Characteristic function: Difference between revisions

Revision as of 02:49, 10 March 2005

Revision as of 02:52, 10 March 2005

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Some mathematicians use the phrase '''<i>''characteristic function</i>''''' synonymously with "''[[indicator function]]"''. The indicator function of a [[subset]] ''A'' of a [[set]] ''B'' is the [[function (mathematics)|function]] with ___domain ''B'', whose value is 1 at each point in ''A'' and 0 at each point that is in ''B'' but not in ''A''.

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= \int_{-\infty}^{\infty} f_X(x)\, e^{itx}\,dx.</math>

Here ''t'' is a [[real number]], E denotes the [[expected value]] and ''F'' is the [[cumulative distribution function]]. The last form is valid only when ''f''--the [[probability density function]]-- ''f'' exists. The form preceding it is a [[Riemann-Stieltjes integral]] and is valid regardless of whether a density function exists.

If ''X'' is a [[vector space|vector]]-valued random variable, one takes the argument ''t'' to be a vector and ''tX'' to be a [[dot product]].

A characteristic function exists for any random variable. More than that, there is a [[bijection]] between cumulative probability distribution functions and characteristic functions. In other words, two probability distributions never share the same characteristic function.

Given a characteristic function &phi;, it is possible to reconstruct the corresponding cumulative probability distribution function ''F'':

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::<math>S_n = \sum_{i=1}^n a_i X_i,</math>

:where the ''a'' _''i'' are constants, then the characteristic function for ''S'' _''n'' is given by

::<math>