Indicator Random Variable

Topics: Probability - Random Variable

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(definition)

Let $A$ be an event.

We define the indicator random variable of $A$, denoted $1_A$ as:

$$X:=1_A$$

…where $1_A$ is the indicator function for $A$:

$$1_A=\{\begin{array}{ll}1,& \text{ if }x\in A\\ 0,& \text{ if }x\notin A\end{array}$$

Density Function and Expected Value §

The density function of $X=1_A$ an indicator random variable is always:

f_X(x) = \begin{cases*} \mathbb{P}(A), \text{ if } x = 1 \\ \mathbb{P}(A^c), \text{ if } x = 0 \\ \end{cases*}

Which tells us that its expected value is:

$$\begin{array}{rl}\mathbb{E}[1_A]=\mathbb{E}[X]& =1\mathbb{P}(A)+0\mathbb{P}(A^c)\\ & =\mathbb{P}(A)\end{array}$$