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In probability theory and statistics, the cumulants κn of a probability distribution are given by
where X is any random variable whose probability distribution is the one whose cumulants are taken. In other words, κn/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" attempts to recover a probability distribution from its sequence of cumulants. In some cases no solution exists; in some cases a unique solution exists; in some cases more than one solution exists.
The first cumulant is shift-equivariant; all of the others are shift-invariant. To state this less tersely, denote by κn(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 κn(X + c) = κn(X) for n≥ 2, i.e., c is added to the first cumulant, but all higher cumulants are unchanged.
The nth cumulant is homogeneous of degree n, i.e. if c is any constant, then
If X and Y are independent random variables then κn(X + Y) = κn(X) + κn(Y).
The cumulants are related to the moments by the following recursion formula:
The nth moment μ′n is an nth-degree polynomial in the first n cumulants, thus:
The "prime" distinguishes the moments μ′n 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.
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
where
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.
The cumulants of the normal distribution with expected value μ and varianceThis article is about mathematics. Alternate meaning: variance (land use). In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically σ2 are κ1 = μ, κ2 = σ2, and κn = 0 for n > 2.
All of the cumulants of the Poisson distributionIn statistics and probability theory, the Poisson distribution is a discrete probability distribution (discovered by Simeon-Denis Poisson ( 1781- 1840) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilite are equal to the expected value.
A distribution with arbitrary given cumulants κn can be approximated through the Gram-Charlier or Edgeworth seriesThe Edgeworth series or Gram-Charlier A series named in honor of Francis Ysidro Edgeworth, are series that approximate a probability distribution in terms of its cumulants. Gram-Charlier A series The key idea of these expansions is to write the characteri.
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