= Indicator random variable
{title2=$\mathbf1_A$}
An <indicator random variable> is the <indicator function> of an <event> $A$ in a <probability space>: $\mathbf1_A$ is one when $A$ occurs and zero otherwise. It has a <Bernoulli distribution> with parameter $p=\mathbb P(A)$, so $\mathbb E[\mathbf1_A]=p$ and $\operatorname{Var}(\mathbf1_A)=p(1-p)$. 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>.
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