Characteristic Function
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 characteristicfunction can therefore be used to generate Moments about 0, | (5) |
, | (6) |
A Distribution is not uniquely specified by its Moments, but isuniquely specified by its characteristic function.
See also Cumulant, Moment, Moment-Generating Function, Probability Density 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.
- Complex Differentiable
- Complex Function
- Complexity (Number)
- Complexity (Sequence)
- Complexity Theory
- Complex Matrix
- Complex Multiplication
- Complex Number
- Complex Plane
- Complex Projective Plane
- Complex Representation
- Complex Structure
- Component
- Composite Knot
- Compositeness Certificate
- Compositeness Test
- Composite Number
- Composite Runs
- Composition
- Composition Series
- Composition Theorem
- Compound Interest
- Compound Polyhedron
- Computability
- Computable Function