Definition:Probability Mass Function

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

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

Then the (probability) mass function or p.m.f. of $$X$$ is the 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 \operatorname{Im} \left({X}\right)\\ 0 & : x \notin \operatorname{Im} \left({X}\right) \end{cases}$$

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

This is usually abbreviated:
 * $$\Pr \left({X = x}\right)$$

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

$$ $$ $$

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

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