Page 140 - Applied statistics and probability for engineers

118 Chapter 4/Continuous Random Variables and Probability Distributions

EXERCISES FOR SECTION 4-5

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
4-50. Suppose that X has a continuous uniform distribu- (b) Mean and variance of the distribution in the previous part
tion over the interval [1.5, 5.5]. Determine the following: (c) Probability that a visitor waits less than 10 minutes for a show
(a) Mean, variance, and standard deviation of X (d) Probability that a visitor waits more than 20 minutes for a show
(
(b) P X < 2 5. ). (c) Cumulative distribution function 4-58. The volume of a shampoo illed into a container is uni-
4-51. Suppose X has a continuous uniform distribution formly distributed between 374 and 380 milliliters.
⎡
over the interval −1 1, ⎦ ⎤. Determine the following: (a) What are the mean and standard deviation of the volume of
⎣
(a) Mean, variance, and standard deviation of X shampoo?
(b) Value for x such that P − ( x < X < x ) = 0 90. (b) What is the probability that the container is illed with less
(c) Cumulative distribution function than the advertised target of 375 milliliters?
4-52. The net weight in pounds of a packaged chemical (c) What is the volume of shampoo that is exceeded by 95% of
herbicide is uniform for 49 75. < < 50 25 pounds. Determine the containers?
.
x
the following: (d) Every milliliter of shampoo costs the producer $0.002. Any
(a) Mean and variance of the weight of packages shampoo more than 375 milliliters in the container is an
extra cost to the producer. What is the mean extra cost?
(b) Cumulative distribution function of the weight of packages
(
(c) P X < 50 1. ) 4-59. An e-mail message will arrive at a time uniformly dis-
tributed between 9:00 a.m. and 11:00 a.m. You check e-mail at
4-53. The thickness of a lange on an aircraft component 9:15 a.m. and every 30 minutes afterward.
is uniformly distributed between 0.95 and 1.05 millimeters. (a) What is the standard deviation of arrival time (in minutes)?
Determine the following: (b) What is the probability that the message arrives less than 10
(a) Cumulative distribution function of lange thickness
minutes before you view it?
(b) Proportion of langes that exceeds 1.02 millimeters
(c) What is the probability that the message arrives more than
(c) Thickness exceeded by 90% of the langes 15 minutes before you view it?
(d) Mean and variance of lange thickness 4-60. Measurement error that is continuous and uniformly
4-54. Suppose that the time it takes a data collection oper- distributed from –3 to +3 millivolts is added to a circuit’s true
ator to ill out an electronic form for a database is uniformly voltage. Then the measurement is rounded to the nearest mil-
between 1.5 and 2.2 minutes. livolt so that it becomes discrete. Suppose that the true voltage
(a) What are the mean and variance of the time it takes an oper- is 250 millivolts.
ator to ill out the form? (a) What is the probability mass function of the measured voltage?
(b) What is the probability that it will take less than two min- (b) What are the mean and variance of the measured voltage?
utes to ill out the form? 4-61. A beacon transmits a signal every 10 minutes (such as
(c) Determine the cumulative distribution function of the time 8:20, 8:30, etc.). The time at which a receiver is tuned to detect
it takes to ill out the form. the beacon is a continuous uniform distribution from 8:00 a.m.
4-55. The thickness of photoresist applied to wafers in to 9:00 a.m. Consider the waiting time until the next signal from
semiconductor manufacturing at a particular location on the the beacon is received.
wafer is uniformly distributed between 0.2050 and 0.2150 (a) Is it reasonable to model the waiting time as a continuous
micrometers. Determine the following: uniform distribution? Explain.
(a) Cumulative distribution function of photoresist thickness (b) What is the mean waiting time?
(b) Proportion of wafers that exceeds 0.2125 micrometers in (c) What is the probability that the waiting time is less than
photoresist thickness 3 minutes?
(c) Thickness exceeded by 10% of the wafers 4-62. An electron emitter produces electron beams with
(d) Mean and variance of photoresist thickness changing kinetic energy that is uniformly distributed between
4-56. An adult can lose or gain two pounds of water in the course three and seven joules. Suppose that it is possible to adjust the
of a day. Assume that the changes in water weight are uniformly upper limit of the kinetic energy (currently set to seven joules).
distributed between minus two and plus two pounds in a day. (a) What is the mean kinetic energy?
What is the standard deviation of a person’s weight over a day? (b) What is the variance of the kinetic energy?
4-57. A show is scheduled to start at 9:00 a.m., 9:30 a.m., (c) What is the probability that an electron beam has a kinetic
and 10:00 a.m. Once the show starts, the gate will be closed. energy of exactly 3.2 joules?
A visitor will arrive at the gate at a time uniformly distributed (d) What should be the upper limit so that the mean kinetic
between 8:30 a.m. and 10:00 a.m. Determine the following: energy increases to eight joules?
(a) Cumulative distribution function of the time (in minutes) (e) What should be the upper limit so that the variance of
between arrival and 8:30 a.m. kinetic energy decreases to 0.75 joules?