Discrete Random Variable

Compact study note.

Summary

One discrete random variable has countable support and is described by a probability mass function. Its CDF is one step function with jumps at support points.^[1]

Prerequisites

• Random Variable

Notation and Assumptions

Use $p_X(x)=P(X=x)$ for the PMF. The support $S$ is finite or countably infinite, and $\sum _{x\in S}{p}_X(x)=1$ .

Essential Result

For any Borel set $C\subseteq \mathbb{R}$ ,

Small Example

If $X$ counts heads in two fair flips, $p_X(0)=1/4$ , $p_X(1)=1/2$ , and $p_X(2)=1/4$ .

Common Mistakes

• Calling a PMF one density.
• Forgetting that probabilities over the support must sum to one.

Connections

• Cumulative Distribution Function
• Bernoulli Distribution
• Binomial Distribution

References