# Derivation of bivariate probability density functions with exponential marginals

## Authors

Originals

- Accepted:

DOI: 10.1007/BF01544178

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

## Abstract

A vivariate probability density function (pdf), where ρ(

*f(x*_{1},*x*_{2}), admissible for two random variables (*X*_{1},*X*_{2}), is of the form$$f(x_1 x_2 ) = f_1 (x_1 )f_2 (x_2 )[1 + \rho \{ F_1 (x_1 ),F_2 (x_2 )\} ]$$

*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 and*F*_{1}(*x*_{1}) and*F*_{2}(*x*_{2}) are cumulative distribution functions (cdf) of marginal probability density functions*f*_{1}(*x*_{1}) and*f*_{2}(*x*_{2}). The purpose of this study is to determine*f*(*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 distributions*F*_{1}(*x*_{1}),*F*(*x*_{1}/*x*_{2}),*F*_{2}(*x*_{2}), and*F*(*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