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Derivation of bivariate probability density functions with exponential marginals

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Stochastic Hydrology and Hydraulics Aims and scope Submit manuscript

Abstract

A vivariate probability density function (pdf),f(x 1,x 2), admissible for two random variables (X 1,X 2), is of the form

$$f(x_1 x_2 ) = f_1 (x_1 )f_2 (x_2 )[1 + \rho \{ F_1 (x_1 ),F_2 (x_2 )\} ]$$

where ρ(u, v) (u=F 1(x 1),v=F 2(x 2)) is any function on the unit square that is 0-marginal and bounded below by−1 andF 1(x 1) andF 2(x 2) are cumulative distribution functions (cdf) of marginal probability density functionsf 1(x 1) andf 2(x 2). The purpose of this study is to determinef(x 1,x 2) for different forms of ρ(u,v). By considering the rainfall intensity and the corresponding depths as dependent random variables, observed and computed probability distributionsF 1(x 1),F(x 1/x 2),F 2(x 2), andF(x 2/x 1) are compared for various forms of ρ(u,v). Subsequently, the best form of ρ(u,v) is specified.

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Authors and Affiliations

  1. Dept. of Mathematics, Louisiana State University, 70803-6405, Baton Rouge, LA, USA

    K. Singh

  2. Dept. of Civil Engineering, Louisiana State University, 70803-6405, Baton Rouge, LA, USA

    V. P. Singh

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  1. K. Singh
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  2. V. P. Singh
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Singh, K., Singh, V.P. Derivation of bivariate probability density functions with exponential marginals. Stochastic Hydrol Hydraul 5, 55–68 (1991). https://doi.org/10.1007/BF01544178

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  • DOI: https://doi.org/10.1007/BF01544178

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