Indicator random variables

We have our sample space, omega. And within that sample space, we have a certain event, A.

Our random variable is defined so that it takes a value of 1, whenever the outcome of the experiment lies in A. And it takes a value of 0 whenever the outcome of the experiment lies outside the event A, i.e. A complement. This random variable is called the indicator random variable of the event A.

Indicator random variables are very useful because they allow us to translate a manipulation of events to a manipulation of random variables.