Some mathematicians use the phrase characteristic function
synonymously with "indicator function
". The indicator function of a subset A
of a set B
is the function
with domain B
, whose value is 1 at each point in A
and 0 at each point that is in B
but not in A
In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:
is a real number
, E denotes the expected value
, and f
is the probability density function
If X is a vector-valued random variable, one takes the argument t to be a vector and tX to be a dot product.
Related concepts include the moment-generating function and the probability-generating function.
The characteristic function is closely related to the Fourier transform:
the characteristic function of a distribution with density function f is proportional to the inverse Fourier transform of f.