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Indicator functions

The indicator function of an event is a random variable that takes value 1 when the event happens and value 0 when the event does not happen. Indicator functions are often used in probability theory to simplify notation and to prove theorems.

Table of Contents

Table of contents

  1. Definition

  2. Properties

    1. Powers

    2. Expected value

    3. Variance

    4. Intersections

    5. Indicators of zero-probability events

  3. Solved exercises

    1. Exercise 1

    2. Exercise 2

    3. Exercise 3

Definition

The following is a formal definition.

Definition Let Omega be a sample space and $Esubseteq Omega $ be an event. The indicator function (or indicator random variable) of the event E, denoted by $1_{E}$, is a random variable defined as follows:[eq1]

While the indicator of an event E is usually denoted by $1_{E}$, sometimes it is also denoted by[eq2]where $chi $ 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[eq3]Define the event [eq4]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 E. The case-by-case definition of this indicator is[eq5]

From the above definition, it can easily be seen that $1_{E}$ is a discrete random variable with support [eq6]and probability mass function[eq7]

Properties

Indicator functions enjoy the following properties.

Powers

The n-th power of $1_{E}$ is equal to $1_{E}$:[eq8]because $1_{E}$ can be either 0 or 1 and[eq9]

Expected value

The expected value of $1_{E}$ is equal to [eq10]:[eq11]

Variance

The variance of $1_{E}$ is equal to [eq12]. Thanks to the usual variance formula and the powers property above, we obtain[eq13]

Intersections

If E and F are two events, then[eq14]because:

  1. if $omega in Ecap F$, then [eq15]and[eq16]

  2. if [eq17], then[eq18]and[eq19]

Indicators of zero-probability events

Let E be a zero-probability event and X an integrable random variable. Then,[eq20]While a rigorous proof of this fact is beyond the scope of this introductory exposition, this property should be intuitive. The random variable [eq21] is equal to zero for all sample points omega except possibly for the points $omega in E$. 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 [eq22] must be zero.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

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

Express Y using the indicator functions of the events [eq24] and [eq25].

Solution

Denote by [eq26]the indicator of the event [eq27] and denote by [eq28]the indicator of the event [eq25]. We can write Y as[eq30]

Exercise 2

Let X be a positive random variable, that is, a random variable that can take on only positive values. Let $c$ be a constant. Prove that [eq31]where [eq32] is the indicator of the event [eq33].

Solution

First note that the sum of the indicators [eq32] and [eq35] is always equal to 1:[eq36]As a consequence, we can write[eq37]Now, note that [eq38] is a positive random variable and that the expected value of a positive random variable is positive:[eq39]Thus,[eq40]

Exercise 3

Let E be an event and denote its indicator function by $1_{E}$. Let $E^{c}$ be the complement of E 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 1:[eq41]Therefore,[eq42]

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