In probability theory and statistics, the **cumulants** κ_{n} of a probability distribution are given by

The "problem of cumulants" seeks characterizations of sequences that are cumulants of some probability distribution.

The *n*th cumulant is homogeneous of degree *n*, i.e. if *c* is any constant, then

If *X* and *Y* are independent random variables then κ_{n}(*X* + *Y*) = κ_{n}(*X*) + κ_{n}(*Y*).

The cumulants are related to the moments by the following recursion formula:

The coefficients are precisely those that occur in Faà di Bruno's formula.

These polynomials have a remarkable combinatorial interpretation: the coefficients count certain partitions of sets. A general form of these polynomials is

- π runs through the list of all partitions of a set of size
*n*; - "
*B*∈ π" means*B*is one of the "blocks" into which the set is partitioned; and - |
*B*| is the size of the set*B*.

The cumulants of the normal distribution with expected value μ and variance σ^{2} are κ_{1} = μ, κ_{2} = σ^{2}, and κ_{n} = 0 for *n* > 2.

All of the cumulants of the Poisson distribution are equal to the expected value.

The **joint cumulant** of several random variables *X*_{1}, ..., *X*_{n} is

The combinatorial meaning of the expression of moments in terms of cumulants is easier to understand than that of cumulants in terms of moments:

The law of total expectation and the law of total variance generalize naturally to conditional cumulants. The case *n* = 3, expressed in the language of (central) moments rather than that of cumulants, says

Cumulants were first introduced by the Danish astronomer, actuary, mathematician, and statistician Thorvald N. Thiele (1838 - 1910) in 1889. Thiele called them *half-invariants*. They were first called *cumulants* in a 1931 paper, *The derivation of the pattern formulae of two-way partitions from those of simpler patterns*, Proceedings of the London Mathematical Society, Series 2, v. 33, pp. 195-208, by the great statistical geneticist Sir Ronald Fisher and the statistician John Wishart, eponym of the Wishart distribution. In another paper published in 1929, Fisher had called them *cumulative moment functions*.

In combinatorics, the *n*th Bell number is the number of partitions of a set of size *n*. All of the cumulants of the sequence of Bell numbers are equal to 1. The Bell numbers are the moments of the Poisson distribution with expected value 1.

For any sequence { κ_{n} : *n* = 1, 2, 3, ... } of scalars in a field of characteristic zero, being considered formal cumulants, there is a corresponding sequence { μ ′ : *n* = 1, 2, 3, ... } of formal moments, given by the polynomials above. For those polynomials, construct a polynomial sequence in the following way. Out the polynomial

This sequence of polynomials is of binomial type. In fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of cumulants.