If f is the ‘Cumulative Distribution Function’ (CDF) for some random variable X that takes on real values, then f(z) is the probability that X is less than z.
For a random variable X=(X1, … Xn) with n-tuple values, i.e. in an n-dimensional vector space, it has been found convenient to introduce a CDF f such that f(x1, … xn) is the probability that Xi < xi for each i.
In 2D we may want the probability p that a<X1<b and c<X2<d.
We have p = f(b, d) − f(a, d) − f(b, c) + f(a, c).
The expression in n dimensions involves 2n terms—a term for each corner of the sub-box.