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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


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

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

We can write $f(x)$ as

\begin{displaymath}f(x)=(1+x)I_{[-1,0)}(x)+(1-x)I_{[0,1)}(x), \end{displaymath}

or more concisely

\begin{displaymath}f(x)=(1-\vert x\vert)I_{[-1,1]}(x) \end{displaymath}



Bob Murison 2000-10-31