Derivation of bivariate probability density functions with exponential marginals. (English) Zbl 0722.60020
Summary: A bivariate 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 \(\rho\) (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 \(\rho\) (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 \(\rho\) (u,v). Subsequently, the best form of \(\rho\) (u,v) is specified.
MSC:
| 60E99 | Distribution theory |
References:
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