Indicator variable

A Random variable that has a support of $\{0,1\}$.

The PMF of an indicator usually has $1$ as the event that $A$ occurs, and $0$ if it doesn’t. If $P(A)$ is nonzero, then the indicator has a Bernoulli distribution.

A “fundamental bridge” exists between the probability of event $A$ occurring and the Expectation of this indicator. This follows from our definition of an indicator’s PMF above.

$$\mathbb{E}(I_A)=0\cdot P(A^{\complement })+1\cdot P(A)=P(A)$$

This allows us to calculate the expectation of a r.v. using indicator variables, for example.