Index > Fundamentals of probability

Indicator function

by Marco Taboga, PhD

The indicator function of an event is a random variable that takes:

value 1 when the event happens;
value 0 when the event does not happen.

Indicator functions are also called indicator random variables.

Table of contents

Things to remember
Definition
Example
Indicators are discrete variables
Properties
Very similar concepts
Solved exercises

Things to remember

To understand the following definition, you need to remember that a random variable is a function:

from the sample space (the set of all possible random outcomes)
to the set of real numbers.

If is one of the possible outcomes, then is the value taken by when the realized outcome is .

Also remember that an event is a subset of the sample space .

Definition

Here is the definition.

Definition Let be a sample space and be an event. The indicator function of , denoted by $1_{E}$ , is the random variable defined as [eq2]

Sometimes we also use the notationwhere is the Greek letter Chi.

Example

We toss a die and one of the six numbers from 1 to 6 can appear face up.

The sample space is

Define the event described by the sentence "An even number appears face up".

A random variable that takes value 1 when an even number appears face up and value 0 otherwise is an indicator of the event .

The case-by-case definition of this indicator is [eq6]

Indicators are discrete variables

From the above definition, it can easily be seen that $1_{E}$ is a discrete random variable with support and probability mass function [eq8]

Properties

Indicator functions enjoy the following properties.

Powers

The -th power of $1_{E}$ is equal to $1_{E}$ :

Proof

This is a consequence of the facts that $1_{E}$ can be either or , and [eq10]

Expected value

The expected value of $1_{E}$ is equal to

Proof

The proof is as follows: [eq12]

Variance

The variance of $1_{E}$ is equal to

Proof

Thanks to the usual variance formula and the powers property above, we obtain [eq14]

Intersections

If and are two events, then

Proof

If , then and [eq17] If , thenand [eq20]

Indicators of zero-probability events

Let be a zero-probability event and an integrable random variable. Then,

Proof

While a rigorous proof of this fact is beyond the scope of this introductory exposition, this property should be intuitive. The random variable $X1_{E}$ is equal to zero for all the sample points , except possibly for the points . The expected value is a weighted average of the values $X1_{E}$ can take on, where each value is weighted by its respective probability. The non-zero values $X1_{E}$ can take on are weighted by zero probabilities, so must be zero.

Very similar concepts

In probability theory and statistics, there are two important concepts that are almost identical to that of an indicator variable:

the Bernoulli distribution;
the dummy variable.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Consider a random variable and another random variable defined as a function of . [eq23]

Express using the indicator functions of the events and .

Solution

Denote by the indicator of the event and denote by the indicator of the event . We can write as

Exercise 2

Let be a positive random variable, that is, a random variable that can take on only positive values.

Let be a constant.

Prove that where is the indicator of the event .

Solution

First note that the sum of the indicators and is always equal to :As a consequence, we can write [eq37] Now, note that is a positive random variable and that the expected value of a positive random variable is positive:Thus,

Exercise 3

Let be an event and denote its indicator function by $1_{E}$ .

Let $E^{c}$ be the complement of and denote its indicator function by $1_{E^{c}}$ .

Can you express $1_{E^{c}}$ as a function of $1_{E}$ ?

Solution

The sum of the two indicators is always equal to :Therefore,

How to cite

Please cite as:

Taboga, Marco (2021). "Indicator function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/indicator-functions.