Page 174 - Applied statistics and probability for engineers

152 Chapter 4/Continuous Random Variables and Probability Distributions

4-202. The size of silver particles in a photographic emulsion (b) What is the mean of the number of grid cells that need to be
is known to have a log normal distribution with a mean of 0.001 viewed to observe 10 that contain ibers?
mm and a standard deviation of 0.002 mm. (c) What is the standard deviation of the number of grid cells
(a) Determine the parameter values for the lognormal distribution. that need to be viewed to observe 10 that contain ibers?
(b) What is the probability of a particle size greater than 0.005 mm? 4-209. Without an automated irrigation system, the height of
4-203. Suppose that f x ( ) = .0 5 x −1 for 2 < x < 4. Determine plants two weeks after germination is normally distributed with a
the following: mean of 2.5 centimeters and a standard deviation of 0.5 centimeter.
(
. (
(
(a) P X < 2 5. ) (b) P X >3) (c) P 2 5 < X <3 5) (a) What is the probability that a plant’s height is greater than
.
(d) Determine the cumulative distribution function of the ran- 2.25 centimeters?
dom variable. (b) What is the probability that a plant’s height is between 2.0
(e) Determine the mean and variance of the random variable. and 3.0 centimeters?
4-204. The time between calls is exponentially distrib- (c) What height is exceeded by 90% of the plants?
uted with a mean time between calls of 10 minutes. 4-210. With an automated irrigation system, a plant grows to a
(a) What is the probability that the time until the irst call is height of 3.5 centimeters two weeks after germination. Without an
less than ive minutes? automated system, the height is normally distributed with mean
(b) What is the probability that the time until the irst call is and standard deviation 2.5 and 0.5 centimeters, respectively.
between 5 and 15 minutes? (a) What is the probability of obtaining a plant of this height or
(c) Determine the length of an interval of time such that the greater without an automated system?
probability of at least one call in the interval is 0.90. (b) Do you think the automated irrigation system increases the
(d) If there has not been a call in 10 minutes, what is the proba- plant height at two weeks after germination?
bility that the time until the next call is less than 5 minutes? 4-211. The thickness of a laminated covering for a wood
(e) What is the probability that there are no calls in the inter- surface is normally distributed with a mean of ive millimeters
vals from 10:00 to 10:05, from 11:30 to 11:35, and from and a standard deviation of 0.2 millimeter.
2:00 to 2:05? (a) What is the probability that a covering thickness is more
(f) What is the probability that the time until the third call is than 5.5 millimeters?
greater than 30 minutes? (b) If the speciications require the thickness to be between 4.5
(g) What is the mean time until the ifth call? and 5.5 millimeters, what proportion of coverings does not
4-205. The CPU of a personal computer has a lifetime meet speciications?
that is exponentially distributed with a mean lifetime of six (c) The covering thickness of 95% of samples is below what
years. You have owned this CPU for three years. value?
(a) What is the probability that the CPU fails in the next three 4-212. The diameter of the dot produced by a printer is
years? normally distributed with a mean diameter of 0.002 inch.
(b) Assume that your corporation has owned 10 CPUs for three (a) Suppose that the speciications require the dot diameter to
years, and assume that the CPUs fail independently. What be between 0.0014 and 0.0026 inch. If the probability that
is the probability that at least one fails within the next three a dot meets speciications is to be 0.9973, what standard
years? deviation is needed?
4-206. Suppose that X has a lognormal distribution with (b) Assume that the standard deviation of the size of a dot
2
parameters θ = 0 and ω = 4. Determine the following: is 0.0004 inch. If the probability that a dot meets speci-
(a) P 10 ( < X < 50) ications is to be 0.9973, what speciications are needed?
(
0
(b) Value for x such that P X < x) = .05 Assume that the speciications are to be chosen symmetri-
(c) Mean and variance of X cally around the mean of 0.002.
4-207. Suppose that X has a lognormal distribution and 4-213. The waiting time for service at a hospital emergency
that the mean and variance of X are 50 and 4000, respectively. department follows an exponential distribution with a mean of
Determine the following: three hours. Determine the following:
2
(a) Parameters θ and ω of the lognormal distribution (a) Waiting time is greater than four hours
(b) Probability that X is less than 150 (b) Waiting time is greater than six hours given that you have
already waited two hours
4-208. Asbestos ibers in a dust sample are identiied by an
(c) Value x (in hours) exceeded with probability 0.25
electron microscope after sample preparation. Suppose that the
number of ibers is a Poisson random variable and the mean 4-214. The life of a semiconductor laser at a constant power is
number of ibers per square centimeter of surface dust is 100. A normally distributed with a mean of 7000 hours and a standard
sample of 800 square centimeters of dust is analyzed. Assume deviation of 600 hours.
that a particular grid cell under the microscope represents (a) What is the probability that a laser fails before 5800 hours?
1/160,000 of the sample. (b) What is the life in hours that 90% of the lasers exceed?
(a) What is the probability that at least one iber is visible in (c) What should the mean life equal for 99% of the lasers to
the grid cell? exceed 10,000 hours before failure?