A finite probability distribution is a function \(f : A \to \mathbb {R}\) from a finite type \(A\), such that \(f\) is nonnegative and the sum of \(f\) is 1.

The uniform distribution on a nonempty set \(A\), \(\operatorname{\operatorname {Uniform}}(A)\), assigns \(\frac1{|A|}\) to all values in \(A\) and \(0\) to other values.

Given two finite probability distributions \(f: A \to \mathbb {R}, g : B \to \mathbb {R}\), we have a probability distribution from \(A \times B\) defines as \((f \times g)(x, y) = f(x) g(y)\).

Given a finite probability distribution \(f: A \to \mathbb {R}\) and a function \(g : A \to B\), we can apply \(g\) to the random variable represented by \(f\). This gives the distribution \(g \# f\).

Given two finite probability distributions \(f: A \to \mathbb {R}, g : A \to \mathbb {R}\), we have a probability distribution defines as \(f-g = s \# (f \times g)\) with \(s(x, y) = x-y\).

Given two finite probability distributions \(f: A \to \mathbb {R}, g : A \to \mathbb {R}\), we have a probability distribution defines as \(f+g = a \# (f \times g)\) with \(a(x, y) = x+y\).

Given a finite probability distribution \(f: A \to \mathbb {R}\), we have a probability distribution defines as \(-f = n \# f\) with \(n(x) = -x\).

Given a finite probability distribution \(f: A \to \mathbb {R}\) and a list of finite probability distributions on \(B\), indexed by elements of \(A\), \(g\), we can define \(g(f)\) as the probability distribution obtained by sampling an element from \(f\), and then sampling an elemente from the corresponding distribution in \(g\).

We say that a distribution \(a\) is \(\varepsilon \)-close to \(N\) entropy if for all sets \(|A| \leq N\), \(\sum _{x \in A} a(x) \leq \varepsilon \). Note that this is a bit different than the usual definition.