DERIVATION OF BIVARIATE PROBABILITY DENSITY-FUNCTIONS WITH EXPONENTIAL MARGINALS

A bivariate probability density function (pdf), f(x1,x2), admissible for two random variables (X1,X2), is of the form f(x1,x2) = f1(x1)f2(x2)[1+rho{F1(x1)F2(x2)}] where rho(u,v) (u=F1(x1), v=F2(x2)) is any function on the unit square that is O-marginal and bounded below by -1 and F1(x1) and F2(x2) are cumulative distribution functions (cdf) of marginal probability density functions f1(x1) and f2(x2). The purpose of this study is to determine f(x1,x2) for different forms of rho(u,v). By considering the rainfall intensity and the corresponding depths as dependent random variables, observed and computed probability distributions F1(x1), F(x1\x2), F2(x2\x1) are compared for various forms of rho(u,v). Subsequently, the best form of rho(u,v) is specified.