Statistics - Maple Help

Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Quantities : Statistics/CharacteristicFunction

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).

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.

 > $T≔\mathrm{Distribution}\left(\mathrm{PDF}=\left(t→\mathrm{piecewise}\left(t<0,0,t<1,6t\left(1-t\right),0\right)\right)\right):$
 > $X≔\mathrm{RandomVariable}\left(T\right):$
 > $\mathrm{CDF}\left(X,t\right)$
 ${{}\begin{array}{cc}{0}& {t}{\le }{0}\\ {-}{2}{}{{t}}^{{3}}{+}{3}{}{{t}}^{{2}}& {t}{\le }{1}\\ {1}& {1}{<}{t}\end{array}$ (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)
 > $\mathrm{CharacteristicFunction}\left(Y,t\right)$
 ${2}{}{\mathrm{π}}{}\left(\genfrac{}{}{0}{}{{\mathrm{invfourier}}{}\left({f}{}\left({u}\right){,}{u}{,}{v}\right)}{\phantom{{v}{=}{t}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{{\mathrm{invfourier}}{}\left({f}{}\left({u}\right){,}{u}{,}{v}\right)}}{{v}{=}{t}}\right)$ (5)

References

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