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.
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 .
Here is the definition.
Definition Let be a sample space and be an event. The indicator function of , denoted by , is the random variable defined as
Sometimes we also use the notationwhere is the Greek letter Chi.
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
Indicator functions enjoy the following properties.
The -th power of is equal to :
This is a consequence of the facts that can be either or , and
The expected value of is equal to
The proof is as follows:
The variance of is equal to
Thanks to the usual variance formula and the powers property above, we obtain
If and are two events, then
If , then andIf , thenand
While a rigorous proof of this fact is beyond the scope of this introductory exposition, this property should be intuitive. The random variable is equal to zero for all the sample points , except possibly for the points . The expected value is a weighted average of the values can take on, where each value is weighted by its respective probability. The non-zero values can take on are weighted by zero probabilities, so must be zero.
In probability theory and statistics, there are two important concepts that are almost identical to that of an indicator variable:
Below you can find some exercises with explained solutions.
Consider a random variable and another random variable defined as a function of .
Express using the indicator functions of the events and .
Denote by the indicator of the event and denote by the indicator of the event . We can write as
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 .
First note that the sum of the indicators and is always equal to :As a consequence, we can writeNow, note that is a positive random variable and that the expected value of a positive random variable is positive:Thus,
Let be an event and denote its indicator function by .
Let be the complement of and denote its indicator function by .
Can you express as a function of ?
The sum of the two indicators is always equal to :Therefore,
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.
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