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.

Given two finite probability distributions, assign the product of their probability masses to each ordered pair. The support is the Cartesian product of the two positive-mass supports. Distributivity gives ${\sum }_{x,y}{p}_1(x)p_2(y)=({\sum }_x{p}_1(x))({\sum }_y{p}_2(y))=1$. This is the joint distribution for independent coordinate outcomes; it is a construction from the marginals, not a claim that arbitrary jointly observed coordinates are independent.

For a nonempty finite sample, assign each distinct observed value its relative frequency. The observed values form the finite positive-mass support; the counts sum to the sample size, so the probabilities sum to one. Repeated observations increase the same outcome's mass rather than adding repeated support entries.

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.