# Discrete Random Variables - Indicator Variables

An **indicator variable** is a random variable that takes the value 1 for some desired outcome, and the value 0 for all other outcomes. They indicate (hence the name) whether a subject belongs to a specific category or not. More specifically, an indicator variable \(X\) is defined by

\[X = \begin{cases} 1 & \text{desired event} \\ 0 & \text{other event} \end{cases}\]

## Properties of indicator variables

Indicator variables satisfy an important property: if \(X,Y\) are indicator variables, then

\[XY = \begin{cases} 1 & \text{*both* desired events} \\ 0 & \text{other event} \end{cases} \]

Indicator variables can be used in several other ways as well. For instance,

\[c^X = \begin{cases} c & \text{desired event} \\ 1 & \text{other event} \end{cases} \]

which is useful for constructing a multiplicative factor; for instance, \(2^X(\ldots)\) doubles the result if the desired event occurs, and leaves it unchanged otherwise. Similarly,

\[cX = \begin{cases} c & \text{desired event} \\ 0 & \text{other event} \end{cases} \]

which is useful for constructing an additive factor; for instance, \(cX+(\ldots)\) adds \(c\) to the result if the desired event occurs, and leaves it unchanged otherwise.

## Constructing formulae with indicator variables

Indicator variables are very useful in constructing formulas involving cases, since they vanish when the criteria for their case is not satisfied. For instance,

At a restaurant, a meal costs $10, dessert costs $5, and a drink costs $3. However, if one purchases all three, then the total price is discounted by $2. What is the price of a trip to this restaurant, in terms of the indicator variables \(X_{\text{meal}}, X_{\text{dessert}}, X_{\text{drink}}\)?

A patron of this restaurant would spend $10 on the meal if \(X_{\text{meal}}=1\), and $0 on the meal otherwise. Hence this can be modeled by \(10X_{\text{meal}}\). Similarly, the dessert and drink can be modeled by \(5X_{\text{dessert}}\) and \(3X_{\text{drink}}\), respectively. This gives an intermediate result of

\[10X_{\text{meal}}+5X_{\text{dessert}}+3X_{\text{drink}}\]

However, this formula fails to account for the $2 discount in the case of all three being purchased. This can be expressed by an indicator variable

\[X_{\text{meal}, \text{dessert}, \text{drink}} = X_{\text{meal}}X_{\text{dessert}}X_{\text{drink}}\]

so the final result is

\[10X_{\text{meal}}+5X_{\text{dessert}}+3X_{\text{drink}}-2X_{\text{meal}}X_{\text{dessert}}X_{\text{drink}}\]

Indicator variables can deal with multiplicative modifiers as well:

As part of a promotion, the same restaurant decides to give 50% off the total price if a customer purchases a meal, a drink, and a dessert, instead of the flat $2 discount. Construct a formula for the new cost of a visit, using the same indicator variables.

As before, the cost without the discount can be written as

\[10X_{\text{meal}}+5X_{\text{dessert}}+3X_{\text{drink}}\]

This should be halved if the customer purchases all 3 items, which can (as before) be modeled by the indicator variable \(X_{\text{meal}}X_{\text{dessert}}X_{\text{drink}}\). Using the multiplicative strategy from the last section, the final result is thus

\[(\frac{1}{2})^{X_{\text{meal}}X_{\text{dessert}}X_{\text{drink}}}(10X_{\text{meal}}+5X_{\text{dessert}}+3X_{\text{drink}})\]

## See Also

**Cite as:**Discrete Random Variables - Indicator Variables.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/discrete-random-variables-indicator-variables/