3.1 Discrete random variables

Definition:

A function $X:\Omega \rightarrow M$ is called a random variable.

If the codomain $M$ is a finite or countable set, then $X$ is called a discrete random variable.

Definition:

Probability mass function

Let $\mathbb P$ be a probability defined on the sample space $\Omega$, and $X:\Omega \rightarrow M$ be a discrete random variable.

The function $f_X:\mathbb R \rightarrow[0,1]$ defined by

$$ f_X(x):= \left\{ \begin{array}{ll} \mathbb P(X=x) &, if\ x \in M, \\ 0 &, if\ x \notin M;\\ \end{array} \right. $$

is called the probability mass function (p.m.f) of the random variable $X$.

• Example:

Theorem:

Properties of probability mass functions:

Let $X:\Omega \rightarrow M$ be a discrete random variable with p.m.f. $f_X$.

Then the following hold:

1. Positivity(non-negative): $f_X(x) \geq 0, \forall x\in \mathbb R.$