Finite probability distribution ($p\ge 0,{\sum }_{x}p(x)=1$)

A probability mass function with finite positive-mass support, nonnegative masses and total mass one. Listing its support means listing exactly positive-mass outcomes once, excluding zero-probability endpoint outcomes. A computational interface returning floating-point masses generally approximates exact rational or real probabilities.

For a finite probability distribution $p$ with support $S$ and a total function $f:S\to Y$, assign each image value the combined mass of its preimages. Its support is exactly the finite image $f(S)$, listed without repetitions. Every listed mass is positive because it contains at least one positive source mass.

The fiber of a function construction gives a set partition of $S$, so finite summation gives ${\sum }_{y}q(y)={\sum }_{x\in S}p(x)=1$. This proves nonnegativity and normalization without assuming that $f$ is injective. In code, mapping support values and then overwriting repeated image keys is incorrect; repeated keys must have their masses added. For example, mapping a uniform distribution on four consecutive integers to parity produces two outcomes with mass one half each.