Fundamentals of Probability and Statistics for Engineers

238 Fundamentals of Probability and Statistics for EngineersOne final remark to be made is that asymptotic distributions of maximum andminimum values from the same initial distribution may not be of the same type.For example, for a gamma initial distribution, its asymptotic maximum-valuedistribution is of Type I whereas the minimum-value distribution falls into TypeIII. With reference to system time-to-failure models, a system having n componentsin series with independent gamma life distributions for its components will have atime-to-failure distribution belonging to the Type-III asymptotic minimum-valuedistribution as n becomes large. The corresponding model for a system having ncomponents in parallel is the Type-I asymptotic maximum-value distribution.7.7 SUMMARYAs in Chapter 6, it is useful to summarize the important properties associatedwith some of the important continuous distributions discussed in this chapter.These are given in Table 7.1.REFERENCESGumbel, E.J., 1958, Statistics of Extremes, Columbia University Press, New York.Kramer, M., 1940, ‘‘Frequency Surfaces in Two Variables Each of Which is UniformlyDistributed’’, Amer. J. of Hygiene32 45–64.Lindberg, J.W., 1922, ‘‘Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeifsrechnung’’,Mathematische Zeitschrift 15 211–225.Weibull, W., 1939, ‘‘A Statistical Theory of the Strength of Materials’’, Proc. RoyalSwedish Inst. for Engr. Res., Stockholm No. 151.Wilks, S., 1942, ‘‘Statistical Prediction with Special Reference to the Problem of ToleranceLimits’’, Ann. Math. Stat . . 13 400.FURTHER READING AND COMMENTSAs we mentioned in Section 7.2.1, the central limit theorem as stated may be generalizedin several directions. Extensions since the 1920s include cases in which random variableY in Equation (7.14) is a sum of dependent and not necessarily identically distributedrandom variables. See, for example, the following two references:Loe´ve, M., 1955, Probability Theory, Van Nostrand, New York.Parzen, E., 1960, Modern Probability Theory and its Applications, John Wiley & SonsInc., New York.Extensive probability tables exist in addition to those given in Appendix A. Probabilitytables for lognormal, gamma, beta, chi-squared, and extreme-value distributionscan be found in some of the references cited in Chapter 6. In particular, the followingreferences are helpful:TLFeBOOK