Note: the term characteristic function is sometimes called anindicator function. This can cause some confusion, asan indicator function is also a specific function used in set theory.In addition, if you’re looking for the indicator functions used in probability and set theory (that take on values of 1 or 0), see: What is an indicator function? Otherwise, read on.

**Contents:**

## What is a Characteristic Function?

**A characteristic function completely defines a probability distribution.** Completely defining a probability distribution involves defining a *complex function *(a mix of real numbers and imaginary numbers).

The characteristic function φ_{(X)} of a random variable X is:

φ

_{(X)}= E(e^{itX}) = E(cos(tX)) + iE(sin(tX))

**Where:**

- t = a real number
- I = an imaginary unit
- E = the expected value.

## Finding Characteristic Functions

Some of the more common functions (e.g. the normal distribution and binomial distribution) have already been defined (adapted from Lee & Lee, 2010):

Finding others can be more challenging, but some rules have been formulated (much in the same way a set of rules for finding derivatives of functions have been found in calculus). For example, if a function is the sum of two independent random variables X and Y, then φ_{(X,Y)} = X + Y (Battin, 1999).

## Fourier Transform

**Outside of probability** (e.g. in quantum mechanics or signal processing), a characteristic function is called the Fourier transform. The Fourier transform in this context is defined as as “a function derived from a given function and representing it by a series of sinusoidal functions.” In other words, it’s a recipe (made up from sinusoidal functions) for a specific function of interest.

## Characteristic Function vs. MGF

A characteristic function is almost the same as a moment generating function (MGF), and in fact, they use the same symbol φ —which can be confusing. Furthermore, the difference is that the “t” in the MGF definition E(e^{tx}) is replaced by “*i*t”. In other words, the imaginary number is not present in the definition of an MGF. Therefore, the characteristic function has the advantage that it always exists—even when there is no MGF.

**References:**

Richard H. Battin. (1999). An Introduction to the Mathematics and Methods of Astrodynamics. AIAA.

Cheng-Few Lee, John Lee. (2010). Handbook of Quantitative Finance and Risk Management. Springer Science & Business Media.