{"history":[],"article":{"id":"633bd8fb-272a-44ff-a77e-5d7445117a30","title":"Cumulant","slug":"cumulant","content":":::summary\n:::\n\n# \nIn [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics), the **cumulants** κn of a [probability distribution](https://bliptext.com/articles/probability-distribution) are a set of quantities that provide an alternative to the [moments](https://bliptext.com/articles/moment-mathematics) of the distribution.\nAny two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.\nThe first cumulant is the [mean](https://bliptext.com/articles/mean), the second cumulant is the [variance](https://bliptext.com/articles/variance), and the third cumulant is the same as the third [central moment](https://bliptext.com/articles/central-moment).\nBut fourth and higher-order cumulants are not equal to central moments.\nIn some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments.\nIn particular, when two or more random variables are [statistically independent](https://bliptext.com/articles/statistically-independent), the\nn\nth-order cumulant of their sum is equal to the sum of their\nn\nth-order cumulants.\nAs well, the third and higher-order cumulants of a [normal distribution](https://bliptext.com/articles/normal-distribution) are zero, and it is the only distribution with this property.\nJust as for moments, where joint moments are used for collections of random variables, it is possible to define joint cumulants.\n\n# Definition\nThe cumulants of a random variable X are defined using the **cumulant-generating function**\nK(t)\n, which is the [natural logarithm](https://bliptext.com/articles/natural-logarithm) of the [moment-generating function](https://bliptext.com/articles/moment-generating-function):\nThe cumulants κn are obtained from a [power series](https://bliptext.com/articles/power-series) expansion of the cumulant generating function:\nThis expansion is a [Maclaurin series](https://bliptext.com/articles/maclaurin-series), so the nth cumulant can be obtained by differentiating the above expansion n times and evaluating the result at zero:\nIf the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later.\n\n## Alternative definition of the cumulant generating function\nSome writers prefer to define the cumulant-generating function as the natural logarithm of the [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory), which is sometimes also called the second [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory),\nAn advantage of\nH(t)\n—in some sense the function\nK(t)\nevaluated for purely imaginary arguments—is that\nE[eitX]\nis well defined for all real values of\nt\neven when\nE[etX]\nis not well defined for all real values of\nt\n, such as can occur when there is \"too much\" probability that\nX\nhas a large magnitude.\nAlthough the function\nH(t)\nwill be well defined, it will nonetheless mimic\nK(t)\nin terms of the length of its [Maclaurin series](https://bliptext.com/articles/maclaurin-series), which may not extend beyond (or, rarely, even to) linear order in the argument\nt\n, and in particular the number of cumulants that are well defined will not change.\nNevertheless, even when\nH(t)\ndoes not have a long Maclaurin series, it can be used directly in analyzing and, particularly, adding random variables.\nBoth the [Cauchy distribution](https://bliptext.com/articles/cauchy-distribution) (also called the Lorentzian) and more generally, [stable distributions](https://bliptext.com/articles/stable-distribution) (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.\n\n# Some basic properties\nThe nth cumulant \\kappa_n(X) of (the distribution of) a random variable X enjoys the following properties:\nThe cumulative property follows quickly by considering the cumulant-generating function:\nso that each cumulant of a sum of independent random variables is the sum of the corresponding cumulants of the [addends](https://bliptext.com/articles/addend).\nThat 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.\nA distribution with given cumulants\nκn\ncan be approximated through an [Edgeworth series](https://bliptext.com/articles/edgeworth-series).\n\n## First several cumulants as functions of the moments\nAll 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.\n\n# Cumulants of some discrete probability distributions\n. The cumulant generating function is\n. The first cumulant is\nand the other cumulants are zero,\n.\np\nof success). The cumulant generating function is\n. The first cumulants are\nand\n. The cumulants satisfy a recursion formula\np\nof success on each trial). The cumulant generating function is\n. The first cumulants are\n, and\n. Substituting\ngives\nand\n.\n. All cumulants are equal to the parameter:\n.\nn\n[independent](https://bliptext.com/articles/statistical-independence) trials with probability\np\nof success on each trial). The special case\nis a Bernoulli distribution.\nEvery cumulant is just\nn\ntimes the corresponding cumulant of the corresponding Bernoulli distribution.\nThe cumulant generating function is\n. The first cumulants are\nand\n. Substituting\ngives\nand\n. The limiting case\nn−1 = 0\nis a Poisson distribution.\nr\nsuccesses with probability\np\nof success on each trial). The special case\nis a geometric distribution.\nEvery cumulant is just\nr\ntimes the corresponding cumulant of the corresponding geometric distribution.\nThe derivative of the cumulant generating function is\nK′(t) = r·((1 − p)−1·e−t−1)−1\n. The first cumulants are\nκ1 = K′(0) = r·(p−1−1)\n, and\nκ2 = K′′(0) = κ1·p−1\n. Substituting\np = (μ·r−1+1)−1\ngives\nK′(t) = ((μ−1 + r−1)e−t − r−1)−1\nand\n. Comparing these formulas to those of the binomial distributions explains the name 'negative binomial distribution'.\nThe [limiting case](https://bliptext.com/articles/limiting-case-mathematics)\nr−1 = 0\nis a Poisson distribution.\nIntroducing the [variance-to-mean ratio](https://bliptext.com/articles/variance-to-mean-ratio)\nthe above probability distributions get a unified formula for the derivative of the cumulant generating function:\nThe second derivative is\nconfirming that the first cumulant is\nand the second cumulant is\n.\nThe constant random variables\nhave\n.\nThe binomial distributions have\nso that\n0 < ε < 1\n.\nThe Poisson distributions have\n.\nThe negative binomial distributions have\nso that\nε > 1\n.\nNote the analogy to the classification of [conic sections](https://bliptext.com/articles/conic-sections) by [eccentricity](https://bliptext.com/articles/eccentricity-mathematics): circles\n, ellipses\n0 < ε < 1\n, parabolas\n, hyperbolas\nε > 1\n.\n\n# Cumulants of some continuous probability distributions\nμ\nand [variance](https://bliptext.com/articles/variance)\nσ2\n, the cumulant generating function is\n. The first and second derivatives of the cumulant generating function are\nK′(t) = μ + σ2·t\nand\nK′′(t) = σ2\n. The cumulants are\n,\n, and\n. The special case\nis a constant random variable\n.\n[−1, 0]\nare\n, where\nBn\nis the\nn\nth [Bernoulli number](https://bliptext.com/articles/bernoulli-number).\nλ\nare\n.\n\n# Some properties of the cumulant generating function\nThe cumulant generating function\nK(t)\n, if it exists, is [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable) and [convex](https://bliptext.com/articles/convex-function), and passes through the origin.\nIts first derivative ranges monotonically in the open interval from the [infimum](https://bliptext.com/articles/infimum) to the [supremum](https://bliptext.com/articles/supremum) of the support of the probability distribution, and its second derivative is strictly positive everywhere it is defined, except for the [degenerate distribution](https://bliptext.com/articles/degenerate-distribution) of a single point mass.\nThe cumulant-generating function exists if and only if the tails of the distribution are majorized by an [exponential decay](https://bliptext.com/articles/exponential-decay), that is, (see [Big O notation](https://bliptext.com/articles/big-o-notation))\nwhere F is the [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function).\nThe cumulant-generating function will have [vertical asymptote](https://bliptext.com/articles/vertical-asymptote)(s) at the negative [supremum](https://bliptext.com/articles/supremum) of such\nc\n, if such a [supremum](https://bliptext.com/articles/supremum) exists, and at the [supremum](https://bliptext.com/articles/supremum) of such\nd\n, if such a supremum exists, otherwise it will be defined for all real numbers.\nIf the [support](https://bliptext.com/articles/support-mathematics) of a random variable\nX\nhas finite upper or lower bounds, then its cumulant-generating function\ny = K(t)\n, if it exists, approaches [asymptote](https://bliptext.com/articles/asymptote)(s) whose slope is equal to the supremum or infimum of the support,\nrespectively, lying above both these lines everywhere.\n(The [integrals](https://bliptext.com/articles/integral)\nyield the [[y-intercept|\ny\n-intercepts]] of these asymptotes, since\nK(0) = 0\n.)\nFor a shift of the distribution by\nc\n, For a degenerate point mass at\nc\n, the cumulant generating function is the straight line K_c(t)=ct, and more generally, if and only if\nX\nand\nY\nare independent and their cumulant generating functions exist; ([subindependence](https://bliptext.com/articles/subindependence) and the existence of second moments sufficing to imply independence. )\nThe [natural exponential family](https://bliptext.com/articles/natural-exponential-family) of a distribution may be realized by shifting or translating\nK(t)\n, and adjusting it vertically so that it always passes through the origin: if\nf\nis the pdf with cumulant generating function and f|\\theta is its natural exponential family, then and\nIf\nK(t)\nis finite for a range\nt1 < Re(t) < t2\nthen if\nt1 < 0 < t2\nthen\nK(t)\nis analytic and infinitely differentiable for\nt1 < Re(t) < t2\n. Moreover for\nt\nreal and\nt1 < t < t2 K(t)\nis strictly convex, and\nK′(t)\nis strictly increasing.\n\n# Further properties of cumulants\n\n## A negative result\nGiven the results for the cumulants of the [normal distribution](https://bliptext.com/articles/normal-distribution), it might be hoped to find families of distributions for which\nκm = κm+1 = ⋯ = 0\nfor some\nm > 3\n, with the lower-order cumulants (orders 3 to\nm − 1\n) being non-zero. There are no such distributions. The underlying result here is that the cumulant generating function cannot be a finite-order polynomial of degree greater than 2.\n\n## Cumulants and moments\nThe [moment generating function](https://bliptext.com/articles/moment-generating-function) is given by:\nSo the cumulant generating function is the logarithm of the moment generating function\nThe first cumulant is the [expected value](https://bliptext.com/articles/expected-value); the second and third cumulants are respectively the second and third [central moments](https://bliptext.com/articles/central-moment) (the second central moment is the [variance](https://bliptext.com/articles/variance)); but the higher cumulants are neither moments nor [central moments](https://bliptext.com/articles/central-moment), but rather more complicated polynomial functions of the moments.\nThe moments can be recovered in terms of cumulants by evaluating the\nn\nth derivative of \\exp(K(t)) at t=0,\nLikewise, the cumulants can be recovered in terms of moments by evaluating the\nn\nth derivative of \\log M(t) at t=0,\nThe explicit expression for the\nn\nth moment in terms of the first\nn\ncumulants, and vice versa, can be obtained by using [Faà di Bruno's formula](https://bliptext.com/articles/fa-di-bruno-s-formula) for higher derivatives of composite functions.\nIn general, we have\nwhere B_{n,k} are incomplete (or partial) [Bell polynomials](https://bliptext.com/articles/bell-polynomials).\nIn the like manner, if the mean is given by \\mu, the central moment generating function is given by\nand the\nn\nth central moment is obtained in terms of cumulants as\nAlso, for\nn > 1\n, the\nn\nth cumulant in terms of the central moments is\nThe\nn\nth [moment](https://bliptext.com/articles/moment-mathematics)\nμ′n\nis an\nn\nth-degree polynomial in the first\nn\ncumulants.\nThe first few expressions are:\nThe \"prime\" distinguishes the moments\nμ′n\nfrom the [central moments](https://bliptext.com/articles/moment-about-the-mean)\nμn\n. To express the central moments as functions of the cumulants, just drop from these polynomials all terms in which\nκ1\nappears as a factor:\nSimilarly, the\nn\nth cumulant\nκn\nis an\nn\nth-degree polynomial in the first\nn\nnon-central moments.\nThe first few expressions are:\nIn general, the cumulant is the determinant of a matrix:\nTo express the cumulants\nκn\nfor\nn > 1\nas functions of the central moments, drop from these polynomials all terms in which μ'1 appears as a factor:\nThe cumulants can be related to the moments by [differentiating](https://bliptext.com/articles/differentiation-mathematics) the relationship\nlog M(t) = K(t)\nwith respect to\nt\n, giving\nM′(t) = K′(t) M(t)\n, which conveniently contains no exponentials or logarithms.\nEquating the coefficient of\ntn−1 / (n−1)!\non the left and right sides and using\nμ′0 = 1\ngives the following formulas for\nn ≥ 1\nThese allow either \\kappa_n or \\mu'_n to be computed from the other using knowledge of the lower-order cumulants and moments.\nThe corresponding formulas for the central moments \\mu_n for n \\ge 2 are formed from these formulas by setting and replacing each \\mu'_n with \\mu_n for n \\ge 2:\n\n## Cumulants and set-partitions\nThese polynomials have a remarkable [combinatorial](https://bliptext.com/articles/combinatorics) interpretation: the coefficients count certain [partitions of sets](https://bliptext.com/articles/partition-of-a-set).\nA general form of these polynomials is\nwhere\nπ\nruns through the list of all partitions of a set of size\nn\nB ∈ π\n\" means\nB\nis one of the \"blocks\" into which the set is partitioned; and\nis the size of the set\nB\n.\nThus each [monomial](https://bliptext.com/articles/monomial) is a constant times a product of cumulants in which the sum of the indices is\nn\n(e.g., in the term\nκ3 κ22 κ1\n, 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](https://bliptext.com/articles/integer)\nn\ncorresponds to each term.\nThe coefficient in each term is the number of partitions of a set of\nn\nmembers that collapse to that partition of the integer\nn\nwhen the members of the set become indistinguishable.\n\n## Cumulants and combinatorics\nFurther connection between cumulants and combinatorics can be found in the work of [Gian-Carlo Rota](https://bliptext.com/articles/gian-carlo-rota), where links to [invariant theory](https://bliptext.com/articles/invariant-theory), [symmetric functions](https://bliptext.com/articles/symmetric-function), and binomial sequences are studied via [umbral calculus](https://bliptext.com/articles/umbral-calculus).\n\n# Joint cumulants\nThe **joint cumulant**\nκ\nof several random variables\nX1, ..., Xn\nis defined as the coefficient\nκ1,...,1(X1, ..., Xn)\nin the Maclaurin series of the multivariate cumulant generating function, see Section 3.1 in,\nNote that\nand, in particular\nAs with a single variable, the generating function and cumulant can instead be defined via\nin which case\nand\n\n## Repeated random variables and relation between the coefficients κk 1, ..., kn\nObserve that can also be written as\nfrom which we conclude that\nFor example\nand\nIn particular, the last equality shows that the cumulants of a single random variable are the joint cumulants of multiple copies of that random variable.\n\n## Relation with mixed moments\nThe joint cumulant or random variables can be expressed as an alternate sum of products of their [mixed moments](https://bliptext.com/articles/moment-mathematics), see Equation (3.2.7) in,\nwhere π runs through the list of all partitions of\nB\nruns through the list of all blocks of the partition π; and where\nis the number of parts in the partition.\nFor example,\nis the expected value of X,\nis the [covariance](https://bliptext.com/articles/covariance) of X and Y, and\nFor zero-mean random variables, any mixed moment of the form vanishes if \\pi is a partition of which contains a singleton B=\\{k\\}.\nHence, the expression of their joint cumulant in terms of mixed moments simplifies.\nFor example, if X,Y,Z,W are zero mean random variables, we have\nMore generally, any coefficient of the Maclaurin series can also be expressed in terms of mixed moments, although there are no concise formulae.\nIndeed, as noted above, one can write it as a joint cumulant by repeating random variables appropriately, and then apply the above formula to express it in terms of mixed moments.\nFor example\nIf some of the random variables are independent of all of the others, then any cumulant involving two (or more) independent random variables is zero.\nThe combinatorial meaning of the expression of mixed moments in terms of cumulants is easier to understand than that of cumulants in terms of mixed moments, see Equation (3.2.6) in:\nFor example:\n\n## Further properties\nAnother important property of joint cumulants is multilinearity:\nJust as the second cumulant is the variance, the joint cumulant of just two random variables is the [covariance](https://bliptext.com/articles/covariance).\nThe familiar identity\ngeneralizes to cumulants:\n\n## Conditional cumulants and the law of total cumulance\nThe [law of total expectation](https://bliptext.com/articles/law-of-total-expectation) and the [law of total variance](https://bliptext.com/articles/law-of-total-variance) generalize naturally to conditional cumulants.\nThe case\nn = 3\n, expressed in the language of (central) [moments](https://bliptext.com/articles/moment-mathematics) rather than that of cumulants, says\nIn general,\nwhere\nof indices, and\nκ(Xπ m )\nindicates that the joint cumulant of the random variables whose indices are in that block of the partition.\n\n## Conditional cumulants and the conditional expectation\nFor certain settings, a derivative identity can be established between the conditional cumulant and the conditional expectation.\nFor example, suppose that\nY = X + Z\nwhere\nZ\nis standard normal independent of\nX\n, then for any\nX\nit holds that\nThe results can also be extended to the exponential family.\n\n# Relation to statistical physics\nIn [statistical physics](https://bliptext.com/articles/statistical-physics) many [extensive quantities](https://bliptext.com/articles/extensive-quantities) – that is quantities that are proportional to the volume or size of a given system – are related to cumulants of random variables.\nThe 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.\nThe 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.\nA system in equilibrium with a thermal bath at temperature\nT\nhave a fluctuating internal energy\nE\n, which can be considered a random variable drawn from a distribution E\\sim p(E).\nThe [partition function](https://bliptext.com/articles/partition-function-statistical-mechanics) of the system is\nwhere [[Thermodynamic beta|\nβ\n]] =\n1/(kT)\nand\nk\nis the [Boltzmann constant](https://bliptext.com/articles/boltzmann-constant) and the notation has been used rather than for the expectation value to avoid confusion with the energy,\nE\n. Hence the first and second cumulant for the energy\nE\ngive the average energy and heat capacity.\nThe [Helmholtz free energy](https://bliptext.com/articles/helmholtz-free-energy) expressed in terms of\nfurther connects thermodynamic quantities with cumulant generating function for the energy.\nThermodynamics properties that are derivatives of the free energy, such as its [internal energy](https://bliptext.com/articles/internal-energy), [entropy](https://bliptext.com/articles/entropy), and [specific heat](https://bliptext.com/articles/specific-heat) capacity, all can be readily expressed in terms of these cumulants.\nOther free energy can be a function of other variables such as the magnetic field or chemical potential \\mu, e.g.\nwhere\nN\nis the number of particles and \\Omega is the grand potential.\nAgain the close relationship between the definition of the free energy and the cumulant generating function implies that various derivatives of this free energy can be written in terms of joint cumulants of\nE\nand\nN\n.\n\n# History\nThe history of cumulants is discussed by [Anders Hald](https://bliptext.com/articles/anders-hald).\nCumulants were first introduced by [Thorvald N. Thiele](https://bliptext.com/articles/thorvald-n-thiele), in 1889, who called them semi-invariants.\nThey were first called cumulants in a 1932 paper by [Ronald Fisher](https://bliptext.com/articles/ronald-fisher) and [John Wishart](https://bliptext.com/articles/john-wishart-statistician).\nFisher was publicly reminded of Thiele's work by Neyman, who also notes previous published citations of Thiele brought to Fisher's attention.\n[Stephen Stigler](https://bliptext.com/articles/stephen-stigler) has said that the name cumulant was suggested to Fisher in a letter from [Harold Hotelling](https://bliptext.com/articles/harold-hotelling).\nIn a paper published in 1929, Fisher had called them cumulative moment functions.\nThe partition function in statistical physics was introduced by [Josiah Willard Gibbs](https://bliptext.com/articles/josiah-willard-gibbs) in 1901.\nThe free energy is often called Gibbs free energy.\nIn [statistical mechanics](https://bliptext.com/articles/statistical-mechanics), cumulants are also known as [Ursell functions](https://bliptext.com/articles/ursell-function) relating to a publication in 1927.\n\n# Cumulants in generalized settings\n\n## Formal cumulants\nMore generally, the cumulants of a sequence\n{ mn : n = 1, 2, 3, ...\n}\n, not necessarily the moments of any probability distribution, are, by definition,\nwhere the values of\nκn\nfor\nn = 1, 2, 3, ...\nare found formally, i.e., by algebra alone, in disregard of questions of whether any series converges.\nAll of the difficulties of the \"problem of cumulants\" are absent when one works formally.\nThe 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.\nFormal cumulants are subject to no such constraints.\n\n## Bell numbers\nIn [combinatorics](https://bliptext.com/articles/combinatorics), the\nn\nth [Bell number](https://bliptext.com/articles/bell-number) is the number of partitions of a set of size\nn\n. All of the [cumulants of the sequence of Bell numbers are equal to 1](https://bliptext.com/articles/bell-number).\nThe Bell numbers are the [moments of the Poisson distribution with expected value 1](https://bliptext.com/articles/moment-generating-function).\n\n## Cumulants of a polynomial sequence of binomial type\nFor any sequence\nof [scalars](https://bliptext.com/articles/scalar-mathematics) in a [field](https://bliptext.com/articles/field-mathematics) of characteristic zero, being considered formal cumulants, there is a corresponding sequence\nof formal moments, given by the polynomials above.\nFor those polynomials, construct a [polynomial sequence](https://bliptext.com/articles/polynomial-sequence) in the following way.\nOut of the polynomial\nmake a new polynomial in these plus one additional variable\nx\nand then generalize the pattern.\nThe pattern is that the numbers of blocks in the aforementioned partitions are the exponents on\nx\n. Each coefficient is a polynomial in the cumulants; these are the [Bell polynomials](https://bliptext.com/articles/bell-polynomials), named after [Eric Temple Bell](https://bliptext.com/articles/eric-temple-bell).\nThis sequence of polynomials is of [binomial type](https://bliptext.com/articles/binomial-type).\nIn fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of formal cumulants.\n\n## Free cumulants\nIn the above moment-cumulant formula\\\nfor joint cumulants, one sums over all partitions of the set\n. If instead, one sums only over the [noncrossing partitions](https://bliptext.com/articles/noncrossing-partition), then, by solving these formulae for the \\kappa in terms of the moments, one gets **free cumulants** rather than conventional cumulants treated above.\nThese free cumulants were introduced by Roland Speicher and play a central role in [free probability](https://bliptext.com/articles/free-probability) theory.\nIn that theory, rather than considering [independence](https://bliptext.com/articles/statistical-independence) of [random variables](https://bliptext.com/articles/random-variable), defined in terms of [tensor products of algebras](https://bliptext.com/articles/tensor-product-of-algebras) of [random variables](https://bliptext.com/articles/random-variable), one considers instead free [independence](https://bliptext.com/articles/statistical-independence) of [random variables](https://bliptext.com/articles/random-variable), defined in terms of [free products](https://bliptext.com/articles/free-product) of algebras.\nThe ordinary cumulants of degree higher than 2 of the [normal distribution](https://bliptext.com/articles/normal-distribution) are zero.\nThe free cumulants of degree higher than 2 of the [Wigner semicircle distribution](https://bliptext.com/articles/wigner-semicircle-distribution) are zero.\nThis 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.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:19:32.045Z","updated_at":"2025-01-25T19:19:32.045Z","search_vector":"'cumul':1","search_content":" \n \n\n# \nIn [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics), the **cumulants** κn of a [probability distribution](https://bliptext.com/articles/probability-distribution) are a set of quantities that provide an alternative to the [moments](https://bliptext.com/articles/moment-mathematics) of the distribution.\nAny two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.\nThe first cumulant is the [mean](https://bliptext.com/articles/mean), the second cumulant is the [variance](https://bliptext.com/articles/variance), and the third cumulant is the same as the third [central moment](https://bliptext.com/articles/central-moment).\nBut fourth and higher-order cumulants are not equal to central moments.\nIn some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments.\nIn particular, when two or more random variables are [statistically independent](https://bliptext.com/articles/statistically-independent), the\nn\nth-order cumulant of their sum is equal to the sum of their\nn\nth-order cumulants.\nAs well, the third and higher-order cumulants of a [normal distribution](https://bliptext.com/articles/normal-distribution) are zero, and it is the only distribution with this property.\nJust as for moments, where joint moments are used for collections of random variables, it is possible to define joint cumulants.\n\n# Definition\nThe cumulants of a random variable X are defined using the **cumulant-generating function**\nK(t)\n, which is the [natural logarithm](https://bliptext.com/articles/natural-logarithm) of the [moment-generating function](https://bliptext.com/articles/moment-generating-function):\nThe cumulants κn are obtained from a [power series](https://bliptext.com/articles/power-series) expansion of the cumulant generating function:\nThis expansion is a [Maclaurin series](https://bliptext.com/articles/maclaurin-series), so the nth cumulant can be obtained by differentiating the above expansion n times and evaluating the result at zero:\nIf the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later.\n\n## Alternative definition of the cumulant generating function\nSome writers prefer to define the cumulant-generating function as the natural logarithm of the [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory), which is sometimes also called the second [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory),\nAn advantage of\nH(t)\n—in some sense the function\nK(t)\nevaluated for purely imaginary arguments—is that\nE[eitX]\nis well defined for all real values of\nt\neven when\nE[etX]\nis not well defined for all real values of\nt\n, such as can occur when there is \"too much\" probability that\nX\nhas a large magnitude.\nAlthough the function\nH(t)\nwill be well defined, it will nonetheless mimic\nK(t)\nin terms of the length of its [Maclaurin series](https://bliptext.com/articles/maclaurin-series), which may not extend beyond (or, rarely, even to) linear order in the argument\nt\n, and in particular the number of cumulants that are well defined will not change.\nNevertheless, even when\nH(t)\ndoes not have a long Maclaurin series, it can be used directly in analyzing and, particularly, adding random variables.\nBoth the [Cauchy distribution](https://bliptext.com/articles/cauchy-distribution) (also called the Lorentzian) and more generally, [stable distributions](https://bliptext.com/articles/stable-distribution) (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.\n\n# Some basic properties\nThe nth cumulant \\kappa_n(X) of (the distribution of) a random variable X enjoys the following properties:\nThe cumulative property follows quickly by considering the cumulant-generating function:\nso that each cumulant of a sum of independent random variables is the sum of the corresponding cumulants of the [addends](https://bliptext.com/articles/addend).\nThat 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.\nA distribution with given cumulants\nκn\ncan be approximated through an [Edgeworth series](https://bliptext.com/articles/edgeworth-series).\n\n## First several cumulants as functions of the moments\nAll 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.\n\n# Cumulants of some discrete probability distributions\n. The cumulant generating function is\n. The first cumulant is\nand the other cumulants are zero,\n.\np\nof success). The cumulant generating function is\n. The first cumulants are\nand\n. The cumulants satisfy a recursion formula\np\nof success on each trial). The cumulant generating function is\n. The first cumulants are\n, and\n. Substituting\ngives\nand\n.\n. All cumulants are equal to the parameter:\n.\nn\n[independent](https://bliptext.com/articles/statistical-independence) trials with probability\np\nof success on each trial). The special case\nis a Bernoulli distribution.\nEvery cumulant is just\nn\ntimes the corresponding cumulant of the corresponding Bernoulli distribution.\nThe cumulant generating function is\n. The first cumulants are\nand\n. Substituting\ngives\nand\n. The limiting case\nn−1 = 0\nis a Poisson distribution.\nr\nsuccesses with probability\np\nof success on each trial). The special case\nis a geometric distribution.\nEvery cumulant is just\nr\ntimes the corresponding cumulant of the corresponding geometric distribution.\nThe derivative of the cumulant generating function is\nK′(t) = r·((1 − p)−1·e−t−1)−1\n. The first cumulants are\nκ1 = K′(0) = r·(p−1−1)\n, and\nκ2 = K′′(0) = κ1·p−1\n. Substituting\np = (μ·r−1+1)−1\ngives\nK′(t) = ((μ−1 + r−1)e−t − r−1)−1\nand\n. Comparing these formulas to those of the binomial distributions explains the name 'negative binomial distribution'.\nThe [limiting case](https://bliptext.com/articles/limiting-case-mathematics)\nr−1 = 0\nis a Poisson distribution.\nIntroducing the [variance-to-mean ratio](https://bliptext.com/articles/variance-to-mean-ratio)\nthe above probability distributions get a unified formula for the derivative of the cumulant generating function:\nThe second derivative is\nconfirming that the first cumulant is\nand the second cumulant is\n.\nThe constant random variables\nhave\n.\nThe binomial distributions have\nso that\n0 1\n.\nNote the analogy to the classification of [conic sections](https://bliptext.com/articles/conic-sections) by [eccentricity](https://bliptext.com/articles/eccentricity-mathematics): circles\n, ellipses\n0 1\n.\n\n# Cumulants of some continuous probability distributions\nμ\nand [variance](https://bliptext.com/articles/variance)\nσ2\n, the cumulant generating function is\n. The first and second derivatives of the cumulant generating function are\nK′(t) = μ + σ2·t\nand\nK′′(t) = σ2\n. The cumulants are\n,\n, and\n. The special case\nis a constant random variable\n.\n[−1, 0]\nare\n, where\nBn\nis the\nn\nth [Bernoulli number](https://bliptext.com/articles/bernoulli-number).\nλ\nare\n.\n\n# Some properties of the cumulant generating function\nThe cumulant generating function\nK(t)\n, if it exists, is [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable) and [convex](https://bliptext.com/articles/convex-function), and passes through the origin.\nIts first derivative ranges monotonically in the open interval from the [infimum](https://bliptext.com/articles/infimum) to the [supremum](https://bliptext.com/articles/supremum) of the support of the probability distribution, and its second derivative is strictly positive everywhere it is defined, except for the [degenerate distribution](https://bliptext.com/articles/degenerate-distribution) of a single point mass.\nThe cumulant-generating function exists if and only if the tails of the distribution are majorized by an [exponential decay](https://bliptext.com/articles/exponential-decay), that is, (see [Big O notation](https://bliptext.com/articles/big-o-notation))\nwhere F is the [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function).\nThe cumulant-generating function will have [vertical asymptote](https://bliptext.com/articles/vertical-asymptote)(s) at the negative [supremum](https://bliptext.com/articles/supremum) of such\nc\n, if such a [supremum](https://bliptext.com/articles/supremum) exists, and at the [supremum](https://bliptext.com/articles/supremum) of such\nd\n, if such a supremum exists, otherwise it will be defined for all real numbers.\nIf the [support](https://bliptext.com/articles/support-mathematics) of a random variable\nX\nhas finite upper or lower bounds, then its cumulant-generating function\ny = K(t)\n, if it exists, approaches [asymptote](https://bliptext.com/articles/asymptote)(s) whose slope is equal to the supremum or infimum of the support,\nrespectively, lying above both these lines everywhere.\n(The [integrals](https://bliptext.com/articles/integral)\nyield the [[y-intercept|\ny\n-intercepts]] of these asymptotes, since\nK(0) = 0\n.)\nFor a shift of the distribution by\nc\n, For a degenerate point mass at\nc\n, the cumulant generating function is the straight line K_c(t)=ct, and more generally, if and only if\nX\nand\nY\nare independent and their cumulant generating functions exist; ([subindependence](https://bliptext.com/articles/subindependence) and the existence of second moments sufficing to imply independence. )\nThe [natural exponential family](https://bliptext.com/articles/natural-exponential-family) of a distribution may be realized by shifting or translating\nK(t)\n, and adjusting it vertically so that it always passes through the origin: if\nf\nis the pdf with cumulant generating function and f|\\theta is its natural exponential family, then and\nIf\nK(t)\nis finite for a range\nt1 3\n, with the lower-order cumulants (orders 3 to\nm − 1\n) being non-zero. There are no such distributions. The underlying result here is that the cumulant generating function cannot be a finite-order polynomial of degree greater than 2.\n\n## Cumulants and moments\nThe [moment generating function](https://bliptext.com/articles/moment-generating-function) is given by:\nSo the cumulant generating function is the logarithm of the moment generating function\nThe first cumulant is the [expected value](https://bliptext.com/articles/expected-value); the second and third cumulants are respectively the second and third [central moments](https://bliptext.com/articles/central-moment) (the second central moment is the [variance](https://bliptext.com/articles/variance)); but the higher cumulants are neither moments nor [central moments](https://bliptext.com/articles/central-moment), but rather more complicated polynomial functions of the moments.\nThe moments can be recovered in terms of cumulants by evaluating the\nn\nth derivative of \\exp(K(t)) at t=0,\nLikewise, the cumulants can be recovered in terms of moments by evaluating the\nn\nth derivative of \\log M(t) at t=0,\nThe explicit expression for the\nn\nth moment in terms of the first\nn\ncumulants, and vice versa, can be obtained by using [Faà di Bruno's formula](https://bliptext.com/articles/fa-di-bruno-s-formula) for higher derivatives of composite functions.\nIn general, we have\nwhere B_{n,k} are incomplete (or partial) [Bell polynomials](https://bliptext.com/articles/bell-polynomials).\nIn the like manner, if the mean is given by \\mu, the central moment generating function is given by\nand the\nn\nth central moment is obtained in terms of cumulants as\nAlso, for\nn > 1\n, the\nn\nth cumulant in terms of the central moments is\nThe\nn\nth [moment](https://bliptext.com/articles/moment-mathematics)\nμ′n\nis an\nn\nth-degree polynomial in the first\nn\ncumulants.\nThe first few expressions are:\nThe \"prime\" distinguishes the moments\nμ′n\nfrom the [central moments](https://bliptext.com/articles/moment-about-the-mean)\nμn\n. To express the central moments as functions of the cumulants, just drop from these polynomials all terms in which\nκ1\nappears as a factor:\nSimilarly, the\nn\nth cumulant\nκn\nis an\nn\nth-degree polynomial in the first\nn\nnon-central moments.\nThe first few expressions are:\nIn general, the cumulant is the determinant of a matrix:\nTo express the cumulants\nκn\nfor\nn > 1\nas functions of the central moments, drop from these polynomials all terms in which μ'1 appears as a factor:\nThe cumulants can be related to the moments by [differentiating](https://bliptext.com/articles/differentiation-mathematics) the relationship\nlog M(t) = K(t)\nwith respect to\nt\n, giving\nM′(t) = K′(t) M(t)\n, which conveniently contains no exponentials or logarithms.\nEquating the coefficient of\ntn−1 / (n−1)!\non the left and right sides and using\nμ′0 = 1\ngives the following formulas for\nn ≥ 1\nThese allow either \\kappa_n or \\mu'_n to be computed from the other using knowledge of the lower-order cumulants and moments.\nThe corresponding formulas for the central moments \\mu_n for n \\ge 2 are formed from these formulas by setting and replacing each \\mu'_n with \\mu_n for n \\ge 2:\n\n## Cumulants and set-partitions\nThese polynomials have a remarkable [combinatorial](https://bliptext.com/articles/combinatorics) interpretation: the coefficients count certain [partitions of sets](https://bliptext.com/articles/partition-of-a-set).\nA general form of these polynomials is\nwhere\nπ\nruns through the list of all partitions of a set of size\nn\nB ∈ π\n\" means\nB\nis one of the \"blocks\" into which the set is partitioned; and\nis the size of the set\nB\n.\nThus each [monomial](https://bliptext.com/articles/monomial) is a constant times a product of cumulants in which the sum of the indices is\nn\n(e.g., in the term\nκ3 κ22 κ1\n, 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](https://bliptext.com/articles/integer)\nn\ncorresponds to each term.\nThe coefficient in each term is the number of partitions of a set of\nn\nmembers that collapse to that partition of the integer\nn\nwhen the members of the set become indistinguishable.\n\n## Cumulants and combinatorics\nFurther connection between cumulants and combinatorics can be found in the work of [Gian-Carlo Rota](https://bliptext.com/articles/gian-carlo-rota), where links to [invariant theory](https://bliptext.com/articles/invariant-theory), [symmetric functions](https://bliptext.com/articles/symmetric-function), and binomial sequences are studied via [umbral calculus](https://bliptext.com/articles/umbral-calculus).\n\n# Joint cumulants\nThe **joint cumulant**\nκ\nof several random variables\nX1, ..., Xn\nis defined as the coefficient\nκ1,...,1(X1, ..., Xn)\nin the Maclaurin series of the multivariate cumulant generating function, see Section 3.1 in,\nNote that\nand, in particular\nAs with a single variable, the generating function and cumulant can instead be defined via\nin which case\nand\n\n## Repeated random variables and relation between the coefficients κk 1, ..., kn\nObserve that can also be written as\nfrom which we conclude that\nFor example\nand\nIn particular, the last equality shows that the cumulants of a single random variable are the joint cumulants of multiple copies of that random variable.\n\n## Relation with mixed moments\nThe joint cumulant or random variables can be expressed as an alternate sum of products of their [mixed moments](https://bliptext.com/articles/moment-mathematics), see Equation (3.2.7) in,\nwhere π runs through the list of all partitions of\nB\nruns through the list of all blocks of the partition π; and where\nis the number of parts in the partition.\nFor example,\nis the expected value of X,\nis the [covariance](https://bliptext.com/articles/covariance) of X and Y, and\nFor zero-mean random variables, any mixed moment of the form vanishes if \\pi is a partition of which contains a singleton B=\\{k\\}.\nHence, the expression of their joint cumulant in terms of mixed moments simplifies.\nFor example, if X,Y,Z,W are zero mean random variables, we have\nMore generally, any coefficient of the Maclaurin series can also be expressed in terms of mixed moments, although there are no concise formulae.\nIndeed, as noted above, one can write it as a joint cumulant by repeating random variables appropriately, and then apply the above formula to express it in terms of mixed moments.\nFor example\nIf some of the random variables are independent of all of the others, then any cumulant involving two (or more) independent random variables is zero.\nThe combinatorial meaning of the expression of mixed moments in terms of cumulants is easier to understand than that of cumulants in terms of mixed moments, see Equation (3.2.6) in:\nFor example:\n\n## Further properties\nAnother important property of joint cumulants is multilinearity:\nJust as the second cumulant is the variance, the joint cumulant of just two random variables is the [covariance](https://bliptext.com/articles/covariance).\nThe familiar identity\ngeneralizes to cumulants:\n\n## Conditional cumulants and the law of total cumulance\nThe [law of total expectation](https://bliptext.com/articles/law-of-total-expectation) and the [law of total variance](https://bliptext.com/articles/law-of-total-variance) generalize naturally to conditional cumulants.\nThe case\nn = 3\n, expressed in the language of (central) [moments](https://bliptext.com/articles/moment-mathematics) rather than that of cumulants, says\nIn general,\nwhere\nof indices, and\nκ(Xπ m )\nindicates that the joint cumulant of the random variables whose indices are in that block of the partition.\n\n## Conditional cumulants and the conditional expectation\nFor certain settings, a derivative identity can be established between the conditional cumulant and the conditional expectation.\nFor example, suppose that\nY = X + Z\nwhere\nZ\nis standard normal independent of\nX\n, then for any\nX\nit holds that\nThe results can also be extended to the exponential family.\n\n# Relation to statistical physics\nIn [statistical physics](https://bliptext.com/articles/statistical-physics) many [extensive quantities](https://bliptext.com/articles/extensive-quantities) – that is quantities that are proportional to the volume or size of a given system – are related to cumulants of random variables.\nThe 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.\nThe 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.\nA system in equilibrium with a thermal bath at temperature\nT\nhave a fluctuating internal energy\nE\n, which can be considered a random variable drawn from a distribution E\\sim p(E).\nThe [partition function](https://bliptext.com/articles/partition-function-statistical-mechanics) of the system is\nwhere [[Thermodynamic beta|\nβ\n]] =\n1/(kT)\nand\nk\nis the [Boltzmann constant](https://bliptext.com/articles/boltzmann-constant) and the notation has been used rather than for the expectation value to avoid confusion with the energy,\nE\n. Hence the first and second cumulant for the energy\nE\ngive the average energy and heat capacity.\nThe [Helmholtz free energy](https://bliptext.com/articles/helmholtz-free-energy) expressed in terms of\nfurther connects thermodynamic quantities with cumulant generating function for the energy.\nThermodynamics properties that are derivatives of the free energy, such as its [internal energy](https://bliptext.com/articles/internal-energy), [entropy](https://bliptext.com/articles/entropy), and [specific heat](https://bliptext.com/articles/specific-heat) capacity, all can be readily expressed in terms of these cumulants.\nOther free energy can be a function of other variables such as the magnetic field or chemical potential \\mu, e.g.\nwhere\nN\nis the number of particles and \\Omega is the grand potential.\nAgain the close relationship between the definition of the free energy and the cumulant generating function implies that various derivatives of this free energy can be written in terms of joint cumulants of\nE\nand\nN\n.\n\n# History\nThe history of cumulants is discussed by [Anders Hald](https://bliptext.com/articles/anders-hald).\nCumulants were first introduced by [Thorvald N. Thiele](https://bliptext.com/articles/thorvald-n-thiele), in 1889, who called them semi-invariants.\nThey were first called cumulants in a 1932 paper by [Ronald Fisher](https://bliptext.com/articles/ronald-fisher) and [John Wishart](https://bliptext.com/articles/john-wishart-statistician).\nFisher was publicly reminded of Thiele's work by Neyman, who also notes previous published citations of Thiele brought to Fisher's attention.\n[Stephen Stigler](https://bliptext.com/articles/stephen-stigler) has said that the name cumulant was suggested to Fisher in a letter from [Harold Hotelling](https://bliptext.com/articles/harold-hotelling).\nIn a paper published in 1929, Fisher had called them cumulative moment functions.\nThe partition function in statistical physics was introduced by [Josiah Willard Gibbs](https://bliptext.com/articles/josiah-willard-gibbs) in 1901.\nThe free energy is often called Gibbs free energy.\nIn [statistical mechanics](https://bliptext.com/articles/statistical-mechanics), cumulants are also known as [Ursell functions](https://bliptext.com/articles/ursell-function) relating to a publication in 1927.\n\n# Cumulants in generalized settings\n\n## Formal cumulants\nMore generally, the cumulants of a sequence\n{ mn : n = 1, 2, 3, ...\n}\n, not necessarily the moments of any probability distribution, are, by definition,\nwhere the values of\nκn\nfor\nn = 1, 2, 3, ...\nare found formally, i.e., by algebra alone, in disregard of questions of whether any series converges.\nAll of the difficulties of the \"problem of cumulants\" are absent when one works formally.\nThe 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.\nFormal cumulants are subject to no such constraints.\n\n## Bell numbers\nIn [combinatorics](https://bliptext.com/articles/combinatorics), the\nn\nth [Bell number](https://bliptext.com/articles/bell-number) is the number of partitions of a set of size\nn\n. All of the [cumulants of the sequence of Bell numbers are equal to 1](https://bliptext.com/articles/bell-number).\nThe Bell numbers are the [moments of the Poisson distribution with expected value 1](https://bliptext.com/articles/moment-generating-function).\n\n## Cumulants of a polynomial sequence of binomial type\nFor any sequence\nof [scalars](https://bliptext.com/articles/scalar-mathematics) in a [field](https://bliptext.com/articles/field-mathematics) of characteristic zero, being considered formal cumulants, there is a corresponding sequence\nof formal moments, given by the polynomials above.\nFor those polynomials, construct a [polynomial sequence](https://bliptext.com/articles/polynomial-sequence) in the following way.\nOut of the polynomial\nmake a new polynomial in these plus one additional variable\nx\nand then generalize the pattern.\nThe pattern is that the numbers of blocks in the aforementioned partitions are the exponents on\nx\n. Each coefficient is a polynomial in the cumulants; these are the [Bell polynomials](https://bliptext.com/articles/bell-polynomials), named after [Eric Temple Bell](https://bliptext.com/articles/eric-temple-bell).\nThis sequence of polynomials is of [binomial type](https://bliptext.com/articles/binomial-type).\nIn fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of formal cumulants.\n\n## Free cumulants\nIn the above moment-cumulant formula\\\nfor joint cumulants, one sums over all partitions of the set\n. If instead, one sums only over the [noncrossing partitions](https://bliptext.com/articles/noncrossing-partition), then, by solving these formulae for the \\kappa in terms of the moments, one gets **free cumulants** rather than conventional cumulants treated above.\nThese free cumulants were introduced by Roland Speicher and play a central role in [free probability](https://bliptext.com/articles/free-probability) theory.\nIn that theory, rather than considering [independence](https://bliptext.com/articles/statistical-independence) of [random variables](https://bliptext.com/articles/random-variable), defined in terms of [tensor products of algebras](https://bliptext.com/articles/tensor-product-of-algebras) of [random variables](https://bliptext.com/articles/random-variable), one considers instead free [independence](https://bliptext.com/articles/statistical-independence) of [random variables](https://bliptext.com/articles/random-variable), defined in terms of [free products](https://bliptext.com/articles/free-product) of algebras.\nThe ordinary cumulants of degree higher than 2 of the [normal distribution](https://bliptext.com/articles/normal-distribution) are zero.\nThe free cumulants of degree higher than 2 of the [Wigner semicircle distribution](https://bliptext.com/articles/wigner-semicircle-distribution) are zero.\nThis 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.","revisionCount":0},"hasMore":false,"nextCursor":null}