# Probability , Copula, rewrite – Mathematics Stack Exchange

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 .

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.