Definition:Probability Mass Function

Definition
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 \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 function $$X$$ 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:
 * $$\sum_{x \in \R} p_X \left({x}\right) = 1$$

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

Joint Probability Mass Function
Let $$X: \Pr \to \R$$ and $$Y: \Pr \to \R$$ both be discrete random variables on $$\left({\Omega, \Sigma, \Pr}\right)$$.

Then the joint (probability) mass function of $$X$$ and $$Y$$ is function $$p_{X, Y}: \R^2 \to \left[{0 \,. \, . \, 1}\right]$$ defined as:
 * $$\forall \left({x, y}\right) \in \R^2: p_{X, Y} \left({x, y}\right) = \begin{cases}

\Pr \left({\left\{{\omega \in \Omega: X \left({\omega}\right) = x \and Y \left({\omega}\right) = y}\right\}}\right) & : x \in \Omega_X \text { and } y \in \Omega_Y \\ 0 & : \text {otherwise} \end{cases}$$

That is, $$p_{X, Y} \left({x, y}\right)$$ is the probability that the function $$X$$ takes the value $$x$$ at the same time that the function $$Y$$ takes the value $$y$$.

$$p_{X, Y} \left({x, y}\right)$$ can also be written:
 * $$\Pr \left({X = x, Y = y}\right)$$

Similarly to the individual mass functions of $$X$$ and $$Y$$, we have:

$$ $$ $$

The latter is usually written:
 * $$\sum_{x \in \R} p_{X, Y} \left({x, y}\right) = 1$$

Generalized Definition
Let $$X = \left\{{X_1, X_2, \ldots, X_n}\right\}$$ be a set of discrete random variables on $$\left({\Omega, \Sigma, \Pr}\right)$$.

Then the joint (probability) mass function of $$X$$ is function $$p_{X}: \R^n \to \left[{0 \,. \, . \, 1}\right]$$ defined as:
 * $$\forall x = \left({x_1, x_2, \ldots, x_n}\right) \in \R^n: p_{X} \left({x}\right) = \Pr \left({X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n}\right)$$

The properties of the two-element case can be appropriately applied.