ApproxUncertainty in Engr
ApproxUncertainty in Engr CIVL 3103
University of Memphis
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Date Created: 10/23/15
THE UNIVERSITY OF M E M Dreamers Thinkers Doers CIVL 3103 Continuous Distributions Part II The Normal Distribution THE UNIVERSITY OF M E M Dreamers Thinkers Doers Learning Objectives Continuous Distributions o Define continuous distributions and identify common distributions applicable to engineering problems o Identify the appropriate distribution ie uniform exponential normal for use in solving a problem o Apply continuous distribution models to solve engineeringoriented problems THE UNIVERSITY OF Dreamers Thinkers Doers The Normal Distribution Thenormaldjsm39bution39 39 ofmany 39 39 itdoesaccumtelyquot 39L L 439 39L 39 F 1 r L 4 The normal distribution is bellshaped symmetrical about the mean 4 andranges from m to ea f6 THE UNIVERSITY OF M M Dreamers Thinkers Doers The Normal Distribution The probability density inciion is given by mlt ltm 11x mltiiltm 6v 039 We nyuidr l rihinna 391 l mini X NLH53 This says the random variablc g is normally distributed with mean u and variance 0 3quot The cumulative distribution function is given by Fm THE UNIVERSITY OF M E M Dreamers Thinkers Doers The Normal Distribution autumnwitgiuairmail THE UNIVERSITY OF M M Dreamers Thinkers Doers The Standard Normal Distribution lf39X is a normally disiribuicd random variable wiih mean I and variance 5quot than 7 i1 fn is a normally distributed random variable with zero mean and unit variance in shorthand noiaLiou le vzaquot then 1 N0l quot 7 For convenience wc define ihc Z Sl lislic as 9 we can writs THE UNIVERSITY OF M E M P H Dreamers Thinkers Doers The Standard Normal Distribution HZ THE UNIVERSITY OF Dreamers Thinkers Doers The Standard Normal Distribution 0 lquot J PM E j F717F THE UNIVERSITY OF M E M Dreamers Thinkers Doers Example 1 PZ lt 255 1 PZ gt 126 autumnwitgiuairmail THE UNIVERSITY OF M E M Dreamers Thinkers Doers Example 1 PZgt 137 1 P 125 lt z lt 037 autumnwitgiuairmail Dreamers Thinkers Doers Exa m p le 5n an industrial process the diameter of a ball bearing is an important component part The buyer sets specifications on the diameter to be 30 1 001 cm The implication is that no part falling outside these specifications will be accepted It is known that in the process the diameter of a ball bearing has a normal distribution with mean 30 and standard deviation of 0005 On the average how many manufactured ball bearings will be scrapped THE UNIVERSITY OF M E M Dreamers Thinkers Doers Example 6 Gauges are used to reject all components where a certain dimension is not within the specification 15 1 d It is known that this measurement is normally distributed with mean 150 and standard deviation 02 Determine the value d such that the specifications cover 95 of the measurements THE UNIVERSITY OF M E M Dreamers Thinkers Doers Example The paper Reliability Enhancement of a New Computer ESS Proceedings 1995 describes a program to enhance the reliability of the motherboard in a massively parallel RISC computer system Part of the paper concentrates on a specific parity error that seems to be temperature dependent The temperatures at which the error occurs follow a normal distribution with a mean of 80 C and a standard deviation of 20 C What is the probability that an error occurs at a temperature of less than 77 C What is the probability of an error occurring between 77 C and 83 C Over what range of temperatures will 2 3 of the errors occur What is the probability that an error will occur at a temperature within 25 of the mean 53 33 THE UNIVERSITY OF Dreamers Thinkers Doers Normal Approx of the Binomial We saidcarlicr w 1 v Lhcp L V V ummumcr V 4 l W I ltiaz r 1 w l bctau c Of he sheer number of ms in the summation Since he binomial distribution approaches he V V 0 normal distribution for largc n we can approximate the probability above using the normal distr iution and forget about calculating all Ihasc terms Recall that the mean oflhc binomial dieribuiion is it 7 Up and mo variance is a um lg follows a binomial distribution and n is large actually ifupg is greater than about 10 Him 1 i L up THE UNIVERSITY OF Dreamers Thinkers Doers Functions in Excel You can use Exccl39s Function Wizard lo iinplcmcnl ilic cxponcnual disin uuiion and lhc no disiribulicn Excel can give you hulli um value of me probabilin density ruucuoqu and the value of the cumulative pmbab39iljly distribution F43 Jusi make the lasi argumcnl of he funciian TRUE if you wam FIX and FALSE iryuu warni If you don39t wam lo usc lhc function wizard you can simply iypc me funclions inn 2 cell Just like any other funclion rmal Exponential Dislributiun 1m Exroxmsm lambda FALSE m Exroxnlsnx lambda TRUE Numml Distribulian f r Nommlsnx mean sldey FALSE m Nommlsnx mean mm TR ULquot Excel also leis y ou lmplcmcni hc cumulative standard normal dismbulion The function has jusi one argument which is z Slundard Normal Distribution u NORMSDISTz