Cumulant

Cumulants of probability distributions

In probability theory and statistics, the cumulants κ_n of a probability distribution are given by

$E\left(e^{tX}\right)=\exp \left(\sum _{n=1}^{\infty }\kappa _{n}t^{n}/n!\right)$

where X is any random variable whose probability distribution is the one whose cumulants are taken. In other words, κ_n/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. The logarithm of the moment-generating function is therefore called the cumulant-generating function.

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

Cumulants of particular probability distributions

The cumulants of the normal distribution with expected value μ and variance σ² are κ₁ = μ, κ₂ = σ², and κ_n = 0 for n > 2.

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

Some properties of cumulants

The first cumulant is shift-equivariant; all of the others are shift-invariant. To state this less tersely, denote by κ_n(X) the nth cumulant of the probability distribution of the random variable X. The statement is that if c is constant then κ₁(X + c) = κ₁(X) + c and κ_n(X + c) = κ_n(X) for n≥ 2, i.e., c is added to the first cumulant, but all higher cumulants are unchanged.

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:

$\kappa _{n}=\mu '_{n}-\sum _{k=1}^{n-1}{n-1 \choose k-1}\kappa _{k}\mu _{n-k}'.$

The nth moment μ′_n is an nth-degree polynomial in the first n cumulants, thus:

$\mu '_{1}=\kappa _{1}$

$\mu '_{2}=\kappa _{2}+\kappa _{1}^{2}$

$\mu '_{3}=\kappa _{3}+3\kappa _{2}\kappa _{1}+\kappa _{1}^{3}$

$\mu '_{4}=\kappa _{4}+4\kappa _{3}\kappa _{1}+3\kappa _{2}^{2}+6\kappa _{2}\kappa _{1}^{2}+\kappa _{1}^{4}$

$\mu '_{5}=\kappa _{5}+5\kappa _{4}\kappa _{1}+10\kappa _{3}\kappa _{2}+10\kappa _{3}\kappa _{1}^{2}+15\kappa _{2}^{2}\kappa _{1}+10\kappa _{2}\kappa _{1}^{3}+\kappa _{1}^{5}$

$\mu '_{6}=\kappa _{6}+6\kappa _{5}\kappa _{1}+15\kappa _{4}\kappa _{2}+15\kappa _{4}\kappa _{1}^{2}+10\kappa _{3}^{2}+60\kappa _{3}\kappa _{2}\kappa _{1}+20\kappa _{3}\kappa _{1}^{3}+15\kappa _{2}^{3}+45\kappa _{2}^{2}\kappa _{1}^{2}+15\kappa _{2}\kappa _{1}^{4}+\kappa _{1}^{6}$

The "prime" distinguishes the moments μ′_n from the central moments μ_n.

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

$\mu '_{n}=\sum _{\pi }\prod _{B\in \pi }\kappa _{\left|B\right|}$

where

• 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.

Thus each monomial is a constant times a product of 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.

"Formal" cumulants

More generally, the cumulants of a sequence { m_n : n = 1, 2, 3, ... }, not necessarily the moments of any probability distribution, are given by

$1+\sum _{n=1}^{\infty }m_{n}t^{n}/n!=\exp \left(\sum _{n=1}^{\infty }\kappa _{n}t^{n}/n!\right)$

where the values of κ_n for n = 1, 2, 3, ... are found "formally", i.e., by algebra alone, in disregard of questions of whether any series converges.

One well-known example

In combinatorics, the nth 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.

Cumulants of a polynomial sequence of binomial type

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

${\begin{matrix}\mu '_{6}=&\kappa _{6}+6\kappa _{5}\kappa _{1}+15\kappa _{4}\kappa _{2}+15\kappa _{4}\kappa _{1}^{2}+10\kappa _{3}^{2}+60\kappa _{3}\kappa _{2}\kappa _{1}\\\\&+20\kappa _{3}\kappa _{1}^{3}+15\kappa _{2}^{3}+45\kappa _{2}^{2}\kappa _{1}^{2}+15\kappa _{2}\kappa _{1}^{4}+\kappa _{1}^{6}\end{matrix}}$

make a new polynomial in these plus one additional variable x:

${\begin{matrix}p_{6}(x)=&(\kappa _{6})\,x+(6\kappa _{5}\kappa _{1}+15\kappa _{4}\kappa _{2}+10\kappa _{3}^{2})\,x^{2}+(15\kappa _{4}\kappa _{1}^{2}+60\kappa _{3}\kappa _{2}\kappa _{1}+15\kappa _{2}^{3})\,x^{3}\\\\&+(45\kappa _{2}^{2}\kappa _{1}^{2})\,x^{4}+(15\kappa _{2}\kappa _{1}^{4})\,x^{5}+(\kappa _{1}^{6})\,x^{6}\end{matrix}}$

... and generalize the pattern. The pattern is that the numbers of blocks in the aforementioned partitions are the exponents on x.

This sequence of polynomials is of binomial type. But what is important is that no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of cumulants.