## What are Characteristic Functions?

Characteristic functions are an alternative method to simplify probability function distributions instead of calculating the density functions directly. When a random variable has any probability density function, then the “characteristic function” is simply the Fourier transform of the that probability density function.

This is particularly useful when working with cumulative distribution functions, since this approach generates simple results for the characteristic functions of distributions, defined by the weighted sums of the random variables.

Characteristic functions can also be defined by vector or matrix-valued random variables, and not just univariate distributions.

### Practical Uses of Characteristic Functions

Manipulating Probability Distributions

– These functions are quite useful for dealing with linear functions of independent random variables under the central limit theorem.

**Moment Generating Functions**– These functions can generate the moments, or shape of a set of points, for any random variable.

**Data Analysis**– Characteristics form the backbone process for fitting probability distributions to large samples of data.