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138 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

4-127. The time between calls is exponentially distributed (a) What is the probability that a plant’s height is greater than
with a mean time between calls of 10 minutes. 2.25 centimeters?
(a) What is the probability that the time until the first call is (b) What is the probability that a plant’s height is between 2.0
less than 5 minutes? and 3.0 centimeters?
(b) What is the probability that the time until the first call is (c) What height is exceeded by 90% of the plants?
between 5 and 15 minutes? 4-136. Continuation of Exercise 4-135. With an automated
(c) Determine the length of an interval of time such that the irrigation system, a plant grows to a height of 3.5 centimeters
probability of at least one call in the interval is 0.90. two weeks after germination.
4-128. Continuation of Exercise 4-127. (a) What is the probability of obtaining a plant of this height or
(a) If there has not been a call in 10 minutes, what is the proba- greater from the distribution of heights in Exercise 4-135.
bility that the time until the next call is less than 5 minutes? (b) Do you think the automated irrigation system increases
(b) What is the probability that there are no calls in the inter- the plant height at two weeks after germination?
vals from 10:00 to 10:05, from 11:30 to 11:35, and from 4-137. The thickness of a laminated covering for a wood
2:00 to 2:05? surface is normally distributed with a mean of 5 millimeters
4-129. Continuation of Exercise 4-127. and a standard deviation of 0.2 millimeter.
(a) What is the probability that the time until the third call is (a) What is the probability that a covering thickness is greater
greater than 30 minutes? than 5.5 millimeters?
(b) What is the mean time until the fifth call? (b) If the specifications require the thickness to be between
4-130. The CPU of a personal computer has a lifetime that 4.5 and 5.5 millimeters, what proportion of coverings do
is exponentially distributed with a mean lifetime of six years. not meet specifications?
You have owned this CPU for three years. What is the proba- (c) The covering thickness of 95% of samples is below what
bility that the CPU fails in the next three years? value?
4-131. Continuation of Exercise 4-130. Assume that your 4-138. The diameter of the dot produced by a printer is nor-
corporation has owned 10 CPUs for three years, and assume mally distributed with a mean diameter of 0.002 inch.
that the CPUs fail independently. What is the probability that Suppose that the specifications require the dot diameter to be
at least one fails within the next three years? between 0.0014 and 0.0026 inch. If the probability that a dot
4-132. Suppose that X has a lognormal distribution with meets specifications is to be 0.9973, what standard deviation
2
parameters 0 and 4 . Determine the following: is needed?
4-139. Continuation of Exercise 4-138. Assume that the stan-
(a) P110 X 502
(b) The value for x such that P1X x2 0.05 dard deviation of the size of a dot is 0.0004 inch. If the proba-
(c) The mean and variance of X bility that a dot meets specifications is to be 0.9973, what spec-
ifications are needed? Assume that the specifications are to be
4-133. Suppose that X has a lognormal distribution and that
the mean and variance of X are 50 and 4000, respectively. chosen symmetrically around the mean of 0.002.
Determine the following: 4-140. The life of a semiconductor laser at a constant power
(a) The parameters and 2 of the lognormal distribution is normally distributed with a mean of 7000 hours and a stan-

dard deviation of 600 hours.
(b) The probability that X is less than 150
(a) What is the probability that a laser fails before 5,800
4-134. Asbestos fibers in a dust sample are identified by an hours?
electron microscope after sample preparation. Suppose that (b) What is the life in hours that 90% of the lasers exceed?
the number of fibers is a Poisson random variable and the
mean number of fibers per squared centimeter of surface dust 4-141. Continuation of Exercise 4-140. What should the
is 100. A sample of 800 square centimeters of dust is analyzed. mean life equal in order for 99% of the lasers to exceed 10,000
Assume a particular grid cell under the microscope represents hours before failure?
1/160,000 of the sample. 4-142. Continuation of Exercise 4-140. A product contains
(a) What is the probability that at least one fiber is visible in three lasers, and the product fails if any of the lasers fails.
the grid cell? Assume the lasers fail independently. What should the mean
(b) What is the mean of the number of grid cells that need to life equal in order for 99% of the products to exceed 10,000
be viewed to observe 10 that contain fibers? hours before failure?
(c) What is the standard deviation of the number of grid cells 4-143. Continuation of Exercise 140. Rework parts (a) and
that need to be viewed to observe 10 that contain fibers? (b). Assume that the lifetime is an exponential random vari-
4-135. Without an automated irrigation system, the height able with the same mean.
of plants two weeks after germination is normally distributed 4-144. Continuation of Exercise 4-140. Rework parts (a)
with a mean of 2.5 centimeters and a standard deviation of 0.5 and (b). Assume that the lifetime is a lognormal random vari-
centimeters. able with the same mean and standard deviation.