Figure 4-24 Lack of memory property of an exponential distribution.

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A B t2 t1 C D t1 + t 2

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4-9 EXPONENTIAL DISTRIBUTION

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memory property of the exponential distribution implies that the device does not wear out. That is, regardless of how long the device has been operating, the probability of a failure in the next 1000 hours is the same as the probability of a failure in the rst 1000 hours of operation. The lifetime L of a device with failures caused by random shocks might be appropriately modeled as an exponential random variable. However, the lifetime L of a device that suffers slow mechanical wear, such as bearing wear, is better modeled by a dist 0 L t2 increases with t. Distributions such as the Weibull tribution such that P1L t distribution are often used, in practice, to model the failure time of this type of device. The Weibull distribution is presented in a later section.

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EXERCISES FOR SECTION 4-9

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4-72. Suppose X has an exponential distribution with Determine the following: (a) P1X 02 (b) P1X 22 (c) P1X 12 (d) P11 X 22 (e) Find the value of x such that P1X x2 0.05. 2. (b) What is the probability that at least one call arrives within a 10-minute interval (c) What is the probability that the rst call arrives within 5 and 10 minutes after opening (d) Determine the length of an interval of time such that the probability of at least one call in the interval is 0.90. 4-78. The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase an automobile that is six years old, with a working voltage regulator, and plan to own it for six years. (a) What is the probability that the voltage regulator fails during your ownership (b) If your regulator fails after you own the automobile three years and it is replaced, what is the mean time until the next failure 4-79. The time to failure (in hours) of fans in a personal computer can be modeled by an exponential distribution with 0.0003. (a) What proportion of the fans will last at least 10,000 hours (b) What proportion of the fans will last at most 7000 hours 4-80. The time between the arrival of electronic messages at your computer is exponentially distributed with a mean of two hours. (a) What is the probability that you do not receive a message during a two-hour period (b) If you have not had a message in the last four hours, what is the probability that you do not receive a message in the next two hours (c) What is the expected time between your fth and sixth messages 4-81. The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes. (a) What is the probability that you wait longer than one hour for a taxi (b) Suppose you have already been waiting for one hour for a taxi, what is the probability that one arrives within the next 10 minutes 4-82. Continuation of Exercise 4-81. (a) Determine x such that the probability that you wait more than x minutes is 0.10.

4-73. Suppose X has an exponential distribution with mean equal to 10. Determine the following: (a) P1X 102 (b) P1X 202 (c) P1X 302 (d) Find the value of x such that P1X x2 0.95. 4-74. Suppose the counts recorded by a geiger counter follow a Poisson process with an average of two counts per minute. (a) What is the probability that there are no counts in a 30second interval (b) What is the probability that the rst count occurs in less than 10 seconds (c) What is the probability that the rst count occurs between 1 and 2 minutes after start-up 4-75. Suppose that the log-ons to a computer network follow a Poisson process with an average of 3 counts per minute. (a) What is the mean time between counts (b) What is the standard deviation of the time between counts (c) Determine x such that the probability that at least one count occurs before time x minutes is 0.95. 4-76. The time to failure (in hours) for a laser in a cytometry machine is modeled by an exponential distribution with 0.00004. (a) What is the probability that the laser will last at least 20,000 hours (b) What is the probability that the laser will last at most 30,000 hours (c) What is the probability that the laser will last between 20,000 and 30,000 hours 4-77. The time between calls to a plumbing supply business is exponentially distributed with a mean time between calls of 15 minutes. (a) What is the probability that there are no calls within a 30minute interval

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