Definition:Probability Mass Function

Definition
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

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

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

\Pr \left({\left\{{\omega \in \Omega: X \left({\omega}\right) = x}\right\}}\right) & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end{cases}$ where $\Omega_X$ is defined as $\operatorname{Im} \left({X}\right)$, the image of $X$.

That is, $p_X \left({x}\right)$ is the probability that the discrete random variable $X$ takes the value $x$.

$p_X \left({x}\right)$ can also be written:
 * $\Pr \left({X = x}\right)$

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

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

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 $\Delta \left({Z}\right)$.

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