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4 results

  • 2 - Review of Probability Theory

  • Armen Der Kiureghian, University of California, Berkeley
  • Book:
    Structural and System Reliability
    Published online:
    13 January 2022
    Print publication:
    13 January 2022, pp 11-57

  • Summary

    Chapter 2 provides a review of probability theory, focusing on the topics that are essential for the remainder of the book. Included are the elements of set theory, the axioms and basic rules of probability theory, the concept of a random variable, discrete and continuous random variables, univariate and multivariate probability distributions, reliability and hazard functions, expectation and statistical moments, distributions and moments of functions of random variables, and extreme-value distributions. Appendix A presents commonly used probability distribution models with their properties, for easy reference. Thorough mastery of the material in this chapter is essential for understanding the remainder of the book.

  • On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients

  • K. F. Turkman, A. M. Walker
  • Journal:
    Advances in Applied Probability / Volume 16 / Issue 4 / December 1984
    Published online by Cambridge University Press:
    01 July 2016, pp. 819-842
    Print publication:
    December 1984

  • Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let Xn(λ) and Yn(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {Xn(λ)}, (Yn(λ)}, {In(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x], —∞ < x <∞.

    It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.