Fundamentals of Probability and Statistics for Engineers

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Some Important Continuous Distributions 2417.15 If X 1 ,X 2 ,...,X n are independent random variables, all having distribution N(m, 2 ),determine the conditions that must be imposed on c 1 ,c 2 ,...,c n such that the sumY ˆ c 1 X 1 ‡ c 2 X 2 ‡‡ c n X nis also N(m, 2 ). Can all cs be positive?7.16 Let U be the standardized normal random variable, and define X ˆjU j. Then, Xis called the folded standardized normal random variable. Determine f X (x).7.17 The Cauchy distribution has the form1f X …x† ˆ…1 ‡ x 2 † ; 1 < x < 1:(a) Show that it arises from the ratio X 1 /X 2 , where X 1 and X 2 are independent anddistributed as N(0, 2 ).(b) Show that the moments of X do not exist.7.18 Let X 1 and X 2 be independent normal random variables, both with mean 0 andstandard deviation 1. Prove that:Y arctan X 2ˆis uniformly distributed from to .7.19 Verify Equations (7.48) for the lognormal distribution.7.20 The lognormal distribution is found to be a good model for strains in structuralmembers caused by wind loads. Let the strain be represented by X, with m X ˆ 1and 2 X ˆ 0:09.(a) Determine the probability P(X > 1:2).(b) If stress Y in a structural member is related to the strain by Y ˆ a ‡ bX, withb > 0, determine f Y (y) and m Y .7.21 Arrivals at a rural entrance booth to the New York State Thruway are consideredto be Poisson distributed with a mean arrival rate of 20 vehicles per hour. The timeto process an arrival is approximately exponentially distributed with a mean time ofone min.(a) What percentage of the time is the tollbooth operator free to work on operationalreports?(b) How many cars are expected to be waiting to be processed, on average, per hour?(c) What is the average time a driver waits in line before paying the toll?(d) Whenever the average number of waiting vehicles reaches 5, a second tollboothwill be opened. How much will the average hourly rate of arrivals have toincrease to require the addition of a second operator?7.22 The life of a power transmission tower is exponentially distributed, with mean life25 years. If three towers, operated independently, are being erected at the sametime, what is the probability that at least 2 will still stand after 35 years?7.23 For a gamma-distributed random variable, show that:(a) Its mean and variance are those given by Equation (7.57).(b) It has a positive skewness.X 1TLFeBOOK
242 Fundamentals of Probability and Statistics for Engineers7.24 Show that, if is a positive integer, the probability distribution function (PDF) ofa gamma-distributed random variable X can be written as8< 0; for x 0;F X …x† ˆ P 1…x† k e x: 1; for x > 0:k!kˆ0Recognize that the terms in the sum take the form of the Poisson mass function andtherefore can be calculated with the aid of probability tables for Poisson distributions.7.25 The system shown in Figure 7.16 has three redundant components, A–C. Let theiroperating lives (in hours) be denoted by T 1 , T 2 , and T 3 , respectively. If theredundant parts come into operation only when the online component fails (coldredundancy), then the operating life of the system, T, is T ˆ T 1 ‡ T 2 ‡ T 3 .Let T 1 ,T 2 ,and T 3 be independent random variables, each distributed as1100f Tj…t j †ˆe tj=100 ; for t j 0; j ˆ 1; 2; 3;0; otherwise:Determine the probability that the system will operate at least 300 hours.7.26 We showed in Section 7.4.1 that an exponential failure law leads to a constantfailure rate. Show that the converse is also true; that is, if h(t) as defined byEquation (7.65) is a constant then the time to failure T is exponentially distributed.7.27 A shifted exponential distribution is defined as an exponential distribution shiftedto the right by an amount a; that is, if random variable X has an exponentialdistribution withf X …x† ˆ e x ; for x 0;0; elsewhere;random variable Y has a shifted exponential distribution if f Y (y) has the sameshape as f X (x) but its nonzero portion starts at point a rather than zero. Determinethe relationship between X and Y and probability density function (pdf) f Y (y).What are the mean and variance of Y ?ABCFigure 7.16 System of components, for Problem 7.25TLFeBOOK
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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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