Prob_Stat Oct 30

# Prob_Stat Oct 30 - Dealing with uncertainty in design and...

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ME 250 Fall 2007 Dealing with uncertainty in design and manufacturing Applications of probability and statistics October 30, 2007 ME 250 Fall 2007 Coursepack Chapter: Uncertainty in design & manufacturing Sections: 1, 2, 3, 4 and 9

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ME 250 Fall 2007 Auto warranty wars 10 YEAR / 100,000 MILES Covers repair or replacement of powertrain components (i.e. selected Engine and Transmission/Transaxle components), originally manufactured or installed by Hyundai that are defective in material or factory workmanship, under normal use and maintenance. Sept 2006: Extension of GM's warranty. Coverage of the powertrain expands from 5 years/60,000 miles to 5 years/100,000 miles…. ME 250 Fall 2007 “US Auto Warranty Costs Soar,” Ed Garsten, Detroit News , 9/14/04 Auto companies incur ~\$12 Billion annually to fix vehicles under warranty A typical recall can cost the OEM on average \$1M a day Warranty costs shave 1-3% off revenues according to AMR Research Critical issue for OEM: Controlling costs
ME 250 Fall 2007 Reliability Reliability is a measure of the ability of a product / system / service to perform its intended function under a prescribed set of conditions Reliability is a “probability” ME 250 Fall 2007 Reliability and probability Reliability: the probability that a product, service, component or system will perform its intended function under a prescribed set of conditions If a powertrain has 95% reliability then the probability that it will fail within 100,000 miles (warranty) is 5%. In other words, five out of every 100 warranty will cost the OEM \$\$\$ to repair

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ME 250 Fall 2007 Probability Example: ¾ Toss two dice and get a 7 Probability that a certain outcome may occur can be determined by a) Identifying all possible outcomes 2,3,4,5,6,7,3,4,5,6,7,8,4,5,6,7,8,9,5,6,7,8,9,10,6,7,8,9,10,11, 7,8,9,10,11,12 b) Counting all possible outcomes Æ 36 ME 250 Fall 2007 Probability d) Probability of occurrence (of the event) is m 1,2,. .., j outcomes all of number occurrence of times of number = = = N n P j j n j = number of times of occurrence of a particular event N = total number of all outcomes m = number of events e) For the case of rolling the total number of 7 the probability is 6 1 36 6 outcomes all of number occurrence of times of number = = = j P
ME 250 Fall 2007 Probability Distribution A probability distribution is a collection of probabilities for all outcomes and can be represented in a table or graph. Also called a frequency function or probability density . ¾ Note all probabilities should sum to 1 Notice the lower case “f” 0 1/36 1/18 1/12 1/9 5/36 1/6 1234567891 01 11 2 Probability = f(x) R a n d o m V a r i a b l e , x 0 1 1 1 2 T o t a l Probability = f(x) 0 1/36 1/18 1/12 1/9 5/36 1/6 5/36 1/9 1/12 1/18 1/36 1 1 1 = = m j j P ME 250 Fall 2007 Cumulative Probability Function or Distribution Cumulative Probability function or distribution F(x i ) is defined as: Notice the capital F R a n d o m V a r i a b l e , x1234567891 0 1 1 1 2 Cumulative Probability Function, F (x i ) 0 1/36 1/12 1/6 5/18 5/12 7/12 13/18 5/6 11/12 35/36 1 0 1/12 1/6 1/4 1/3 5/12 1/2 7/12 2/3 3/4 5/6 11/12 1 0 1 1 1 2 Cumulative Probability Function, F (x i) < = i j x x j i x f x F ) ( ) (

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ME 250 Fall 2007 Recap reliability If a transmission has a 99% reliability
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## This note was uploaded on 04/04/2008 for the course ME 250 taught by Professor Dutta during the Fall '07 term at University of Michigan.

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Prob_Stat Oct 30 - Dealing with uncertainty in design and...

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