BUILT-IN MATHEMATICA SYMBOL

CharacteristicFunction

gives the characteristic function for the symbolic distribution dist as a function of the variable t.

gives the characteristic function for the multivariate symbolic distribution dist as a function of the variables

,

, ....

MORE INFORMATION

CharacteristicFunction is equivalent to Expectation[Exp[ t x], xdist].

CharacteristicFunction is equivalent to Expectation[Exp[ t.x], xdist] for vectors t and x.

The k moment can be extracted from a characteristic function cf through SeriesCoefficient[cf, {t, 0, k}]k! (-)^k.

EXAMPLES

CLOSE ALL

Basic Examples (4)

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:

In[1]:=

Out[1]=

Characteristic function for the binomial distribution:

In[1]:=

Out[1]=

Characteristic function for the bivariate normal distribution:

In[1]:=

Out[1]=

Characteristic function for the multinomial distribution:

In[1]:=

Out[1]=

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:

Properties & Relations (4)

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:

Possible Issues (1)

Symbolic closed forms do not exist for some distributions: