Fundamentals of Probability and Statistics for Engineers

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Some Important Continuous Distributions 239Arkin,H.,and Colton,R.1963,Tablesfor Statisticians,2nd edn.,Barnesand Noble,NewYork.Beyer, W.H., 1996, Handbook of Tables for Probability and Statistics, Chemical RubberCo., Cleveland, OH.Hald, A., 1952, Statistical Tables and Formulas, John Wiley & Sons Inc. New York.Owen, D., 1962, Handbook of Statistical Tables, Addision-Wesley, Reading,Pearson, E.S., and Harley, H.O. (eds) 1954, Biometrika Tables for Statisticians, Volume 1,Cambridge University Press, Cambridge, England.Additional useful references include:Aitchison, J., and Brown, J.A.C., 1957, The Log-normal Distribution, CambridgeUniversity Press, Cambridge, England.Harter, H.L., 1964, New Tables of the Incomplete Gamma Function Ratio and of PercentagePoints of the Chi-square and Beta Distributions, Aerospace Laboratory; USGovernment Printing office, Washington, DC.National Bureau of Standards, 1954, Tables of the Bivariate Normal Distribution andRelated Functions: Applied Mathematics Series 50, US Government Printing office,Washington, DC.PROBLEMS7.1 The random variables X and Y are independent and uniformly distributed ininterval (0.1). Determine the probability that their product XY is less than 1/2.7.2 The characteristic function (CF) of a random variable X uniformly distributed in theinterval ( 1, 1) is X …t† ˆsin t :t(a) Find the CF of Y , that is uniformly distributed in interval ( a,a).(b) Find the CF of Y if it is uniformly distributed in interval (a,a ‡ b).7.3 A machine component consisting of a rod-and-sleeve assembly is shown in Figure7.15. Owing to machining inaccuracies, the inside diameter of the sleeve is uniformlydistributed in the interval (1.98cm, 2.02cm), and the rod diameter is also uniformlydistributed in the interval (1.95cm, 2.00cm). Assuming independence of these twodistributions, find the probability that:(a) The rod diameter is smaller than the sleeve diameter.(b) There is at least a 0.01 cm clearance between the rod and the sleeve.SleeveRodFigure 7.15 Rod and sleeve arrangement, for Problem 7.3TLFeBOOK
240 Fundamentals of Probability and Statistics for Engineers7.4 Repeat Problem 7.3 if the distribution of the rod diameter remains uniform butthat of the sleeve inside diameter is N(2 cm, 0: 0004 cm 2 ).7.5 The first mention of the normal distribution was made in the work of de Moivre in1733 as one method of approximating probabilities of a binomial distribution whenn is large. Show that this approximation is valid and give an example showingresults of this approximation.7.6 If the distribution of temperature T of a given volume of gas is N(400, 1600),measured in degrees Fahrenheit, find:(a) f T (450);(b) P(T 450);(c) P( jT m T j20);(d) P( jT m T j20jT 300).7.7 If X is a random variable and distributed as N(m, 2 ), show thatEfjX mjg ˆ 2 1=2:7.8 Let random variable X and Y be identically and normally distributed. Show thatrandom variables X ‡ Y and X Y are independent.7.9 Suppose that the useful lives measured in hours of two electronic devices, say T 1and T 2 , have distributions N(40, 36) and N(45, 9), respectively. If the electronicdevice is to be used for a 45-hour period, which is to be preferred? Which ispreferred if it is to be used for a 48-hour period?7.10 Verify Equation (7.13) for normal random variables.7.11 Let random variables X 1 ,X 2 ,...,X n be jointly normal with zero means. Show thatEfX 1 X 2 X 3 gˆ 0;EfX 1 X 2 X 3 X 4 gˆEfX 1 X 2 gEfX 3 X 4 g‡EfX 1 X 3 gEfX 2 X 4 g‡EfX 1 X 4 gEfX 2 X 3 g:Generalize the results above and verify Equation (7.35).7.12 Two rods, for which the lengths are independently, identically, and normallydistributed random variables with means 4 inches and variances 0.02 square inches,are placed end to end.(a) What is the distribution of the total length?(b) What is the probability that the total length will be between 7.9 inches and 8.1inches?7.13 Let random variables X 1 ,X 2 ,and X 3 be independent and distributed accordingto N(0,1), N(1,1), and N(2,1), respectively. Determine probability P(X 1 ‡ X 2 ‡X 3 > 1).7.14 A rope with 100 strands supports a weight of 2100 pounds. If the breaking strengthof each strand is random, with mean equal to 20 pounds and standard deviation 4pounds, and if the breaking strength of the rope is the sum of the independentbreaking strengths of its strands, determine the probability that the rope will notfail under the load. (Assume there is no individual strand breakage before ropefailure.)TLFeBOOK
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ContentsPREFACExiii1 INTRODUCTION 1
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Contentsix6.1.2 Geometric Distribut
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ContentsxiAPPENDIX A: TABLES 365A.1
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PrefaceThis book was written for an
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1IntroductionAt present, almost all
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Introduction 3may be rejected at th
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Part AProbability and Random Variab
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8 Fundamentals of Probability and S
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390 Subject IndexExtreme-value dist
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