---- > [!definition] Definition. ([[indicator random variable]]) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[probability|probability space]] and $A \in \mathcal{F}$ an [[probability|event]]. Then the [[characteristic function]] $1_{A}$ given by $1_{A}(\omega)=\begin{cases} 1 & \omega \in A \\ 0 & \omega \not \in A \end{cases}$ defines a [[random variable|random variable]] $\Omega \to \mathbb{R}$, called the **indicator (random variable)** of the [[probability|event]] $A$. > When $\mathbb{P}(A)<\infty$, $\mathbb{E} 1_{A}$ [[expectation|is defined]]; it equals $\mathbb{P}(A)$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```