of tape need to be inspected so that the
from New York City to Albany, New York, is very popular. probability that at least one flaw is found is 90%?
A small bus can carry four passengers. The time between calls 4-94. Continuation of Exercise 4-93. (More difficult ques-
for tickets is exponentially distributed with a mean of 30 min- tions.)
utes. Assume that each call orders one ticket. What is the prob- (a) What is the probability that the first time the distance be-
ability that the bus is filled in less than 3 hours from the time tween two flaws exceeds 8 meters is at the fifth flaw?
of the fare reduction? (b) What is the mean number of flaws before a distance be-
4-88. The time between arrivals of small aircraft at a county tween two flaws exceeds 8 meters?
airport is exponentially distributed with a mean of one hour. 4-95. Derive the formula for the mean and variance of an
What is the probability that more than three aircraft arrive exponential random variable.
within an hour?

4-10 ERLANG AND GAMMA DISTRIBUTIONS

4-10.1 Erlang Distribution

An exponential random variable describes the length until the first count is obtained in a
Poisson process. A generalization of the exponential distribution is the length until r counts