Stochastic Hydrology and Hydraulics

, Volume 5, Issue 1, pp 55–68

Derivation of bivariate probability density functions with exponential marginals

  • K. Singh
  • V. P. Singh


A vivariate probability density function (pdf),f(x1,x2), admissible for two random variables (X1,X2), 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=F1(x1),v=F2(x2)) is any function on the unit square that is 0-marginal and bounded below by−1 andF1(x1) andF2(x2) are cumulative distribution functions (cdf) of marginal probability density functionsf1(x1) andf2(x2). The purpose of this study is to determinef(x1,x2) for different forms of ρ(u,v). By considering the rainfall intensity and the corresponding depths as dependent random variables, observed and computed probability distributionsF1(x1),F(x1/x2),F2(x2), andF(x2/x1) are compared for various forms of ρ(u,v). Subsequently, the best form of ρ(u,v) is specified.

Key words

Bivariate probability distribution random variables zero marginals Finch-Groblicki method 


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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • K. Singh
    • 1
  • V. P. Singh
    • 2
  1. 1.Dept. of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.Dept. of Civil EngineeringLouisiana State UniversityBaton RougeUSA

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