This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

CharacteristicFunction

 CharacteristicFunctiongives the characteristic function for the symbolic distribution dist as a function of the variable t. CharacteristicFunctiongives the characteristic function for the multivariate symbolic distribution dist as a function of the variables , , ....
• The k moment can be extracted from a characteristic function cf through SeriesCoefficient[cf, {t, 0, k}]k! (-)k.
Characteristic function (cf) for the normal distribution:
Characteristic function for the binomial distribution:
Characteristic function for the bivariate normal distribution:
Characteristic function for the multinomial distribution:
Characteristic function (cf) for the normal distribution:
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Characteristic function for the binomial distribution:
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Characteristic function for the bivariate normal distribution:
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Characteristic function for the multinomial distribution:
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 Scope   (7)
Characteristic function for a specific continuous distribution:
Characteristic function for a specific discrete distribution:
Characteristic function at a particular value:
Characteristic function evaluated numerically:
Obtain a result at any precision:
Compute the characteristic function for a formula distribution:
Find the characteristic function for a parameter mixture distribution:
 Applications   (7)
Compute the raw moments for a Poisson distribution:
First 5 raw moments using derivatives of the characteristic function at the origin:
Use Moment directly:
Compute mixed raw moments for a multivariate distribution:
Use Moment to obtain raw moments directly:
Find raw moments of a Student distribution from its characteristic function:
Compute to extract moments by taking limits from the right:
Evaluate the limits from the left:
Only the first four moments are defined, as confirmed by using Moment directly:
Use inverse Fourier transform to compute the PDF corresponding to a characteristic function:
Illustrate the central limit theorem on the example of symmetric LaplaceDistribution:
Find the characteristic function of the rescaled random variate:
Compute the large limit of the cf of the sum of such i.i.d. random variates:
Compare with the characteristic function of a standard normal variate:
Use smooth characteristic function to construct the upper bound for the distribution density of ErlangDistribution:
Plot the upper bounds and the original density:
Verify that the sum where are independent identically distributed BernoulliDistribution variates tends in distribution to UniformDistribution for large :
Use a combinatorial equality for product :
Evaluate the sum:
Take the limit and compare it to the characteristic function of the UniformDistribution:
CharacteristicFunction is the Expectation of for real :
The characteristic function is related to all other generating functions when they exist:
The PDF is the inverse Fourier transform of the cf for continuous distributions:
The PDF is the inverse Fourier sequence transform of the cf for discrete distributions:
Symbolic closed forms do not exist for some distributions: