Let $X$ be a discrete random variable. The function $f_X\colon\mathbb{R} \to [0,1]$ defined as $f_X(x)=P[X=x]$ is called the discrete probability function of $X$ Sometimes the syntax $p_X(x)$ is used, to mark the difference between this function and the continuous density function.

If $X$ has discrete density function $f_X(x)$ it is said that the random variable $X$ has the distribution or is distributed $f_X(x)$ and this fact is denoted as $X \sim f_X(x)$

Discrete density functions are required to satisfy the following properties:

• $f_X(x) \geq 0$ for all $x$
• $\sum_{x}f_X(x) = 1$