Rectangle Probabilities by Inclusion-Exclusion

Why Four CDF Values Determine a Rectangle Probability

A joint cumulative distribution function records probabilities for regions extending down and to the left from a point. A bounded rectangle is not itself such a region, but it can be assembled from four of them: take the large lower-left region, subtract two strips, then add back their overlap. This is the inclusion-exclusion principle in a form that is especially useful for joint distributions.

The exact boundary convention matters. The usual four-value formula applies directly to rectangles open at their lower edges and closed at their upper edges. If the random vector can put probability on a boundary, changing an open edge to a closed one can change the answer. The corner values must then be adjusted accordingly.

The Four-Corner Formula

The signs follow the same pattern in both coordinates: start with the upper-right corner, subtract the two lower edges, and add back the lower-left corner that was subtracted twice. In particular, if \(X\) and \(Y\) have atoms on the boundary, the formula still gives the probability for exactly the stated half-open rectangle.

The formula can also be proved directly with indicator variables. For any outcome, the difference \(1_{\{X\le b\}}-1_{\{X\le a\}}\) equals 1 exactly when \(a<X\le b\), and equals 0 otherwise. The corresponding statement holds for \(Y\). Their product is therefore the indicator of the rectangle:

Taking expectations and expanding the product gives four probabilities. By the definition of \(F\), those probabilities are \(F(b,d)\), \(F(a,d)\), \(F(b,c)\), and \(F(a,c)\), with signs \(+,-,-,+\), respectively. This proves the result. \(\square\)

Infinite Endpoints

Infinite endpoints are handled by limits of CDF values. A real-valued random variable is finite with probability one, so it is useful to extend the notation by setting

At a lower endpoint of \(-\infty\), the corresponding lower-left event has probability zero. For example, \(P(X\le-\infty,\ Y\le y)=0\). Consequently, if \(a=-\infty\), the four-corner formula reduces to

If \(b=\infty\), the terms at that endpoint use the marginal CDF of \(Y\). For instance,

These conventions follow from probability continuity: probabilities of increasing events converge to the probability of their union, and probabilities of decreasing events converge to the probability of their intersection. Thus finite-threshold CDF values approaching an infinite endpoint converge to the stated marginal or total probability. The same formulas apply when more than one endpoint is infinite, with zero lower-tail terms omitted as appropriate.

Open and Closed Edges When Atoms Are Present

The ordinary joint CDF uses the comparison \(\le\) in both coordinates. It therefore gives the four-corner formula directly for \((a,b]\times(c,d]\), but not automatically for every choice of open and closed edges. To include a lower edge, the probability on that edge must be retained rather than removed. To exclude an upper edge, the probability on that edge must be omitted.

One precise way to keep track of the distinction is to allow strict-threshold probabilities as well. Write \(F_{<,\le}(x,y)=P(X<x,\ Y\le y)\), \(F_{\le,<}(x,y)=P(X\le x,\ Y<y)\), and \(F_{<,<}(x,y)=P(X<x,\ Y<y)\). These are determined by one-sided limits of the ordinary joint CDF:

Here \(F(x-,y)=P(X<x,\ Y\le y)\), and the other left limits have the analogous meaning. For example, \(F(x-,y)=\lim_{u\uparrow x}F(u,y)\). These limits are important when there is an atom at a threshold: in general \(F(x-,y)\ne F(x,y)\).

For example, a rectangle closed at both lower edges and both upper edges, \([a,b]\times[c,d]\), has probability

The left limits occur at the lower edges because the points \(X=a\) and \(Y=c\) must remain in the rectangle. When there are no atoms on the relevant edges, these limits agree with the corresponding CDF values, and changing the boundary convention does not change the probability. When there are atoms, the distinction can be essential.

Common Mistake and Why the Formula Matters

A common mistake is to use \(F(b,d)-F(a,d)-F(b,c)+F(a,c)\) for a closed rectangle \([a,b]\times[c,d]\) without checking for atoms. That calculation gives the probability of \((a,b]\times(c,d]\), not necessarily of the closed rectangle. In the discrete example, it gives \(0.25\), while the closed rectangle has probability \(0.70\). The correction is to use strict-threshold limits at included lower edges, or to add the relevant boundary probabilities explicitly.

The four-corner identity is useful because a joint CDF can encode discrete, continuous, singular, and mixed distributions. Once its four corner values—or the needed one-sided limits—are available, the probability of an axis-aligned rectangle follows without separately summing atoms or integrating a density. Always state the interval convention before substituting values, and check whether endpoint masses can affect the result.