**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 12-2 PM
Franz 1178 Prof. K. D. Stolzenbach
5732J Boelter Hall, 206-7624
[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
Type-I 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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