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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:

Here t is a real number, E denotes the expected value and F is the cumulative distribution function. The last form is valid only when f--the probability density function--exists. The form preceding it is a Riemann-Stieltjes integral and is valid regardless of whether a density function exists.

If X is a vector-valued random variable, one takes the argument t to be a vector and tX to be a dot product.

A characteristic function exists for any random variable. More than that, there is a bijection between cumulative probability functions and characteristic functions. In other words, two probability distributions never share the same characteristic function.

Given a characteristic function φ, it is possible to reconstruct the corresponding cumulative probability distribution function F:

In general this is an improper integral; the function being integrated may be only conditionally integrable rather than Lebesgue-integrable, i.e. the integral of its absolute value may be infinite.

Characteristic functions are used in the most frequently seen proof of the central limit theoremCentral limit theorems are a set of weak-convergence results in probability theory. Intuitively, they all express the fact that any sum of many independent identically distributed random variables is approximately normally distributed. These results expla.

Characteristic functions can also be used to find momentsSee also moment (physics). The concept of moment in mathematics evolved from the concept of moment in physics. The n''th moment of a real-valued function f ''x of a real variable is : The problem of moments seeks characterizations of sequences { μ&prim of random variable. Provided that n-th moment exists, characteristic function can be differentiated n times and

Related concepts include the moment-generating functionIn probability theory and statistics, the moment-generating function of a random variable X is : The moment-generating function generates the moments of the probability distribution, as follows: : If X has a continuous probability density function f ''x t and the probability-generating functionIn probability theory, the probability-generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability-generating functions are often employe.

The characteristic function is closely related to the Fourier transformThe Fourier transform named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes").: the characteristic function of a distribution with density function f is proportional to the inverse Fourier transform of f.

Characteristic functions are particularly useful for dealing with functions of independentIn probability theory, to say that two events are independent intuitively means that knowing whether or not one of them occurs makes it neither more probable nor less probable that the other occurs. For example, the event of getting a "1" when a die is th random variables. For example, if X₁, X₂, ..., X_n is a sequence of independent (and not necessarily identically distributed) random variables, and

where the a _i are constants, then the characteristic function for S _n is given by