Let $Pr\_{(X,Y)}$ be a probability distribution of a random vector $(X,Y)$. Let $F$ be the cumulative distribution function of $(X,Y)$. Define
$$
\mathcal{A}\equiv \{(x,y): x\leq 2 \text{ and }x-y\leq 3\}
$$
Is there a way to express $Pr\_{(X,Y)}(\mathcal{A})$ as
$$
(\*)\quad \sum\_{k=1}^K F(a\_k, b\_k)\times c\_k
$$
for some finite $K$ and $\{a\_k, b\_k, c\_k\}\_{k=1}^K$?




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Note: if $(X,Y)$ had a uniform distribution, then we could write
$$
Pr\_{(X,Y)}(\mathcal{A})=Pr\_{(X,Y)}((-\infty,2]\times [-1,\infty))+\frac{1}{2}Pr\_{(X,Y)}((-\infty,2]\times (-\infty,-1])
$$
which can be rewritten as $(\*)$.
However, I wonder whether we could do something similar for the general case without uniformity.



Probability measure of trapezoidal area
No, this cannot be done in general. Indeed, let $A:=\mathcal A$. You want to express
$$P((X,Y)\in A)=Ef(X,Y)$$
as
$$\sum\_{k=1}^K c\_k F(a\_k,b\_k)=Eg(X,Y),$$
where
$$f(x,y):=1(x\le2,x-y\le3)$$
and
$$g(x,y):=\sum\_{k=1}^K c\_k 1(x\le a\_k,y\le b\_k).$$
However, for any choice of the numbers $a\_k,b\_k,c\_k$, there will be some $(x\_\*,y\_\*)\in\mathbb R^2$ such that $f(x\_\*,y\_\*)\ne g(x\_\*,y\_\*)$. Letting the random pair $(X,Y)$ take value $(x\_\*,y\_\*)$ with probability $1$, we get
$$P((X,Y)\in A)=Ef(X,Y)=f(x\_\*,y\_\*)\ne g(x\_\*,y\_\*)=Eg(X,Y)=\sum\_{k=1}^K c\_k F(a\_k,b\_k),$$
so that $P(A)\ne\sum\_{k=1}^K c\_k F(a\_k,b\_k)$.


