Distribution Induced by a Random Variable

Compact study note.

Summary

One random variable transfers probability from the original sample space to the real line. The resulting pushforward measure is the distribution of the random variable.^[1]

Prerequisites

• Random Variable
• Borel Sigma-Algebra

Notation and Assumptions

For $X:\mathrm{\Omega }\to \mathbb{R}$ , define $P_X(B)=P(X^{-1}(B))$ for every $B\in \mathcal{B}(\mathbb{R})$ .

Essential Result

The induced distribution is a probability measure on the real Borel space. For intervals,

Small Example

If $X$ counts heads in two fair flips, then $P_X(\{1\})=P(\{HT,TH\})=1/2$ .

Common Mistakes

• Confusing the original probability measure $P$ with the induced distribution $P_X$ .
• Using $P_X(B)$ before checking $B$ is Borel.

Connections

• Random Variable
• Cumulative Distribution Function
• Distributions

References