# Distribution Function Evaluated With Inequality

Let's say you have a distribution function F. According to Intro to Mathematical Statistics (Hogg), the following property holds:

Pr($a < X \leq b$, $c < Y \leq d$) = $F(b,d) - F(b,c) - F(a,d) + F(a,c)$

What is the intuition behind this property, and how can you prove this formula?

Thank you!

Note that $Pr(a < X \leq b, c < Y \leq d) = Pr(a < X \leq b, Y \leq d) - Pr(a <X \leq b, Y \leq c)$. To see why this relationship holds, moves $Pr(a <X \leq b, Y \leq c)$ to the left side of the equality and note that $\{a <X \leq b, Y \leq c\}$ and $\{a < X \leq b, c < Y \leq d\}$ are disjoint events.

Now use this trick again to the $X$ term and you are ready to go~

It is just the Principle of Inclusion and Exclusion.

Consider the quadrants on a Venn Diagram : $~{\bbox[blue]{\Box}~\bbox[white]{\Box}\\\bbox[purple]{\Box}~\bbox[red]{\Box}}$

\def\P{\operatorname{\sf P}} \begin{align} \P((X,Y)\in(a;b]{\times}(c;d]) & =\P(\Box) \\[1ex] & = \P\left({\bbox[blue]{\Box}~\bbox[white]{\Box}\\\bbox[purple]{\Box}~\bbox[red]{\Box}}\right)-\P\left({\bbox[blue]{\Box}\\\bbox[purple]{\Box}}\right)-\P({\bbox[purple]{\Box}~\bbox[red]{\Box}})+\P(\bbox[purple]{\Box}) \\[1ex] & = {\P((X,Y)\in(-\infty;b]{\times}(-\infty;d]) \\-\P((X,Y)\in(-\infty;a]{\times}(-\infty;d]) \\ -\P((X,Y)\in(-\infty;b]{\times}(-\infty;c]) \\+\P((X,Y)\in(-\infty;b]{\times}(-\infty;c])} \\[1ex] & = F(b,d)-F(a,d)-F(b,c)+F(a,c)\end{align}