Indicator function

In mathematics, the indicator function is a particular function that finds application in many areas including probability and finance.

Given any set X and any subset A of X we define the indicator function on A from X to $\{0,1\}$ as follows:

$I_{A}(x)=\{_{0\ if\ x\notin A}^{1\ if\ x\in A}$

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The indicator function is a basic tool in probability because of the following relationship

${\mathcal {E}}(I_{A})=\Sigma _{a\in A}{\mathcal {P}}(a)={\mathcal {P}}(A)$

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where ${\mathcal {E}}$ is the expectation operator and ${\mathcal {P}}$ the probability function. The result holds for events A in any probability space but the one line proof above holds only for discrete spaces.