Consider two uniform random variables $U, V$ . If $ U \sim \mbox{Uniform}(0,1)$ and $V = 1 -U$ how would we show that the joint distribution of $(U, V)$ is given by $\max \{ u + v -1,0 \}$?
The reason I ask is because the joint distribution is the lower Frechet-Hoeffding bound from the theory of Copulas. This makes sense intuitively given that the Frechet-Hoeffding bounds correspond to cases of extreme dependency and if we set $V = 1 - U$, the random variables are perfectly negatively correlated. However, how can we show this result explicitly?