 Indicator random variable

An indicator random variable is the indicator function of an event $A$ in a probability space: $1_A$ is one when $A$ occurs and zero otherwise. It has a Bernoulli distribution with parameter $p=\mathbb{P}(A)$, so $\mathbb{E}[1_A]=p$ and $\mathrm{Var}(1_A)=p(1-p)$. Even an event described without numbers becomes a numerical random variable in this way.

For a finite collection of events, their indicator random variables add up to the number that occur. Linearity of expectation therefore turns an average count into a sum of probabilities, without requiring independence. The fixed point count of a uniform random permutation provides an example where the indicators are dependent random variables.