CharacteristicFunction - Maple Help
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Statistics

 CharacteristicFunction
 compute the characteristic function

 Calling Sequence CharacteristicFunction(X, t, options)

Parameters

 X - algebraic; random variable or distribution t - algebraic; point options - (optional) equation of the form numeric=value; specifies options for computing the Characteristic function of a random variable

Description

 • The CharacteristicFunction function computes the Characteristic function of the specified random variable at the specified point.
 • The first parameter can be a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

Computation

 • By default, all computations involving random variables are performed symbolically (see option numeric below).
 • For more information about computation in the Statistics package, see the Statistics[Computation] help page.

Options

 The options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.
 • numeric=truefalse -- By default, the Characteristic function is computed using exact arithmetic. To compute the Characteristic function numerically, specify the numeric or numeric = true option.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$

Compute the Characteristic function of the beta distribution with parameters $p$ and $q$.

 > $\mathrm{CharacteristicFunction}\left('\mathrm{Β}'\left(p,q\right),t\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{p}\right]{,}\left[{p}{+}{q}\right]{,}{I}{}{t}\right)$ (1)

Define new distribution.

 > $T≔\mathrm{Distribution}\left(\mathrm{PDF}=\left(t→\mathrm{piecewise}\left(t<0,0,t<1,6t\left(1-t\right),0\right)\right)\right):$
 > $X≔\mathrm{RandomVariable}\left(T\right):$
 > $\mathrm{CDF}\left(X,t\right)$
 $\left\{\begin{array}{cc}{0}& {t}{\le }{0}\\ {-}{2}{}{{t}}^{{3}}{+}{3}{}{{t}}^{{2}}& {t}{\le }{1}\\ {1}& {1}{<}{t}\end{array}\right\$ (2)
 > $\mathrm{CharacteristicFunction}\left(X,t\right)$
 $\frac{{6}{}\left({-}{2}{}{I}{}{{ⅇ}}^{{I}{}{t}}{-}{{ⅇ}}^{{I}{}{t}}{}{t}{+}{2}{}{I}{-}{t}\right)}{{{t}}^{{3}}}$ (3)

Another distribution

 > $U≔\mathrm{Distribution}\left(\mathrm{CDF}=\left(t→F\left(t\right)\right),\mathrm{PDF}=\left(t→f\left(t\right)\right)\right):$
 > $Y≔\mathrm{RandomVariable}\left(U\right):$
 > $\mathrm{CDF}\left(Y,t\right)$
 ${F}{}\left({t}\right)$ (4)
 > $\mathrm{CharacteristicFunction}\left(Y,t\right)$
 ${2}{}{\mathrm{\pi }}{}{\mathrm{invfourier}}{}\left({f}{}\left({u}\right){,}{u}{,}{t}\right)$ (5)

References

 Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998.  Vol. 1: Distribution Theory.