CEE 110 PS4 11 Examples

CEE 110 PS4 11 Examples - UNIVERSITY OF CALIFORNIA, LOS...

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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 12-2 PM Franz 1178 Prof. K. D. Stolzenbach 5732J Boelter Hall, 206-7624 [email protected] Problem Set 4 Examples _____________________________________________________________________________ 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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