In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.

The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the n-th-order cumulant of their sum is equal to the sum of their n-th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property.

Just as for moments, where joint moments are used for collections of random variables, it is possible to define joint cumulants.

The cumulants of a random variable X are defined using the cumulant-generating function K(t), which is the natural logarithm of the moment-generating function:

The cumulants κn are obtained from a power series expansion of the cumulant generating function:

This expansion is a Maclaurin series, so the n-th cumulant can be obtained by differentiating the above expansion n times and evaluating the result at zero:[1]

If the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later.

Some writers[2][3] prefer to define the cumulant-generating function as the natural logarithm of the characteristic function, which is sometimes also called the second characteristic function,[4][5]

An advantage of H(t)—in some sense the function K(t) evaluated for purely imaginary arguments—is that E[eitX] is well defined for all real values of t even when E[etX] is not well defined for all real values of t, such as can occur when there is "too much" probability that X has a large magnitude. Although the function H(t) will be well defined, it will nonetheless mimic K(t) in terms of the length of its Maclaurin series, which may not extend beyond (or, rarely, even to) linear order in the argument t, and in particular the number of cumulants that are well defined will not change. Nevertheless, even when H(t) does not have a long Maclaurin series, it can be used directly in analyzing and, particularly, adding random variables. Both the Cauchy distribution (also called the Lorentzian) and more generally, stable distributions (related to the Lévy distribution) are examples of distributions for which the power-series expansions of the generating functions have only finitely many well-defined terms.

The cumulative property follows quickly by considering the cumulant-generating function:

so that each cumulant of a sum of independent random variables is the sum of the corresponding cumulants of the addends. That is, when the addends are statistically independent, the mean of the sum is the sum of the means, the variance of the sum is the sum of the variances, the third cumulant (which happens to be the third central moment) of the sum is the sum of the third cumulants, and so on for each order of cumulant.

A distribution with given cumulants κn can be approximated through an Edgeworth series.

All of the higher cumulants are polynomial functions of the central moments, with integer coefficients, but only in degrees 2 and 3 are the cumulants actually central moments.

the above probability distributions get a unified formula for the derivative of the cumulant generating function:[citation needed]

confirming that the first cumulant is κ1 = K′(0) = μ and the second cumulant is κ2 = K′′(0) = με.

Note the analogy to the classification of conic sections by eccentricity: circles ε = 0, ellipses 0 < ε < 1, parabolas ε = 1, hyperbolas ε > 1.

The cumulant generating function K(t), if it exists, is infinitely differentiable and convex, and passes through the origin. Its first derivative ranges monotonically in the open interval from the infimum to the supremum of the support of the probability distribution, and its second derivative is strictly positive everywhere it is defined, except for the degenerate distribution of a single point mass. The cumulant-generating function exists if and only if the tails of the distribution are majorized by an exponential decay, that is, (see Big O notation)

If the support of a random variable X has finite upper or lower bounds, then its cumulant-generating function y = K(t), if it exists, approaches asymptote(s) whose slope is equal to the supremum and/or infimum of the support,

If K(t) is finite for a range t1 < Re(t) < t2 then if t1 < 0 < t2 then K(t) is analytic and infinitely differentiable for t1 < Re(t) < t2. Moreover for t real and t1 < t < t2 K(t) is strictly convex, and K′(t) is strictly increasing.[citation needed]

Given the results for the cumulants of the normal distribution, it might be hoped to find families of distributions for which κm = κm+1 = ⋯ = 0 for some m > 3, with the lower-order cumulants (orders 3 to m − 1) being non-zero. There are no such distributions.[7] The underlying result here is that the cumulant generating function cannot be a finite-order polynomial of degree greater than 2.

So the cumulant generating function is the logarithm of the moment generating function

The first cumulant is the expected value; the second and third cumulants are respectively the second and third central moments (the second central moment is the variance); but the higher cumulants are neither moments nor central moments, but rather more complicated polynomial functions of the moments.

The explicit expression for the n-th moment in terms of the first n cumulants, and vice versa, can be obtained by using Faà di Bruno's formula for higher derivatives of composite functions. In general, we have

The n-th moment μn is an n-th-degree polynomial in the first n cumulants. The first few expressions are:

The "prime" distinguishes the moments μn from the central moments μn. To express the central moments as functions of the cumulants, just drop from these polynomials all terms in which κ1 appears as a factor:

Similarly, the n-th cumulant κn is an n-th-degree polynomial in the first n non-central moments. The first few expressions are:

To express the cumulants κn for n > 1 as functions of the central moments, drop from these polynomials all terms in which μ'1 appears as a factor:

To express the cumulants κn for n > 2 as functions of the standardized central moments, also set μ'2=1 in the polynomials:

The cumulants can be related to the moments by differentiating the relationship log M(t) = K(t) with respect to t, giving M′(t) = K′(t) M(t), which conveniently contains no exponentials or logarithms. Equating the coefficient of t n−1 on the left and right sides and using μ′0 = 1 gives the following formulas for n ≥ 1:[8]

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

Thus each monomial is a constant times a product of cumulants in which the sum of the indices is n (e.g., in the term κ3 κ22 κ1, the sum of the indices is 3 + 2 + 2 + 1 = 8; this appears in the polynomial that expresses the 8th moment as a function of the first eight cumulants). A partition of the integer n corresponds to each term. The coefficient in each term is the number of partitions of a set of n members that collapse to that partition of the integer n when the members of the set become indistinguishable.

Further connection between cumulants and combinatorics can be found in the work of Gian-Carlo Rota, where links to invariant theory, symmetric functions, and binomial sequences are studied via umbral calculus.[9]

The joint cumulant of several random variables X1, ..., Xn is defined by a similar cumulant generating function

If any of these random variables are identical, e.g. if X = Y, then the same formulae apply, e.g.

although for such repeated variables there are more concise formulae. For zero-mean random vectors,

The joint cumulant of just one random variable is its expected value, and that of two random variables is their covariance. If some of the random variables are independent of all of the others, then any cumulant involving two (or more) independent random variables is zero. If all n random variables are the same, then the joint cumulant is the n-th ordinary cumulant.

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

Just as the second cumulant is the variance, the joint cumulant of just two random variables is the covariance. The familiar identity

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

In statistical physics many extensive quantities – that is quantities that are proportional to the volume or size of a given system – are related to cumulants of random variables. The deep connection is that in a large system an extensive quantity like the energy or number of particles can be thought of as the sum of (say) the energy associated with a number of nearly independent regions. The fact that the cumulants of these nearly independent random variables will (nearly) add make it reasonable that extensive quantities should be expected to be related to cumulants.

Cumulants were first introduced by Thorvald N. Thiele, in 1889, who called them semi-invariants.[13] They were first called cumulants in a 1932 paper[14] by Ronald Fisher and John Wishart. Fisher was publicly reminded of Thiele's work by Neyman, who also notes previous published citations of Thiele brought to Fisher's attention.[15] Stephen Stigler has said[citation needed] that the name cumulant was suggested to Fisher in a letter from Harold Hotelling. In a paper published in 1929,[16] Fisher had called them cumulative moment functions. The partition function in statistical physics was introduced by Josiah Willard Gibbs in 1901.[citation needed] The free energy is often called Gibbs free energy. In statistical mechanics, cumulants are also known as Ursell functions relating to a publication in 1927.[citation needed]

where the values of κn for n = 1, 2, 3, ... are found formally, i.e., by algebra alone, in disregard of questions of whether any series converges. All of the difficulties of the "problem of cumulants" are absent when one works formally. The simplest example is that the second cumulant of a probability distribution must always be nonnegative, and is zero only if all of the higher cumulants are zero. Formal cumulants are subject to no such constraints.

In combinatorics, the n-th Bell number is the number of partitions of a set of size n. All of the . The Bell numbers are the .

and then generalize the pattern. The pattern is that the numbers of blocks in the aforementioned partitions are the exponents on x. Each coefficient is a polynomial in the cumulants; these are the Bell polynomials, named after Eric Temple Bell.[citation needed]

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 formal cumulants.[citation needed]

The ordinary cumulants of degree higher than 2 of the normal distribution are zero. The free cumulants of degree higher than 2 of the Wigner semicircle distribution are zero.[18] This is one respect in which the role of the Wigner distribution in free probability theory is analogous to that of the normal distribution in conventional probability theory.