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=(X_{1}, … X_{n}) 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(x_{1}, … x_{n}) is the probability that X_{i} < x_{i} for each i.
In 2D we may want the probability p that a<X_{1}<b and c<X_{2}<d.
We have p = f(b, d) − f(a, d) − f(b, c) + f(a, c).
The expression in n dimensions involves 2^{n} terms—a term for each corner of the sub-box.