For a discrete random variable $X$ with possible values $x_1, x_2, x_3, \ldots$ occurring with probabilities $p_1, p_2, p_3, \ldots$, where $p_i = P[X = x_i]$, the rule that assigns each $x_i$ its probability $p_i$ is called the probability mass function of $X$, provided every $p_i \ge 0$ and the probabilities add up to 1, i.e. $\sum p_i = 1$. Listing the pairs $(x_i, p_i)$, often in a two-row table, gives the probability distribution of $X$. Every possible value must genuinely occur with positive probability, and because the values of $X$ between them account for every outcome in the sample space, their probabilities must sum to exactly 1 — this two-part test (non-negativity and sum equal to 1) is exactly what is used to decide whether a given table of numbers can be a valid p.m.f.