Let X and Y be uniform random variables whose dependence function $C(x,y)$ is of the form $phi^{-1}{phi(x)+phi(y)}$ for some convex decreasing function $phi$ on (0,1) with the property that $phi(1)=0$. Set

begin{alignat*}{3}

U=frac{phi(X)}{{phi(X)+phi(Y)}} & quad V=C(X,Y) & quad lambda(v)=frac{phi(v)}{phi^{‘}(v)}

end{alignat*}for $0 <vleq 1$.

Consider $S=phi(X)$, $T=phi(Y)$, $Pr(Uleq u, Vleq v)=Pr(phi(v)-T leq Sleq frac{uT}{1-u})$ and the density of T with $frac{-1}{phi^{‘}(phi^{-1}(t))}$.

How can I rewrite $Pr(Uleq u,Vleq v)$ to

$$ Pr(Uleq u, Vleq v) = int_{(1-u)phi(v)}^{phi(v)}-frac{Pr(phi(v)-t leq S leq frac{ut}{1-u}|T=t)}{phi^{‘}(phi^{-1}(t))}dt\

+ int_{phi(v)}^{phi(0)}-frac{Pr(Sleq frac{ut}{1-u}|T=t)}{phi^{‘}(phi^{-1}(t))} dt$$ ? I would be very grateful for any advice.