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Unformatted text preview: UNIVERSITY OF CALIFORNIA, LOS ANGELES
Civil and Environmental Engineering Department
CEE 110 Introduction to Probability and Statistics for Engineers
Spring Quarter 2011
MW 122 PM
Franz 1178 Prof. K. D. Stolzenbach
5732J Boelter Hall, 2067624
[email protected] Problem Set 4 Examples
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1. A light fixture holds 2 types of light bulbs. Bulb A is the type that has a lifetime that is
normally distributed with mean 800 hours and standard deviation 100 hours. Bulb B has a
lifetime that is normally distributed with mean 900 hours and standard deviation 150 hours.
Assume the lifetimes of the bulbs are independent.
(a) What is the probability that bulb B lasts longer than bulb A? [0.7088]
(b) Another light fixture holds only one bulb. A bulb of type A is installed and when it
burns out a bulb of type B is installed. What is the probability that the total lifetime of
the two bulbs is more than 2000 hours? [0.0485] 2. The annual maximum runoff Q of a certain river can be modeled with a lognormal
distribution. The observed mean and standard deviation of the flow are µ = 400 cfs and
σ = 200 cfs. What is the probability of the flow exceeding 500 cfs in a given year? [0.24] 3. A manufacturer claims that the tensile strength of a certain composite has the lognormal
distribution with µ = 150 MPa and σ = 90 MPa. Let X be the strength of a randomly
sampled specimen of this composite.
(a) If the claim is true, what is the probability that X < 130 MPa? [0.504]
(b) If you observe a tensile strength of 20 MPa, would this be convincing evidence that the
claim is false? [P(X 20 MPa = 0.0003, so it is unlikely the claim is true] 4. The breaking strength R of a cable can be assumed to be lognormally distributed with a
mean value of R of 100 kip and a standard deviation of 30 kip.
(a) If a load P of magnitude 80 kip is hung from the cable, what is the probability of failure
of the cable? [0.28]
(b) If P can not be determined with certainty, but has a 30% probability of being 50 kips and
a 70% probability of being 80 kips, what is the probability of failure of the cable? [0.20]
(c) If the cable breaks, what is the probability that the load was 50 kips? [0.023] 5. Suppose the life of an integrated circuit can be modeled by an exponential distribution with
an average lifetime of 3 years.
(a) What is the probability that a new circuit lasts longer than 4 years? [0.26]
(b) If the circuit is now 5 years old and is still functioning, what is the probability that it
functions for 4 more years? [0.26] 6. Suppose the life of a light bulb can be modeled by an exponential distribution with an
average lifetime of 12 months. Suppose a maintenance worker checks the bulbs every 6
months.
(a) What is the probability that a newly installed light bulb will need to be replaced at the
next scheduled inspection? [0.39]
(b) If the lightbulb is in good condition during the first scheduled inspection, what is the
probability that it will be in good condition at the next scheduled inspection? [0.61]
(c) If there are 10 light bulbs in a room, what is the probability that at least one of them
needs replacement at the first scheduled inspection? [0.993] 7. Of 50 buildings in an industrial park, 12 have electrical code violations.
(a) If 10 buildings are selected at random for inspection, calculate the probability that
exactly 3 of the 10 have code violations using the hypergeometric distribution. [0.27]
(b) The result in part (a) assumes that the probability of an electrical code violation is
known. If the 12 violations out of 50 buildings is considered to be just a result based on
one sample of 50 buildings, calculate the probability that exactly 3 of the 10 have code
violations using the hyperbinomial distribution. [0.23] 8. In a large reinforced concrete construction project, the contract requires that no more than
2% of the concrete delivered is defective. The acceptance /rejection criterion requires that at
least 9 of 10 cylinders selected at random from a total of 100 collected each day must have
a required minimum strength. Is this testing procedure adequate to detect an if the actual
percentage of poor mixes is 2%? [Probability of detecting 2% defects is 0.0091] 9. The daily water consumption of a city may be assumed to be normally distributed with a
mean of 600,000 gal/day (gpd) and a standard deviation of 200,000 gpd. The daily water
supply is either 750,000 gallons or 900,000 gallons with probabilities 0.8 and 0.2,
respectively.
(a) What is the probability of a water shortage on a given day? [0.195]
(b) Assuming that conditions between days are statistically independent, what is the
probability of at least one shortage in any given week? [0.781]
(c) On the average, how often would a shortage occur? [5.13 days]
(d) If the occurrence of a water shortage is a Poisson process, what would be the probability
of at least one shortage in a week? [0.745 – compare to answer in (b)]
(e) If the city engineer wants the probability of a shortage to be no more than 0.01 on any
given day, how much water supply is required? [1,065,000 gpd] 10. The annual maximum stage height in a river channel is modeled using a Type I extreme
value distribution of the largest value with a mean of 30 ft and a coefficient of variance
(COV) of 10%. The stage height at which flooding will occur is 40 ft. What is the
probability of flooding each year? [0.0078] 11. The maximum daily gasoline demand during the month of May in Los Angeles follows the
TypeI maximum value distribution with µ = 3000 gallons and σ = 2000 gallons.
(a) Determine the probability that the demand will exceed 5000 gallons in any day during
the month of May. [0.145]
(b) What is the supply level (gallons per day) that has a 95% probability of not being
exceeded by demand in any given day in May. [6739 gallons] SOLUTIONS
1. (b) 2. 3. (a) (b) 4. (a) (b) (c) 5. (a) (b) 6. (a) (b) (c)
7. (a) (b) 8. 6. (a) (b) (c) (d) (e) 10. , and, , 11. (a) (b) ...
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This note was uploaded on 10/22/2011 for the course ENG 101 taught by Professor Yukov during the Spring '11 term at UCLA.
 Spring '11
 Yukov
 Environmental Engineering

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