Definition:Characteristic Function
Definition
A characteristic function is a member of the class of functions which in some way allows one to sum up a number of characteristics of a particular mathematical object.
Disambiguation
This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.
Characteristic Function may refer to:
Set theory
Characteristic Function of Set
Let $E \subseteq S$.
The characteristic function of $E$ is the function $\chi_E: S \to \set {0, 1}$ defined as:
- $\map {\chi_E} x = \begin {cases} 1 & : x \in E \\ 0 & : x \notin E \end {cases}$
That is:
- $\map {\chi_E} x = \begin {cases} 1 & : x \in E \\ 0 & : x \in \relcomp S E \end {cases}$
where $\relcomp S E$ denotes the complement of $E$ relative to $S$.
Characteristic Function of Relation
The concept of a characteristic function of a subset carries over directly to relations.
Let $\RR \subseteq S \times T$ be a relation.
The characteristic function of $\RR$ is the function $\chi_\RR: S \times T \to \set {0, 1}$ defined as:
- $\map {\chi_\RR} {x, y} = \begin {cases} 1 & : \tuple {x, y} \in \RR \\ 0 & : \tuple {x, y} \notin \RR \end{cases}$
It can be expressed in Iverson bracket notation as:
- $\map {\chi_\RR} {x, y} = \sqbrk {\tuple {x, y} \in \RR}$
More generally, let $\ds \mathbb S = \prod_{i \mathop = 1}^n S_i = S_1 \times S_2 \times \ldots \times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \ldots, S_n$.
Let $\RR \subseteq \mathbb S$ be an $n$-ary relation on $\mathbb S$.
The characteristic function of $\RR$ is the function $\chi_\RR: \mathbb S \to \set {0, 1}$ defined as:
- $\map {\chi_\RR} {s_1, s_2, \ldots, s_n} = \begin {cases} 1 & : \tuple {s_1, s_2, \ldots, s_n} \in \RR \\ 0 & : \tuple {s_1, s_2, \ldots, s_n} \notin \RR \end {cases}$
It can be expressed in Iverson bracket notation as:
- $\map {\chi_\RR} {s_1, s_2, \ldots, s_n} = \sqbrk {\tuple {s_1, s_2, \ldots, s_n} \in \RR}$
Probability theory
Characteristic Function of Random Variable
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
The characteristic function of $X$ is the mapping $\phi: \R \to \C$ defined by:
- $\map \phi t = \expect {e^{i t X} }$
where:
- $i$ is the imaginary unit
- $\expect \cdot$ denotes expectation.