Statistics
CharacteristicFunction
compute the characteristic function
Calling Sequence
Parameters
Description
Computation
Options
Examples
References
CharacteristicFunction(X, t, options)
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
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]).
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.
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.
with⁡Statistics:
Compute the Characteristic function of the beta distribution with parameters p and q.
CharacteristicFunction⁡'Β'⁡p,q,t
hypergeom⁡p,p+q,I⁢t
Define new distribution.
T ≔ Distribution⁡PDF=t→piecewise⁡t<0,0,t<1,6⁢t⁢1−t,0:
X ≔ RandomVariable⁡T:
CDF⁡X,t
{0t≤0−2⁢t3+3⁢t2t≤111<t
CharacteristicFunction⁡X,t
6⁢−2⁢I⁢ⅇI⁢t−ⅇI⁢t⁢t+2⁢I−tt3
Another distribution
U ≔ Distribution⁡CDF=t→F⁡t,PDF=t→f⁡t:
Y ≔ RandomVariable⁡U:
CDF⁡Y,t
F⁡t
CharacteristicFunction⁡Y,t
2⁢π⁢invfourier⁡f⁡u,u,vv=t|invfourier⁡f⁡u,u,vv=t
Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.
See Also
Statistics[Computation]
Statistics[Distributions]
Statistics[RandomVariables]
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