Indicator Functions

The random variable $\mathbb{1}_A$ takes the value 1 if $\omega \in A$ and 0 otherwise.
$\mathbb{E}[\mathbb{1}_A] = P(A)$

Indicator functions are very useful for computations as the next example will show.

Example 1.
Consider the hat problem we solved using the inclusion exclusion principle. Let $X$ be the random variable, the number of people who get back their own hat. $X = 1_{A_1} + 1_{A_2}+…+1_{A_n}$ We can calculate the mean and variance of $X$ using linearity of the expectation operator. $$\begin{align} \mathbb{E}[X] = \sum_i^n \mathbb{E}[1_{A_i}] = \sum_{i=1}^n P(A_i) = 1 \end{align}$$ $$\begin{align} X^2 = \sum_i^n 1_{A_i}^2 + 2\sum_{i < j} 1_{A_i} 1_{A_j} \end{align}$$ so that $$\begin{align} \mathbb{E}[X^2] &= \sum_i^n P(A_i) + 2\sum_{i < j} P(A_i \cap A_j) \\ &= 1 + 2 \frac{n (n-1)}{2} \frac{1}{n}\frac{1}{n-1} \\ &=2 \end{align}$$ so $\text{Var}(X) = 1$