Stochastic Hydrology and Hydraulics

, Volume 5, Issue 1, pp 55–68

Derivation of bivariate probability density functions with exponential marginals


  • K. Singh
    • Dept. of MathematicsLouisiana State University
  • V. P. Singh
    • Dept. of Civil EngineeringLouisiana State University

DOI: 10.1007/BF01544178

Cite this article as:
Singh, K. & Singh, V.P. Stochastic Hydrol Hydraul (1991) 5: 55. doi:10.1007/BF01544178


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.

Key words

Bivariate probability distribution random variables zero marginals Finch-Groblicki method

Copyright information

© Springer-Verlag 1991