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Statistics

  

CharacteristicFunction

  

compute the characteristic function

 

Calling Sequence

Parameters

Description

Computation

Options

Examples

References

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

withStatistics:

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

CharacteristicFunction'Β'p,q,t

hypergeomp,p+q,It

(1)

Define new distribution.

TDistributionPDF&equals;t&rarr;piecewiset<0&comma;0&comma;t<1&comma;6t1t&comma;0&colon;

XRandomVariableT&colon;

CDFX&comma;t

&lcub;0t02t3&plus;3t2t111<t

(2)

CharacteristicFunctionX&comma;t

62I&ExponentialE;It&ExponentialE;Itt&plus;2Itt3

(3)

Another distribution

UDistributionCDF&equals;t&rarr;Ft&comma;PDF&equals;t&rarr;ft&colon;

YRandomVariableU&colon;

CDFY&comma;t

Ft

(4)

CharacteristicFunctionY&comma;t

2&pi;invfourierfu&comma;u&comma;vv&equals;t|invfourierfu&comma;u&comma;vv&equals;t

(5)

References

  

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

See Also

Statistics

Statistics[Computation]

Statistics[Distributions]

Statistics[RandomVariables]

 


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