Is the indicator function of the set ${x+y\geq0}$ a distribution function of some random vector?

Question

Is $1_{x+y\geq0}$ a distribution function of some random vector? If the answer is yes, what is the probability $P(x<-1,y<-1)$?

Answer 1

The answer is: no.

If a CDF $F_{X,Y}(x,y)$ is an indicator function then it only takes values in $\{0,1\}$. Then it can be deduced that also the marginals $F_X(x)$ and $F_Y(y)$ only take values in $\{0,1\}$. This tells us that $X$ and $Y$ are both degenerated random variables with $P(X=a)=1=P(Y=b)$ for constants $a,b\in\mathbb R$. Then consequently $F_{X,Y}$ is the indicator function of the set $[a,\infty)\times[b,\infty)$.

So only indicator function of the form $\mathsf1_{[a,\infty)\times[b,\infty)}$ can serve as CDF.