Page 255 - Fundamentals of Probability and Statistics for Engineers

238 Fundamentals of Probability and Statistics for Engineers

One final remark to be made is that asymptotic distributions of maximum and
minimum values from the same initial distribution may not be of the same type.
For example, for a gamma initial distribution, its asymptotic maximum-value
distribution is of Type I whereas the minimum-value distribution falls into Type
III. With reference to system time-to-failure models, a system having n components
in series with independent gamma life distributions for its components will have a
time-to-failure distribution belonging to the Type-III asymptotic minimum-value
distribution as n becomes large. The corresponding model for a system having n
components in parallel is the Type-I asymptotic maximum-value distribution.

7.7 SUMMARY

As in Chapter 6, it is useful to summarize the important properties associated
with some of the important continuous distributions discussed in this chapter.
These are given in Table 7.1.

REFERENCES

Gumbel, E.J., 1958, Statistics of Extremes, Columbia University Press, New York.

Kramer, M., 1940, ‘‘Frequency Surfaces in Two Variables Each of Which is Uniformly

Distributed’’, Amer. J. of Hygiene32 45–64.
Lindberg, J.W., 1922, ‘‘Eine neue Herleitung des Exponentialgesetzes in der Wahrschein-

lichkeifsrechnung’’, Mathematische Zeitschrift 15 211–225.
Weibull, W., 1939, ‘‘A Statistical Theory of the Strength of Materials’’, Proc. Royal

Swedish Inst. for Engr. Res., Stockholm No. 151.
Wilks, S., 1942, ‘‘Statistical Prediction with Special Reference to the Problem of Toler-
.
ance Limits’’, Ann. Math. Stat 13 400.

.
FURTHER READING AND COMMENTS
As we mentioned in Section 7.2.1, the central limit theorem as stated may be generalized
in several directions. Extensions since the 1920s include cases in which random variable
Y in Equation (7.14) is a sum of dependent and not necessarily identically distributed
random 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 & Sons
Inc., New York.
Extensive probability tables exist in addition to those given in Appendix A. Prob-
ability tables for lognormal, gamma, beta, chi-squared, and extreme-value distributions
can be found in some of the references cited in Chapter 6. In particular, the following
references are helpful:

TLFeBOOK