Discrete Random Variable

probability-theory

Definition

Discrete Random Variable

A random variable $X:\Omega \to \mathbb{R}$ is called discrete if the image $X(\Omega )\subseteq \mathbb{R}$ is finite or countable.

Probability Mass Function

The probability mass function $f:\mathbb{R}\to \mathbb{R}$ of a discrete random variable $X$ is given by:

$$f(x)=P(X=x)=P(X^{-1}(\{x\}))$$

Properties:

• $\forall x\in \mathbb{R}:0\le f(x)\le 1$
• $x\notin f(\Omega )⟹f(x)=0$