Definition:Probability Mass Function

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X: \Omega \to \R$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.

Then the (probability) mass function of $X$ is the (real-valued) function $p_X: \R \to \closedint 0 1$ defined as:
 * $\forall x \in \R: \map {p_X} x = \begin{cases}

\map \Pr {\set {\omega \in \Omega: \map X \omega = x} } & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end{cases}$ where $\Omega_X$ is defined as $\Img X$, the image of $X$.

That is, $\map {p_X} x$ is the probability that the discrete random variable $X$ takes the value $x$.

$\map {p_X} x$ can also be written:
 * $\map \Pr {X = x}$

Note that for any discrete random variable $X$, the following applies:

The latter is usually written:
 * $\displaystyle \sum_{x \mathop \in \R} \map {p_X} x = 1$

Thus it can be seen by definition that a probability mass function is an example of a normalized weight function.

The set of probability mass functions on a finite set $Z$ can be seen denoted $\map \Delta Z$.

Also known as
A (probability) mass function is often seen abbreviated p.m.f., pmf or PMF.