Exact Distribution of the Max/Min of Two Correlated Random Variables
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Exact Distribution of the Max/Min of Two Correlated Random Variables Y. Zhang1 · S. Nadarajah1
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Statistical static timing analysis involves the distributions of the maximum and minimum of correlated random variables. Nadarajah and Kotz (IEEE Trans Very Large Scale Integr Syst 16:210–2012, 2008) derived closed form expressions for the distributions when the random variables are Gaussian. Here, we extend the work when the random variables follow a wide range of non-Gaussian distributions. Keywords Moment generating function · Moments · Statistical static time analysis
1 Introduction Statistical static time analysis (SSTA) is a stock analysis algorithm for the design of digital circuits. Maximum and minimum of correlated random variables arise naturally with respect to SSTA. Traditionally, SSTA involves maximum and minimum of correlated Gaussian random variables. Various authors have suggested approximations for the distributions of the maximum and minimum, see [2, 6, 7, 13, 27, 29] and [21]. Nadarajah and Kotz [19] pointed out that closed form expressions do exist for the distributions of the maximum and minimum of two correlated Gaussian random variables, hence that there is no need for approximations. Nadarajah and Kotz [19] gave closed form expressions for the probability density function, cumulative distribution function, moment generating function and the first two moments for the distributions of the maximum and minimum of two correlated Gaussian random variables. Recently, approaches for SSTA have been based on non-Gaussian distributions, that is, involving maximum and minimum of correlated non-Gaussian random variables: [1, 4, 5, 8–10, 15, 26, 30, 32, 34] and [33] used non-Gaussian distributions [24, 31] and [11] used skew-normal distributions; [28] used half triangular distributions; [16] and [22] used arbitrary distributions. The proposed approaches are again based on approximations.
* S. Nadarajah [email protected] Y. Zhang [email protected] 1
Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
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Y. Zhang, S. Nadarajah
The aim of this paper is to show that closed form expressions for the distributions of the maximum and minimum of two correlated random variables can be derived for a wide range of non-Gaussian distributions. Hence, the need for approximations can be avoided. Let FX,Y (x, y) = Pr ≤ (X ≤ x, Y ≤ y) , the joint cumulative distribution function of (X, Y). Let U denote the smaller of the two random variables. Let V denote the larger of the two. We give closed form expressions for the cumulative distribution function of U, the probability density function of U, the nth moment of U, the moment generating function of U, the cumulative distribution function of V, the probability density function of V, the nth moment of V and the moment generating function of V. Note that the cumulative distribution functions of U and V are
FU (u) = FX,Y (u
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