Indicator Functions

A class of functions known as indicator functions is useful in statistics.

Definition 2..1

Suppose $\Omega$ is a set with typical element $\omega$, and let $A$ be a subset of $\Omega$. The indicator function of $A$, denoted by $I_A(\cdot)$, is defined by

$$I_A(w)= \left\{ \begin{array}{l} 1 \ \ \ \ \mbox{if} \ \ \... ...0 \ \ \ \ \mbox{if} \ \ \omega \notin A. \end{array} \right.$$ (2.1)

That is, $I_A(\cdot)$ indicates the set $A$. Some properties of the indicator function are listed.

(a)

$I_A(\omega )=1-I_{\bar{A}}(\omega )$ where $\bar{A}$ is the complement of $A$.

(b)

$I_A^2(\omega )=I_A(\omega )$

(c)

$I_{A\cap B}(\omega )=I_A(w).I_B(\omega )$

(d)

$I_{A\cup B}(\omega )=I_A(\omega )+I_B(\omega )-I_{A\cap B}(\omega )$

(e)

$I_{A_1\cup A_2 \cup \dots \cup A_n}(\omega )=\max\{ I_{A_1}(\omega ),\, \dots,\, I_{A_n}(\omega )\}$

The following example shows a use for indicator functions.

Example 2..1
Suppose random variable $X$ has pdf given by

$$f(x)= \left\{ \begin{array}{ll} 0 \ , & x <-1\\ 1+x \ , &... ... \ , & 0 \leq x < 1\\ 0 \ , & x \geq 1 \end{array} \right.$$

We can write $f(x)$ as

$$f(x)=(1+x)I_{[-1,0)}(x)+(1-x)I_{[0,1)}(x),$$

or more concisely

$$f(x)=(1-\vert x\vert)I_{[-1,1]}(x)$$