---
title: "probability mass function"
type: concept
related: [Random Variables, Probability Mass Function]
source: https://www.jemoka.com/posts/kbhprobability_mass_function/
confidence: high
status: active
---

PMF is a function that maps possible outcomes of a discrete random variables to the corresponding actual probabilities.
For random variable \(Y\), we have:
\begin{equation} f(k) = P(Y=k) \end{equation}
and \(f\) is a function that is the PMF, which is the mapping between a random variable and a value it takes on to the probability that the random variable takes on that value.
Shorthand \begin{equation} P(Y=k) = p(y), where\ y=k \end{equation}
its written smaller \(y\) represents a case of \(Y\) where \(Y=y\).
Properties of PMD \begin{equation} 0 \leq P(x) \leq 1 \end{equation}
and
\begin{equation} \sum_{}^{} P(x) = 1 \end{equation}
