Characteristic function: Difference between revisions

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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>\varphivarphi_X(t) = \operatorname{E}\left(e^{itX}\right)

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

= \int_\Omega e^{itx}\, dF_X(x)

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

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

Here ''t'' is a [[real number]], E denotes the [[expected value]], and ''fF'' is the [[cumulative distribution function]]. The last equation is only valid when ''f''--the [[probability 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]].

Characteristic function exists for any random variable.

More than that, there is a bijection between cumulative probability functions and characteristic functions.

In other words, each cumulative probability function has one and only one characteristic function that corresponds to it.

Given a characteristic function ''f'', it is possible to reconstruct the corresponding cumulative probability function:

:<math>F_X(y) - F_X(x) = \lim_{\tau \to +\infty} \frac{1} {2\pi}

\int_{-\tau}^{+\tau} \frac{e^{-itx} - e{-ity}} {it}\, \varphi_X(t)\, dt</math>

Characteristic function can also be used to find [[moment (mathematics)|moments]] of random variable. Provided that ''n''-th moment exists, ''f'' can be differentiated ''n'' times and

:<math>\operatorname{E}\left(X^n\right) = i^n\, \varphi_X^{(n)}(0)

= i^n\, \left.\frac{d^n}{dt^n}\right|_{t=0} \varphi_X(t)</math>

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