Indicator Random Variable¶

Definition¶

For an event $A$, the indicator random variable $\indicator_{A}$ equals $1$ if $A$ occurs and $0$ otherwise.

Interpretation¶

An indicator turns an event into a numerical yes-or-no observation: $\indicator_A$ is $1$ on the outcomes in $A$ and $0$ on all others. Counting how many events occur becomes adding random variables: the number of the events $A_1,\ldots,A_n$ that occur is $\indicator_{A_1}+\cdots+\indicator_{A_n}$${\text{𝟙}}_{A_n}$, whether or not the events are independent.

In the figure the outcomes $\omega_1$ and $\omega_2$ lie in $A$ and are sent to $1$, while $\omega_3$ and $\omega_4$ are sent to $0$. So $\indicator_A$ takes only the values $0$ and $1$, with $\mathbb{P}(\indicator_A=1)=\mathbb{P}(A)$$\mathbb{P}(A)$: a Bernoulli variable with parameter $\mathbb{P}(A)$, as in the Key Results.

Operations on events become arithmetic: $\indicator_A\indicator_B=1$ exactly when both $A$ and $B$ occur, $1-\indicator_A=1$ exactly when $A$ fails, and $\indicator_{A\cup B}=\indicator_A+\indicator_B-\indicator_A\indicator_B$ is the indicator form of the inclusion-exclusion principle. Combined with the fundamental bridge $\mathbb{E}[\indicator_A]=\mathbb{P}(A)$$\mathbb{P}(A)$, a sum of indicators turns an expected count into a sum of probabilities.

Key Results¶

• $\indicator_A\sim\operatorname{Bern}(\mathbb{P}(A))$$\mathrm{Bern}(\mathbb{P}(A))$.
• $\indicator_{A\cap B}=\indicator_A\indicator_B$ and $\indicator_{A^c}=1-\indicator_A$${\text{𝟙}}_A$.
• A sum of indicators counts the number of events that occur, even when those events are dependent.

Examples¶

• A fair-toss heads indicator is $1$ on heads and $0$ on tails.
• The number of heads in three tosses is $\indicator_{A_1}+\indicator_{A_2}+\indicator_{A_3}$${\text{𝟙}}_{A_3}$.

Prerequisites¶

• Event
• Random Variable

Related Concepts¶

• Inclusion-Exclusion Principle
• Expectation
• Bernoulli Distribution
• Linearity of Expectation

Sources¶

• Probability Theory for EOR · Week 3
• Probability Theory for EOR · Week 4
• Nicky van Foreest · Probability Distributions · Refresher

Aliases: indicator r.v., indicator variable Keywords: event indicator Symbols: intersection, omega, union

Last updated: 20 October 2026 03:02