A value of the joint CDF of two random variables from other known values

Question

Let $X$ and $Y$ two continuous random variables for which I know $F_X$, $F_Y$ (for any value of $X$ and $Y$'s support) and $F_{XY}(a,b')$, $F_{XY}(a',b)$ and $F_{XY}(a',b')$, where $F_{XY}$ is the joint CDF of $X$ and $Y$, $a<a'$ and $b<b'$. With such amount of information, can I obtain $F_{XY}(a,b)$?.

I know that $F_{XY}(a,b)=F_X(a)+F_Y(b)-P(X>a,Y>b)$, but I don't know how to calculate the last probability with the tools I have. Any hints?

Answer 1

No, that info is not enough to obtain $F_{X,Y}(a,b)$.

If we define:

• $\alpha=P(a<X\leq a',b<Y\leq b')$
• $\beta=P(X\leq a,b<Y\leq b')$
• $\gamma=P(a<X\leq a',Y\leq b)$
• $\delta=P(X\leq a,Y\leq b)$

then we have the equalities:

• $\alpha+\beta+\gamma+\delta=F_{X,Y}(a',b')$
• $\beta+\delta= F_{X,Y}(a,b')$
• $\gamma+\delta= F_{X,Y}(a',b)$
• $\delta=F_{X,Y}(a,b)$

But knowledge $\alpha+\beta+\gamma+\delta$, $\beta+\delta$ and $\gamma+\delta$ does in general not enable us to find $\delta$.