On the transitivity of the comonotonic and countermonotonic comparison of random variables (Q860341)

For any two random variables \(X\) and \(Y\) define \[ Q(X,Y)=\text{Prob}\{Y<X\}+\frac{1}{2}\,\text{Prob}\{X=Y\}. \] The quantity \(Q(X,Y)\) depends on the joint distribution of \(X\) and \(Y\). The authors model it by postulating the copula that is associated with that joint distribution. When this copula is either the lower or the upper Fréchet-Hoeffding bound, the authors derive explicit formulae to compute \(Q(X,Y)\), for the case when \(X\) and \(Y\) are discrete, and for the case when \(X\) and \(Y\) are continuous. Some transitivity properties of \(Q\) are derived for general random variables. Finally, explicit formulae are obtained for \(Q(X,Y)\) when \(X\) and \(Y\) are normal random variables, and the coupling between them is made by the lower or by the upper Fréchet-Hoeffding bound. The transitivity properties of these couplings are analyzed.