Indicator Random Variables

Definition

The indicator r.v. $I_A$ or $I(A)$ for an event $A$ is defined to be $1$ if occurs and $0$ otherwise.

Properties

Let $A$ and $B$ be events. Then the following properties hold

1. $(I_A{)}^k=I_A$ for any positive integer $k$
2. $I_{A^c}=1-I_A$
3. $I_{A\cap B}=I_A{I}_B$
4. $I_{A\cup B}=I_A+I_B-I_A{I}_B$

Fundamental Bridge between Probability and Expectation

The probability of an event $A$ is the expected value of its indicator r.v. $I_A$:

$$P(A)=\mathrm{E}[I_A]$$