Characteristic function: Difference between revisions

Revision as of 18:40, 9 November 2003

Revision as of 15:13, 15 January 2004

Line 4:

In [[probability theory]], the '''characteristic function''' of any [[probability distribution]] on the [[real number|real]] line is given by the following formula, where ''X'' is any [[random variable]] with the distribution in question:

:<math>\varphi(t)=E\left(e^{itX}\right).</math>

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

Here ''t'' is a [[real number]] and, E denotes the [[expected value]], and ''f'' is the [[probability density function]].

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]].

Related concepts include the [[moment-generating function]] and the [[probability-generating function]].

The characteristic function is closely related to the [[Fourier transform]]:

the characteristic function of a distribution with density function ''f'' is proportional to the inverse Fourier transform of ''f''.