homework7-2011-key

homework7-2011-key - Homework 7.• Homework must be...

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Unformatted text preview: Homework 7.• Homework must be answered in the order shown here (else please make a note telling the reader where it is). • Write your answers neatly • Work must be shown for full credit • No late homework accepted under any circumstances • Staple the homework 1. Particles independently arrive at a location at an average rate of five per minute. Let X be a random variable modeling the number of particles arriving at the location in any 15 ­ minute period and let Y be a random variable modeling the waiting time for the next arrival. (a) What are reasonable random models for these variables? Why? X is a discrete random variable, perhaps best modeled using a Poisson random variable with parameter λ1 = 5×15 = 75. Y is a continuous random variable, perhaps best modeled using an exponential random variable with arrival rate λ2 = 5. (b) Using these models find the probability that there will arrive four particles in any particular 15 ­minute period. Show work. (c ) Using these models find the probability that it takes between 4 and 6 minutes for the next particle to arrive. Show work. 2. ­A piece of equipment costing $10,000 has an exponentially distributed lifetime with an average of 5 years. If the equipment fails during the first year, the manufacturer agrees to give a full refund. If the equipment fails between the first and the second year, the manufacturer agrees to give a half ­price refund. Finally, if the equipment fails after the second year but before the fifth year, the manufacturer will refund $1000. If the manufacturer sells 10 of these pieces of equipment, how much should it expect to pay in refunds? Show work. Let X be the lifetime of the equipment, and Y be the refund paid by the manufacturer for one piece of equipment. Then 1 So for 10 pieces of equipment, the manufacturer should expect to pay about $28,572 in refunds. 3. ­ The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter l = 1/2. What is (a) The probability that a repair time exceeds 2 hours? (b) The conditional probability that a repair takes at least 10 hours, given that its duration exceeds 9 hours? 4.- Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with parameter 1/20. Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability that she would get at least 20,000 additional miles out of it? Repeat under the assumption that the lifetime mileage of the car is not exponentially distributed but rather is (in thousands of miles) uniformly distributed over (0,40). 2 5.- If X is uniformly distributed over (0,1) , find the density function of Y=eX. 6.-Verify that the gamma density function integrates to 1. The gamma is defined as: We need to show that it integrates to 1, i.e., or equivalently This is achieved by changing the variables and hence 7 . A Weibull distribution with α = 5 and β= 125 is suggested as a population model for fracture strength of silicon nitride braze joints. f(x) = (α/βα)xα ­1exp( ­(x/β)α) for x > 0, and 0 otherwise. a.- What are the quartiles of this distribution and what is the value of the IQR? 3 The first quartile is the value of x that leaves 0.25 of the area under the curve below it. So it will be the value q1 such that Doing similar operations, q3= 133.4 So IQR= 133.4-97.43 = 36.01 b.- Suppose that the value of β is changed to 12.5. Determine the values of the quartiles and the value of the IQR. Note: In essense, this amounts to dividing each observation in the population distribution by 10, because β is a “scale” parameter and changing its value stretches or compresses the x scale without changing the shape of the distribution. Ql==9.74 Qu=13.34 IQR=3.6 4 8.- If 65 percent of the population of a large community is in favor of a proposed rise in school taxes, approximate the probability that a random sample of 100 people will contain (a) at least 50 who are in favor of the proposition (b) between 60 and 70 inclusive who are in favor (c) fewer than 75 in favor. Let X denote the number in favor. Then X is binomial with mean 65 and standard deviation . Also let Z be a standard normal random variable. 9.- The weekly downtime X (in hours) for a certain industrial machine has approximately a gamma distribution with α=3.5 and λ =1.5. The loss L (in dollars) to the industrial operation as a result of this downtime is given by L=30X + 2X2 Find the expected value and the variance of L. Show work. Note that So , , , Then 5 10. ­Adam, Beth, Carlos, and Donna are running a one ­mile run. Their completion times in minutes are uniformly distributed in the following intervals: Adam’s time is A ~ Uniform( 8.3; 10.2) Beth’s time is B~ U(8.9; 10.5) Carlo’s time is C ~ U(8.1; 10.2) Donna’s time is D ~ U(8.4; 10.7) (a) Find the probability that the earliest completion time is less than 10 minutes. (b) Find the probability that the latest completion time is less than 10 minutes 6 ...
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This document was uploaded on 03/20/2011.

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