Characteristic function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the term "characteristic function" can refer to any of several distinct concepts:

<ul><li>The <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of a <a href="/facts/Subset/SJGERmbA">subset</a>, that is the <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
 
 
 
 
 
 1
 
 
 A
 
 
 :
 X
 →
 {
 0
 ,
 1
 }
 ,
 
 
 {\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},}
 
 which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.</li>
<li>The <a href="/facts/Characteristic_function_(convex_analysis)/XfTwr4yd">characteristic function</a> in <a href="/facts/Convex_analysis/SGhUNLMK">convex analysis</a>, closely related to the indicator function of a set: 
 
 
 
 
 χ
 
 A
 
 
 (
 x
 )
 :=
 
 
 {
 
 
 
 0
 ,
 
 
 x
 ∈
 A
 ;
 
 
 
 
 +
 ∞
 ,
 
 
 x
 ∉
 A
 .
 
 
 
 
 
 
 
 
 {\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}
 
</li>
<li>In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, the <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> of any <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> on the <a href="/facts/Real_line/6eq42xX5">real line</a> is given by the following formula, where X is any <a href="/facts/Random_variable/TwTBXnLT">random variable</a> with the distribution in question: 
 
 
 
 
 φ
 
 X
 
 
 (
 t
 )
 =
 E
 ⁡
 
 (
 
 e
 
 i
 t
 X
 
 
 )
 
 ,
 
 
 {\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),}
 
 where 
 
 
 
 E
 
 
 {\displaystyle \operatorname {E} }
 
 denotes <a href="/facts/Expected_value/1XV0JKL8">expected value</a>. For <a href="/facts/Joint_probability_distribution/klX2ksGY">multivariate distributions</a>, the product tX is replaced by a <a href="/facts/Scalar_product/tNz8MLjT">scalar product</a> of vectors.</li>
<li>The characteristic function of a <a href="/facts/Cooperative_game_theory/L2KfzepP">cooperative game</a> in <a href="/facts/Game_theory/zkEIj1ya">game theory</a>.</li>
<li>The <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> in <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>.</li>
<li>The <a href="/facts/Characteristic_state_function/AyfBATrx">characteristic state function</a> in <a href="/facts/Statistical_mechanics/6zMsNYYG">statistical mechanics</a>.</li>
<li>The <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a>, a <a href="/facts/Topology/2EqW47cU">topological</a> invariant.</li>
<li>The <a href="/facts/Receiver_operating_characteristic/o9EMmFay">receiver operating characteristic</a> in statistical <a href="/facts/Decision_theory/TbXcQX0v">decision theory</a>.</li>
<li>The point characteristic function in <a href="/facts/Statistics/DNPsKGYU">statistics</a>.</li></ul>

Characteristic function open-in-new

Characteristic function