Indicator function
In
mathematics, the
indicator function (sometimes also called
characteristic function) of a
subset A of a
set X
is a
function from
A into to {0,1} defined as follows:
The term
characteristic function is potentially confusing becase it is also used to denote a quite different concept that is also prevalent in probability theory; see
characteristic function.
The indicator function is a basic tool in probability theory: if X is a probability space with probability measure P and A is a measurable set, then IA becomes a random variable whose expected value is equal to the probability of A:
For discrete spaces the proof may be written more simply as
Furthermore, if
A and
B are two subsets of
X, then