The characteristic function is defined as the Fourier Transform of the Probability Density Function,

(1) | |||

(2) | |||

(3) | |||

(4) |

where (sometimes also denoted ) is the th Moment about 0 and . The characteristic function can therefore be used to generate Moments about 0,

(5) |

(6) |

A Distribution is not uniquely specified by its Moments, but is uniquely specified by its characteristic function.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 928, 1972.

Kenney, J. F. and Keeping, E. S. ``Moment-Generating and Characteristic Functions,'' ``Some Examples of Moment-Generating Functions,''
and ``Uniqueness Theorem for Characteristic Functions.'' §4.6-4.8 in *Mathematics of Statistics, Pt. 2, 2nd ed.*
Princeton, NJ: Van Nostrand, pp. 72-77, 1951.

© 1996-9

1999-05-26