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{{Short description|Set of quantities in probability theory}}
In [[probability theory]] and [[statistics]], the '''cumulants''' {{mvar|κ<sub>n</sub>}} of a [[probability distribution]] are a set of quantities that provide an alternative to the ''[[Moment (mathematics)|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 {{math|''n''}}th-order cumulant of their sum is equal to the sum of their {{math|''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''.

== Definition ==
The cumulants of a random variable {{mvar|X}} are defined using the '''cumulant-generating function''' {{math|''K''(''t'')}}, which is the [[natural logarithm]] of the [[moment-generating function]]:
<math display=block>K(t)=\log\operatorname{E}\left[e^{tX}\right].</math>

The cumulants {{mvar|κ<sub>n</sub>}} are obtained from a [[power series]] expansion of the cumulant generating function:
<math display=block>K(t)=\sum_{n=1}^\infty \kappa_{n} \frac{t^{n}}{n!} =\kappa_1 \frac{t}{1!} + \kappa_2 \frac{t^2}{2!}+ \kappa_3 \frac{t^3}{3!}+ \cdots = \mu t + \sigma^2 \frac{t^2}{2} + \cdots.</math>

This expansion is a [[Maclaurin series]], so the {{mvar|n}}th cumulant can be obtained by differentiating the above expansion {{mvar|n}} times and evaluating the result at zero:<ref>Weisstein, Eric W. "Cumulant". From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/Cumulant.html</ref>
<math display=block>\kappa_{n} = K^{(n)}(0).</math>

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

=== Alternative definition of the cumulant generating function ===
Some writers<ref>Kendall, M. G., Stuart, A. (1969) ''The Advanced Theory of Statistics'', Volume 1 (3rd Edition). Griffin, London. (Section 3.12)</ref><ref>Lukacs, E. (1970) ''Characteristic Functions'' (2nd Edition). Griffin, London. (Page 27)</ref> prefer to define the cumulant-generating function as the natural logarithm of the [[Characteristic function (probability theory)|characteristic function]], which is sometimes also called the '''''second'' characteristic function''',<ref>Lukacs, E. (1970) ''Characteristic Functions'' (2nd Edition). Griffin, London. (Section 2.4)</ref><ref>Aapo Hyvarinen, Juha Karhunen, and Erkki Oja (2001) ''Independent Component Analysis'', [[John Wiley & Sons]]. (Section 2.7.2)</ref>
<math display=block>H(t)=\log\operatorname{E} \left[e^{i t X}\right]=\sum_{n=1}^\infty \kappa_n \frac{(it)^n}{n!}=\mu it - \sigma^2 \frac{ t^2}{2} + \cdots</math>

An advantage of {{math|''H''(''t'')}} — in some sense the function {{math|''K''(''t'')}} evaluated for purely imaginary arguments — is that {{math|E[''e''<sup>''itX''</sup>]}} is well defined for all real values of {{math|''t''}} even when {{math|E[''e''<sup>''tX''</sup>]}} is not well defined for all real values of {{math|''t''}}, such as can occur when there is "too much" probability that {{math|''X''}} has a large magnitude.  Although the function {{math|''H''(''t'')}} will be well defined, it will nonetheless mimic {{math|''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&nbsp;{{math|''t''}}, and in particular the number of cumulants that are well defined will not change.  Nevertheless, even when {{math|''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 distribution]]s (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.

== Some basic properties ==

The <math display=inline>n</math>th cumulant <math display=inline>\kappa_n(X)</math> of (the distribution of) a random variable <math display=inline>X</math> enjoys the following properties:
* If <math display=inline>n>1</math> and <math display=inline>c</math> is constant (i.e. not random) then <math display=inline> \kappa_n(X+c) = \kappa_n(X),</math> i.e. the cumulant is [[translation invariant]]. (If <math display=inline> n=1</math> then we have <math display=inline> \kappa_1(X+c) = \kappa_1(X)+c.) </math>
* If <math display=inline>c</math> is constant (i.e. not random) then <math display=inline> \kappa_n(cX) = c^n\kappa_n(X),</math> i.e. the <math display=inline>n</math>th cumulant is [[homogeneous polynomial|homogeneous]] of degree&nbsp;<math display=inline>n</math>.
* If random variables <math display=inline>X_1,\ldots,X_m</math> are independent then <math display="block"> \kappa_n(X_1+\cdots+X_m) = \kappa_n(X_1) + \cdots + \kappa_n(X_m)\,. </math> That is, the cumulant is cumulative — hence the name.

The cumulative property follows quickly by considering the cumulant-generating function:
<math display=block>\begin{align}
K_{X_1+\cdots+X_m}(t) & =\log\operatorname{E} \left[e^{t(X_1+\cdots+X_m)}\right] \\[5pt]
& = \log \left(\operatorname{E} \left[e^{tX_1}\right] \cdots \operatorname{E} \left[ e^{tX_m} \right] \right) \\[5pt]
& = \log\operatorname{E}\left[e^{tX_1}\right] + \cdots + \log \operatorname{E} \left[ e^{tX_m} \right] \\[5pt]
&= K_{X_1}(t) + \cdots + K_{X_m}(t),
\end{align}</math>
so that each cumulant of a sum of independent random variables is the sum of the corresponding cumulants of the [[addend]]s.  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 {{math|''κ''<sub>''n''</sub>}} can be approximated through an [[Edgeworth series]].

=== First several cumulants as functions of the moments ===

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.
* <math display="inline"> \kappa_1(X) = \operatorname E[X] = {} </math>mean
* <math display="inline"> \kappa_2(X) = \operatorname{var}(X) = \operatorname E\left[{\left(X-\operatorname E[X]\right)}^2\right] ={}</math>the variance, or second central moment.
* <math display="inline"> \kappa_3(X) = \operatorname E\left[{\left(X - \operatorname E(X)\right)}^3\right] = {} </math>the third central moment.
* <math display="inline"> \kappa_4(X) = \operatorname E\left[{\left(X - \operatorname E[X]\right)}^4\right] - 3\left( \operatorname E\left[{\left(X-\operatorname E[X]\right)}^2\right] \right)^2={} </math>the fourth central moment minus three times the square of the second central moment. Thus this is the first case in which cumulants are not simply moments or central moments. The central moments of degree more than 3 lack the cumulative property.
* <math display="inline"> \kappa_5(X) = \operatorname E\left[{\left(X - \operatorname E[X]\right)}^5\right] - 10 \operatorname E\left[ {\left(X - \operatorname E[X]\right)}^3\right] \operatorname E\left[{\left(X - \operatorname E[X]\right)}^2\right].</math>

== Cumulants of some discrete probability distributions ==
* The constant random variables {{math|''X'' {{=}} ''μ''}}. The cumulant generating function is {{math|''K''(''t'') {{=}} ''μt''}}.  The first cumulant is {{math|''κ''<sub>1</sub> {{=}} ''K''′(0) {{=}} ''μ''}} and the other cumulants are zero, {{math|''κ''<sub>2</sub> {{=}} ''κ''<sub>3</sub> {{=}} ''κ''<sub>4</sub> {{=}} &sdot;&sdot;&sdot; {{=}} 0}}.
* The [[Bernoulli distribution]]s, (number of successes in one trial with probability {{math|''p''}} of success). The cumulant generating function is {{math|''K''(''t'') {{=}} log(1 − ''p'' + ''p''e<sup>''t''</sup>)}}. The first cumulants are {{math|''κ''<sub>1</sub> {{=}} ''K'' '(0) {{=}} ''p''}} and {{math|''κ''<sub>2</sub> {{=}} ''K′′''(0) {{=}} ''p''·(1 − ''p'')}}. The cumulants satisfy a recursion formula <math display="block">\kappa_{n+1}=p (1-p) \frac{d\kappa_n}{dp}.</math>
* The [[geometric distribution]]s, (number of failures before one success with probability {{math|''p''}} of success on each trial). The cumulant generating function is {{math|''K''(''t'') {{=}} log(''p'' / (1 + (''p'' − 1)e<sup>''t''</sup>))}}. The first cumulants are {{math|''κ''<sub>1</sub> {{=}} ''K′''(0) {{=}} ''p''<sup>−1</sup> − 1}}, and {{math|''κ''<sub>2</sub> {{=}} ''K′′''(0) {{=}} ''κ''<sub>1</sub>''p''<sup>−1</sup>}}. Substituting {{math|''p'' {{=}} (''μ'' + 1)<sup>−1</sup>}} gives {{math|''K''(''t'') {{=}} −log(1 + ''μ''(1−e<sup>''t''</sup>))}} and {{math|''κ''<sub>1</sub> {{=}} ''μ''}}.
* The [[Poisson distribution]]s. The cumulant generating function is {{math|''K''(''t'') {{=}} ''μ''(e<sup>''t''</sup> − 1)}}. All cumulants are equal to the parameter: {{math|''κ''<sub>1</sub> {{=}} ''κ''<sub>2</sub> {{=}} ''κ''<sub>3</sub> {{=}} ... {{=}} ''μ''}}.
* The [[binomial distribution]]s, (number of successes in {{math|''n''}} [[statistical independence|independent]] trials with probability {{math|''p''}} of success on each trial). The special case {{math|''n'' {{=}} 1}} is a Bernoulli distribution. Every cumulant is just {{math|''n''}} times the corresponding cumulant of the corresponding Bernoulli distribution.  The cumulant generating function is {{math|''K''(''t'') {{=}} ''n'' log(1 − ''p'' + ''p''e<sup>''t''</sup>)}}. The first cumulants are {{math|''κ''<sub>1</sub> {{=}} ''K′''(0) {{=}} ''np''}} and {{math|''κ''<sub>2</sub> {{=}} ''K′′''(0) {{=}} ''κ''<sub>1</sub>(1 − ''p'')}}. Substituting {{math|''p'' {{=}} ''μ''·''n''<sup>−1</sup>}} gives {{math|''K'' '(''t'') {{=}} ((''μ''<sup>−1</sup> − ''n''<sup>−1</sup>)·e<sup>−''t''</sup> + ''n''<sup>−1</sup>)<sup>−1</sup>}} and {{math|''κ''<sub>1</sub> {{=}} ''μ''}}. The limiting case {{math|''n'' → +∞}} is a Poisson distribution.
* The [[negative binomial distribution]]s, (number of failures before {{math|''r''}} successes with probability {{math|''p''}} of success on each trial). The special case {{math|''r'' {{=}} 1}} is a geometric distribution. Every cumulant is just {{math|''r''}} times the corresponding cumulant of the corresponding geometric distribution. The derivative of the cumulant generating function is {{math|1=''K''′(''t'') = ''r''·((1 − ''p'')<sup>−1</sup>·e<sup>−''t''</sup>−1)<sup>−1</sup>}}. The first cumulants are {{math|1=''κ''<sub>1</sub> = ''K''′(0) = ''r''·(''p''<sup>−1</sup>−1)}}, and {{math|1=''κ''<sub>2</sub> = ''K''′′(0) = ''κ''<sub>1</sub>·''p''<sup>−1</sup>}}. Substituting {{math|1=''p'' = (μ·''r''<sup>−1</sup>+1)<sup>−1</sup>}} gives {{math|''K''′(''t'') {{=}} ((''μ''<sup>−1</sup> + ''r''<sup>−1</sup>)''e''<sup>−''t''</sup> − ''r''<sup>−1</sup>)<sup>−1</sup>}} and {{math|''κ''<sub>1</sub> {{=}} ''μ''}}. Comparing these formulas to those of the binomial distributions explains the name 'negative binomial distribution'. The [[limiting case (mathematics)|limiting case]] {{math|''r'' → +∞}}  is a Poisson distribution.

Introducing the [[variance-to-mean ratio]]
<math display=block>\varepsilon=\mu^{-1}\sigma^2=\kappa_1^{-1}\kappa_2,</math>
the above probability distributions get a unified formula for the derivative of the cumulant generating function:{{Citation needed|date=September 2010}}
<math display=block>K'(t)=(1+(e^{-t}-1)\varepsilon)^{-1}\mu</math>

The second derivative is
<math display=block>K''(t)=(\varepsilon-(\varepsilon-1)e^t)^{-2}\mu\varepsilon e^t</math>
confirming that the first cumulant is {{math|''κ''<sub>1</sub> {{=}} ''K′''(0) {{=}} ''μ''}} and the second cumulant is {{math|''κ''<sub>2</sub> {{=}} ''K′′''(0) {{=}} ''με''}}.

The constant random variables {{math|''X'' {{=}} ''μ''}} have {{math|''ε'' {{=}} 0}}.

The binomial distributions have {{math|''ε'' {{=}} 1 − ''p''}} so that {{math|0 < ''ε'' < 1}}.

The Poisson distributions have {{math|''ε'' {{=}} 1}}.

The negative binomial distributions have {{math|''ε'' {{=}} ''p''<sup>−1</sup>}} so that {{math|''ε'' > 1}}.

Note the analogy to the classification of [[conic sections]] by [[eccentricity (mathematics)|eccentricity]]: circles {{math|''ε'' {{=}} 0}}, ellipses {{math|0 < ''ε'' < 1}}, parabolas {{math|''ε'' {{=}} 1}}, hyperbolas {{math|''ε'' > 1}}.

== Cumulants of some continuous probability distributions ==
* For the [[normal distribution]] with [[expected value]] {{math|''μ''}} and [[variance]] {{math|''σ''<sup>2</sup>}}, the cumulant generating function is {{math|''K''(''t'') {{=}} ''μt'' + ''σ''<sup>2</sup>''t''<sup>2</sup>/2}}. The first and second derivatives of the cumulant generating function are {{math|''K''′(''t'') {{=}} ''μ'' + ''σ''<sup>2</sup>·''t''}} and {{math|''K''′′(''t'') {{=}} ''σ''<sup>2</sup>}}. The cumulants are {{math|''κ''<sub>1</sub> {{=}} ''μ''}}, {{math|''κ''<sub>2</sub> {{=}} ''σ''<sup>2</sup>}}, and {{math|''κ''<sub>3</sub> {{=}} ''κ''<sub>4</sub> {{=}} &sdot;&sdot;&sdot; {{=}} 0}}. The special case {{math|''σ''<sup>2</sup> {{=}} 0}} is a constant random variable {{math|''X'' {{=}} ''μ''}}.
* The cumulants of the [[Uniform distribution (continuous)|uniform distribution]] on the interval {{math|[−1, 0]}} are {{math|''κ''<sub>''n''</sub> {{=}} ''B''<sub>''n''</sub> /''n''}}, where {{math|''B''<sub>''n''</sub>}} is the {{math|''n''}}th [[Bernoulli number]].
* The cumulants of the [[exponential distribution]] with rate parameter {{math|''λ''}} are {{math|''κ''<sub>''n''</sub> {{=}} ''λ''<sup>−''n''</sup> (''n'' − 1)!}}.

== Some properties of the cumulant generating function ==
The cumulant generating function {{math|''K''(''t'')}}, if it exists, is [[infinitely differentiable]] and [[convex function|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]]'')
<math display=block>
\begin{align}
& \exists c>0,\,\, F(x)=O(e^{cx}), x\to-\infty; \text{ and} \\[4pt]
& \exists d>0,\,\, 1-F(x)=O(e^{-dx}),x\to+\infty;
\end{align}
</math>
where <math display=inline>F</math> is the [[cumulative distribution function]].  The cumulant-generating function will have [[vertical asymptote]](s) at the negative [[supremum]] of such {{math|''c''}}, if such a supremum exists, and at the [[supremum]] of such {{math|''d''}}, if such a supremum exists, otherwise it will be defined for all real numbers.

If the [[Support (mathematics)|support]] of a random variable {{math|''X''}} has finite upper or lower bounds, then its cumulant-generating function {{math|1=''y'' = ''K''(''t'')}}, if it exists, approaches [[asymptote]](s) whose slope is equal to the supremum or infimum of the support,
<math display=block>
\begin{align}
y & =(t+1)\inf \operatorname{supp} X-\mu(X), \text{ and} \\[5pt]
y & =(t-1)\sup \operatorname{supp}X+\mu(X),
\end{align}
</math>
respectively, lying above both these lines everywhere. (The [[integral]]s
<math display=block>\int_{-\infty}^0 \left[t\inf \operatorname{supp}X-K'(t)\right]\,dt, \qquad \int_{\infty}^0 \left[t\inf \operatorname{supp}X-K'(t) \right]\,dt</math>
yield the [[y-intercept|{{math|''y''}}-intercepts]] of these asymptotes, since&nbsp;{{math|1=''K''(0) = 0}}.)

For a shift of the distribution by {{math|''c''}}, <math display=inline>K_{X+c}(t)=K_X(t)+ct.</math> For a degenerate point mass at {{math|''c''}}, the cumulant generating function is the straight line <math display=inline>K_c(t)=ct</math>, and more generally, <math display=inline>K_{X+Y}=K_X+K_Y</math> if and only if {{math|''X''}} and {{math|''Y''}} are independent and their cumulant generating functions exist; ([[subindependence]] and the existence of second moments sufficing to imply independence.<ref>
{{cite journal
 | journal = Studia Scientiarum Mathematicarum Hungarica
 | title = A note on sub-independent random variables and a class of bivariate mixtures
 | volume = 49
 | issue = 1
 | pages = 19–25 |date = 2012-03-01
 | first1 = G. G.| last1 = Hamedani
 | first2 = Hans | last2 = Volkmer
 | first3 = J. | last3 = Behboodian
 | doi = 10.1556/SScMath.2011.1183
}}</ref>)

The [[natural exponential family]] of a distribution may be realized by shifting or translating {{math|''K''(''t'')}}, and adjusting it vertically so that it always passes through the origin: if {{math|''f''}} is the pdf with cumulant generating function <math display=inline>K(t)=\log M(t),</math> and <math display=inline>f|\theta</math> is its natural exponential family, then <math display=inline>f(x\mid\theta)=\frac1{M(\theta)}e^{\theta x} f(x),</math> and <math display=inline>K(t\mid\theta)=K(t+\theta)-K(\theta).</math>

If {{math|''K''(''t'')}} is finite for a range {{math|''t''<sub>1</sub> < Re(''t'') < ''t''<sub>2</sub>}} then if {{math|''t''<sub>1</sub> < 0 < ''t''<sub>2</sub>}} then {{math|''K''(''t'')}} is analytic and infinitely differentiable for {{math|''t''<sub>1</sub> < Re(''t'') < ''t''<sub>2</sub>}}. Moreover for {{math|''t''}} real and {{math|''t''<sub>1</sub> < ''t'' < ''t''<sub>2</sub> ''K''(''t'')}} is strictly convex, and {{math|''K''&prime;(''t'')}} is strictly increasing. {{Citation needed|date=March 2011}}

== Further properties of cumulants ==

=== A negative result ===
Given the results for the cumulants of the [[normal distribution]], it might be hoped to find families of distributions for which
{{math|1=''κ''<sub>''m''</sub> = ''κ''<sub>''m''+1</sub> = ⋯ = 0}} for some {{math|1=''m'' > 3}}, with the lower-order cumulants (orders 3 to {{math|1=''m'' − 1}}) being non-zero. There are no such distributions.<ref>Lukacs, E. (1970) Characteristic Functions (2nd Edition), Griffin, London. (Theorem 7.3.5)</ref> The underlying result here is that the cumulant generating function cannot be a finite-order polynomial of degree greater than&nbsp;2.

=== Cumulants and moments ===
The [[moment generating function]] is given by:
<math display=block>M(t) = 1+\sum_{n=1}^\infty \frac{\mu'_n t^n}{n!} = \exp \left(\sum_{n=1}^\infty \frac{\kappa_n t^n}{n!}\right) = \exp(K(t)).</math>

So the cumulant generating function is the logarithm of the moment generating function
<math display=block>K(t) = \log M(t).</math>

The first cumulant is the [[expected value]]; the second and third cumulants are respectively the second and third [[central moment]]s (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 moments can be recovered in terms of cumulants by evaluating the {{math|''n''}}th derivative of <math display=inline>\exp(K(t))</math> at {{tmath|1=t=0}},
<math display=block>\mu'_n = M^{(n)}(0) = \left. \frac{\mathrm{d}^n \exp (K(t))}{\mathrm{d}t^n}\right|_{t=0}. </math>

Likewise, the cumulants can be recovered in terms of moments by evaluating the {{math|''n''}}th derivative of <math display=inline>\log M(t)</math> at {{tmath|1=t=0}},
<math display=block>\kappa_n = K^{(n)}(0) = \left. \frac{\mathrm{d}^n \log M(t)}{\mathrm{d}t^n} \right|_{t=0}.</math>

The explicit expression for the {{math|''n''}}th moment in terms of the first {{math|''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
<math display=block>\mu'_n = \sum_{k=1}^n B_{n,k}(\kappa_1,\ldots,\kappa_{n-k+1}) </math>
<math display=block>\kappa_n = \sum_{k=1}^n (-1)^{k-1} (k-1)! B_{n,k}(\mu'_1, \ldots, \mu'_{n-k+1}),</math>
where <math display=inline>B_{n,k}</math> are incomplete (or partial) [[Bell polynomials]].

In the like manner, if the mean is given by <math display=inline>\mu</math>, the central moment generating function is given by
<math display=block>C(t) = \operatorname{E}[e^{t(x-\mu)}] = e^{-\mu t} M(t) = \exp(K(t) - \mu t), </math>
and the {{math|''n''}}th central moment is obtained in terms of cumulants as
<math display=block>\mu_n = C^{(n)}(0) = \left. \frac{\mathrm{d}^n}{\mathrm{d}t^n} \exp (K(t) - \mu t) \right|_{t=0} = \sum_{k=1}^n B_{n,k}(0,\kappa_2,\ldots,\kappa_{n-k+1}).</math>

Also, for {{math|''n'' > 1}}, the {{math|''n''}}th cumulant in terms of the central moments is
<math display=block>
\begin{align}
\kappa_n  & = K^{(n)}(0) = \left. \frac{\mathrm{d}^n}{\mathrm{d}t^n} (\log C(t) + \mu t) \right|_{t=0} \\[4pt]
& = \sum_{k=1}^n (-1)^{k-1} (k-1)! B_{n,k}(0,\mu_2,\ldots,\mu_{n-k+1}).
\end{align}
</math>
 
The {{math|''n''}}th [[moment (mathematics)|moment]] {{math|''μ''′<sub>''n''</sub>}} is an {{math|''n''}}th-degree polynomial in the first {{math|''n''}} cumulants. The first few expressions are:

<!-- NOTE: All coefficients below are POSITIVE.  Only when goes in the opposite direction – expressing cumulants in terms of moments--does one see some negative coefficients. -->
<math display=block>
\begin{align}
\mu'_1 = {} & \kappa_1 \\[5pt]
\mu'_2 = {} & \kappa_2+\kappa_1^2 \\[5pt]
\mu'_3 = {} & \kappa_3+3\kappa_2\kappa_1+\kappa_1^3 \\[5pt]
\mu'_4 = {} & \kappa_4 + 4\kappa_3\kappa_1 + 3\kappa_2^2 + 6\kappa_2\kappa_1^2 + \kappa_1^4 \\[5pt]
\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 \\[5pt]
\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{align}
</math>

The "prime" distinguishes the moments {{math|''μ''′<sub>''n''</sub>}} from the [[moment about the mean|central moments]] {{math|''μ''<sub>''n''</sub>}}.  To express the ''central'' moments as functions of the cumulants, just drop from these polynomials all terms in which {{math|''κ''<sub>1</sub>}} appears as a factor:
<math display=block>
\begin{align}
\mu_1 & =0 \\[4pt]
\mu_2 & =\kappa_2 \\[4pt]
\mu_3 & =\kappa_3 \\[4pt]
\mu_4 & =\kappa_4+3\kappa_2^2 \\[4pt]
\mu_5 & =\kappa_5+10\kappa_3\kappa_2 \\[4pt]
\mu_6 & =\kappa_6+15\kappa_4\kappa_2+10\kappa_3^2+15\kappa_2^3.
\end{align}
</math>

Similarly, the {{math|''n''}}th cumulant {{math|''κ''<sub>''n''</sub>}} is an {{math|''n''}}th-degree polynomial in the first {{math|''n''}} non-central moments. The first few expressions are:
<math display=block>
\begin{align}
\kappa_1 = {} & \mu'_1 \\[4pt]
\kappa_2 = {} & \mu'_2-{\mu'_1}^2 \\[4pt]
\kappa_3 = {} & \mu'_3-3\mu'_2\mu'_1+2{\mu'_1}^3 \\[4pt]
\kappa_4 = {} & \mu'_4-4\mu'_3\mu'_1-3{\mu'_2}^2+12\mu'_2{\mu'_1}^2-6{\mu'_1}^4 \\[4pt]
\kappa_5 = {} & \mu'_5-5\mu'_4\mu'_1-10\mu'_3\mu'_2 + 20\mu'_3{\mu'_1}^2 + 30{\mu'_2}^2\mu'_1-60\mu'_2{\mu'_1}^3 + 24{\mu'_1}^5 \\[4pt]
\kappa_6 = {} & \mu'_6-6\mu'_5\mu'_1-15\mu'_4\mu'_2+30\mu'_4{\mu'_1}^2-10{\mu'_3}^2 + 120\mu'_3\mu'_2\mu'_1 \\
& {} - 120\mu'_3{\mu'_1}^3 + 30{\mu'_2}^3 - 270{\mu'_2}^2 {\mu'_1}^2+360\mu'_2{\mu'_1}^4-120{\mu'_1}^6\,.
\end{align}
</math>

In general,<ref>{{Cite web |last=Bazant |first=Martin |date=February 4, 2005 |title=MIT 18.366 {{!}} Fall 2006 {{!}} Graduate {{!}} Random Walks and Diffusion, Lecture 2: Moments, Cumulants, and Scaling |url=https://ocw.mit.edu/courses/18-366-random-walks-and-diffusion-fall-2006/resources/lec02/ |url-status=live |archive-url=https://web.archive.org/web/20221007061116/https://ocw.mit.edu/courses/18-366-random-walks-and-diffusion-fall-2006/resources/lec02/ |archive-date=2022-10-07 |access-date=2023-09-03 |website=MIT OpenCourseWare |language=en}}</ref> the cumulant is the determinant of a matrix:
<math display="block">\kappa_l = (-1)^{l+1}
\left|\begin{array}{cccccccc}
\mu'_1 & 1 & 0 & 0 & 0 & 0 & \ldots & 0 \\
\mu'_2 & \mu'_1 & 1 & 0 & 0 & 0 & \ldots & 0 \\
\mu'_3 & \mu'_2 & \left(\begin{array}{l}
2 \\
1
\end{array}\right) \mu'_1 & 1 & 0 & 0 & \ldots & 0 \\
\mu'_4 & \mu'_3 & \left(\begin{array}{l}
3 \\
1
\end{array}\right) \mu'_2 & \left(\begin{array}{l}
3 \\
2
\end{array}\right) \mu'_1 & 1 & 0 & \ldots & 0 \\
\mu'_5 & \mu'_4 & \left(\begin{array}{l}
4 \\
1
\end{array}\right) \mu'_3 & \left(\begin{array}{l}
4 \\
2
\end{array}\right) \mu'_2 & \left(\begin{array}{c}
4 \\
3
\end{array}\right) \mu'_1 & 1 & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\
\mu'_{l-1} & \mu'_{l-2} & \ldots & \ldots & \ldots & \ldots & \ddots & 1 \\
\mu'_l & \mu'_{l-1} & \ldots & \ldots & \ldots & \ldots & \ldots & \left(\begin{array}{l}
l-1 \\
l-2
\end{array}\right) \mu'_1
\end{array}\right|</math>

To express the cumulants {{math|''κ''<sub>''n''</sub>}} for {{math|''n'' > 1}} as functions of the central moments, drop from these polynomials all terms in which μ'<sub>1</sub> appears as a factor:
<math display=block>\kappa_2=\mu_2\,</math>
<math display=block>\kappa_3=\mu_3\,</math>
<math display=block>\kappa_4=\mu_4-3{\mu_2}^2\,</math>
<math display=block>\kappa_5=\mu_5-10\mu_3\mu_2\,</math>
<math display=block>\kappa_6=\mu_6-15\mu_4\mu_2-10{\mu_3}^2+30{\mu_2}^3\,.</math>

The cumulants can be related to the moments by [[Differentiation (mathematics)|differentiating]] the relationship {{math|1=log ''M''(''t'') = ''K''(''t'')}} with respect to {{math|''t''}}, giving {{math|1=''M′''(''t'') = ''K′''(''t'') ''M''(''t'')}}, which conveniently contains no exponentials or logarithms.  Equating the coefficient of {{math|''t''<sup> ''n''−1</sup> / (''n''−1)!}} on the left and right sides and using {{math|1=''μ′''<sub>0</sub> = 1}} gives the following formulas for {{math|''n'' ≥ 1}}:<ref>{{cite journal |last1=Smith |first1=Peter J. |date=May 1995 |title=A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa|url=https://www.jstor.org/stable/2684642 |journal=The American Statistician |volume=49 |issue=2 |pages=217–218 |doi=10.2307/2684642|jstor=2684642 |url-access=subscription }}</ref>
<math display=block>
\begin{align}
\mu'_1 = {} & \kappa_1 \\[1pt]
\mu'_2 = {} & \kappa_1\mu'_1+\kappa_2 \\[1pt]
\mu'_3 = {} & \kappa_1\mu'_2+2\kappa_2\mu'_1+\kappa_3 \\[1pt]
\mu'_4 = {} & \kappa_1\mu'_3+3\kappa_2\mu'_2+3\kappa_3\mu'_1+\kappa_4 \\[1pt]
\mu'_5 = {} & \kappa_1\mu'_4+4\kappa_2\mu'_3+6\kappa_3\mu'_2+4\kappa_4\mu'_1+\kappa_5 \\[1pt]
\mu'_6 = {} & \kappa_1\mu'_5+5\kappa_2\mu'_4+10\kappa_3\mu'_3+10\kappa_4\mu'_2+5\kappa_5\mu'_1+\kappa_6 \\[1pt]
\mu'_n = {} & \sum_{m=1}^{n-1}{n-1 \choose m-1}\kappa_m \mu'_{n-m} + \kappa_n\,.
\end{align}
</math>
These allow either <math display=inline>\kappa_n</math> or <math display=inline>\mu'_n</math> to be computed from the other using knowledge of the lower-order cumulants and moments.  The corresponding formulas for the central moments <math display=inline>\mu_n</math> for <math display=inline>n \ge 2</math> are formed from these formulas by setting <math display=inline>\mu'_1 = \kappa_1 = 0</math> and replacing each <math display=inline>\mu'_n</math> with <math display=inline>\mu_n</math> for <math display=inline>n \ge 2</math>:
<math display=block>
\begin{align}
\mu_2 = {} & \kappa_2 \\[1pt]
\mu_3 = {} & \kappa_3 \\[1pt]
\mu_n = {} & \sum_{m=2}^{n-2}{n-1 \choose m-1}\kappa_m \mu_{n-m} + \kappa_n\,.
\end{align}
</math>

=== Cumulants and set-partitions ===
These polynomials have a remarkable [[combinatorics|combinatorial]] interpretation: the coefficients count certain [[partition of a set|partitions of sets]].  A general form of these polynomials is
<math display=block>\mu'_n=\sum_{\pi \, \in \, \Pi} \prod_{B \, \in \, \pi} \kappa_{|B|}</math>
where
* {{math|''π''}} runs through the list of all partitions of a set of size {{math|''n''}};
* "{{math|''B'' ∈ {{pi}}}}" means {{math|''B''}} is one of the "blocks" into which the set is partitioned; and
* {{math|{{abs|''B''}}}} is the size of the set {{math|''B''}}.

Thus each [[monomial]] is a constant times a product of cumulants in which the sum of the indices is {{math|''n''}} (e.g., in the term {{math|1=''κ''<sub>3</sub> ''κ''<sub>2</sub><sup>2</sup> ''κ''<sub>1</sub>}}, 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]] {{math|''n''}}  corresponds to each term.  The ''coefficient'' in each term is the number of partitions of a set of {{math|''n''}} members that collapse to that partition of the integer {{math|''n''}} when the members of the set become indistinguishable.

=== Cumulants and combinatorics ===
Further connection between cumulants and combinatorics can be found in the work of [[Gian-Carlo Rota]], where links to [[invariant theory]], [[symmetric function]]s, and binomial sequences are studied via [[umbral calculus]].<ref>{{cite journal |first1=G.-C. |last1=Rota |first2=J. |last2=Shen |title=On the Combinatorics of Cumulants |journal=Journal of Combinatorial Theory, Series A |volume=91 |issue=1–2 |pages=283–304 |year=2000 |doi=10.1006/jcta.1999.3017 |doi-access=free }}</ref>

== Joint cumulants==
The '''joint cumulant''' {{math|''κ''}} of several random variables {{math|''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>}} is defined as the coefficient {{math|''κ''<sub>1,...,1</sub>(''X''<sub>''1''</sub>, ..., ''X''<sub>''n''</sub>)}} in the Maclaurin series of the multivariate cumulant generating function, see Section 3.1 in,<ref name="link.springer.com">{{Cite journal |last1=Peccati |first1=Giovanni |last2=Taqqu |first2=Murad S. |date=2011 |title=Wiener Chaos: Moments, Cumulants and Diagrams |url=https://link.springer.com/book/10.1007/978-88-470-1679-8 |journal=Bocconi & Springer Series |volume=1 |language=en |doi=10.1007/978-88-470-1679-8 |isbn=978-88-470-1678-1 |issn=2039-1471|url-access=subscription }}</ref>
<math display="block">G(t_1,\dots,t_n)=\log \mathrm{E}(\mathrm{e}^{\sum_{j=1}^n t_j X_j})
=\sum_{k_1,\ldots,k_n} \kappa_{k_1,\ldots,k_n} \frac{t_1^{k_1} \cdots t_n^{k_n}}{k_1! \cdots k_n!} \,.</math>
Note that
<math display="block">\kappa_{k_1,\dots,k_n} 
= \left.\left(\frac{\mathrm{d}}{\mathrm{d} t_1}\right)^{k_1} \cdots \left(\frac{\mathrm{d}}{\mathrm{d} t_n}\right)^{k_n} G(t_1,\dots,t_n) \right|_{t_1 = \dots = t_n = 0}\,,</math>
and, in particular
<math display="block">\kappa(X_1,\ldots,X_n) 
= \left. \frac{\mathrm{d}^n}{\mathrm{d} t_1 \cdots \mathrm{d} t_n} G(t_1,\dots,t_n) \right|_{t_1 = \dots = t_n = 0}\,.</math>
As with a single variable, the generating function and cumulant can instead be defined via 
<math display="block">H(t_1,\dots,t_n)
=\log \mathrm{E}(\mathrm{e}^{\sum_{j=1}^n i t_j X_j})
=\sum_{k_1,\ldots,k_n} \kappa_{k_1,\ldots,k_n} i^{k_1+\cdots+k_n} \frac{t_1^{k_1} \cdots t_n^{k_n}}{k_1! \cdots k_n!}\,,</math>
in which case
<math display="block">\kappa_{k_1,\dots,k_n} = (-i)^{k_1+\cdots+k_n} \left.\left(\frac{\mathrm{d}}{\mathrm{d} t_1}\right)^{k_1} \cdots \left(\frac{\mathrm{d}}{\mathrm{d} t_n}\right)^{k_n} H(t_1,\dots,t_n) \right|_{t_1 = \dots = t_n = 0}\,,</math>
and
<math display="block">\kappa(X_1,\ldots,X_n) 
= \left. (-i)^{n} \frac{\mathrm{d}^n}{\mathrm{d} t_1 \cdots \mathrm{d} t_n} H(t_1,\dots,t_n) \right|_{t_1 = \dots = t_n = 0}\,.</math>

=== Repeated random variables and relation between the coefficients ''&kappa;''<sub>''k''<sub>1</sub>, ..., ''k''<sub>''n''</sub></sub> ===
Observe that  <math display=inline>\kappa_{k_1,\dots,k_n} (X_1,\ldots,X_n)</math> can also be written as
<math display="block">\kappa_{k_1,\dots,k_n} 
= \left. \frac{\mathrm{d}^{k_1}}{\mathrm{d} t_{1,1} \cdots \mathrm{d} t_{1,k_1}} \cdots \frac{\mathrm{d}^{k_n}}{\mathrm{d} t_{n,1} \cdots \mathrm{d} t_{n,k_n}} G\left(\sum_{j=1}^{k_1}t_{1,j},\dots,\sum_{j=1}^{k_n}t_{n,j}\right) \right|_{t_{i,j}=0},</math>
from which we conclude that
<math display="block">\kappa_{k_1,\dots,k_n} (X_1,\ldots,X_n)
= \kappa_{1,\ldots,1} ( \underbrace{X_1,\dots,X_1}_{k_1}, \ldots , \underbrace{X_n,\dots,X_n}_{k_n} ) .</math>
For example
<math display=block>\kappa_{2,0,1}(X,Y,Z) = \kappa(X,X,Z),\,</math>
and
<math display=block>\kappa_{0,0,n,0}(X,Y,Z,T) = \kappa_{n}(Z) = \kappa(\underbrace{Z,\dots,Z}_{n}) .\,</math>
In particular, the last equality shows that the cumulants of a single random variable are the joint cumulants of multiple copies of that random variable.

=== Relation with mixed moments ===
The joint cumulant of random variables can be expressed as an alternate sum of products of their [[Moment (mathematics)#Mixed moments|mixed moments]], see Equation (3.2.7) in,<ref name="link.springer.com"/> 
<math display="block">\kappa(X_1,\dots,X_n)=\sum_\pi (|\pi|-1)!(-1)^{|\pi|-1}\prod_{B\in\pi}E\left(\prod_{i\in B}X_i\right)</math>
where&nbsp;{{pi}} runs through the list of all partitions of {{math|{{mset|1, ..., ''n''}}}}; where&nbsp;{{math|''B''}} runs through the list of all blocks of the partition&nbsp;{{pi}}; and where&nbsp;{{math|{{abs|{{pi}}}}}} is the number of parts in the partition.

For example,
<math display=block>\kappa(X)=\operatorname E(X),</math>
is the expected value of <math display="inline">X</math>,
<math display=block>\kappa(X,Y)=\operatorname E(XY) - \operatorname E(X) \operatorname E(Y),</math>
is the [[covariance]]  of <math display="inline">X</math> and <math display="inline">Y</math>, and
<math display=block>\kappa(X,Y,Z)=\operatorname E(XYZ) - \operatorname E(XY) \operatorname E(Z) - \operatorname E(XZ) \operatorname E(Y) - \operatorname E(YZ) \operatorname E(X) + 2\operatorname E(X)\operatorname E(Y)\operatorname E(Z).\,</math>

For zero-mean random variables <math display="inline">X_1,\ldots,X_n</math>, any mixed moment of the form <math display="inline">\prod_{B\in\pi} E\left(\prod_{i\in B} X_i\right)</math> vanishes if <math display="inline">\pi</math> is a partition of <math display="inline">\{ 1,\ldots,n \}</math> which contains a singleton <math display="inline">B=\{k\}</math>.
Hence, the expression of their joint cumulant in terms of mixed moments simplifies.
For example, if X,Y,Z,W are zero mean random variables, we have 
<math display=block>\kappa(X,Y,Z) = \operatorname E(XYZ).\,</math>
<math display=block>\kappa(X,Y,Z,W) = \operatorname E(XYZW) - \operatorname E(XY) \operatorname E(ZW) - \operatorname E(XZ) \operatorname E(YW) - \operatorname E(XW) \operatorname E(YZ).\,</math>

More generally, any coefficient of the Maclaurin series can also be expressed in terms of mixed moments, although there are no concise formulae.
Indeed, 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. For example
<math display=block>\kappa_{201}(X,Y,Z) = \kappa(X,X,Z)=\operatorname E(X^2Z)  -2\operatorname E(XZ)\operatorname E(X) - \operatorname E(X^2)\operatorname E(Z) + 2\operatorname E(X)^2\operatorname E(Z).\,</math>

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.{{fact|date=December 2023}}

The 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:<ref name="link.springer.com"/>
<math display=block>\operatorname E(X_1\cdots X_n)=\sum_\pi\prod_{B\in\pi}\kappa(X_i : i \in B). </math>

For example:
<math display=block>\operatorname E(XYZ) = \kappa(X,Y,Z) + \kappa(X,Y)\kappa(Z) + \kappa(X,Z)\kappa(Y) + \kappa(Y,Z)\kappa(X) + \kappa(X)\kappa(Y)\kappa(Z).\,</math>

=== Further properties ===
Another important property of joint cumulants is multilinearity:
<math display=block>\kappa(X+Y,Z_1,Z_2,\dots) = \kappa(X,Z_1,Z_2,\ldots) + \kappa(Y,Z_1,Z_2,\ldots).\,</math>

Just as the second cumulant is the variance, the joint cumulant of just two random variables is the [[covariance]].  The familiar identity
<math display=block>\operatorname{var}(X+Y) = \operatorname{var}(X) + 2\operatorname{cov}(X,Y) + \operatorname{var}(Y)\,</math>
generalizes to cumulants:
<math display=block>\kappa_n(X+Y)=\sum_{j=0}^n {n \choose j} \kappa( \, \underbrace{X,\dots,X}_j, \underbrace{Y,\dots,Y}_{n-j}\,).\,</math>

=== Conditional cumulants and the law of total cumulance ===
{{Main|law of total cumulance}}
The [[law of total expectation]] and the [[law of total variance]] generalize naturally to conditional cumulants.  The case {{math|1=''n'' = 3}}, expressed in the language of (central) [[moment (mathematics)|moments]] rather than that of cumulants, says
<math display=block>\mu_3(X) = \operatorname E(\mu_3(X\mid Y)) + \mu_3(\operatorname E(X\mid Y)) + 3 \operatorname{cov}(\operatorname E(X\mid Y), \operatorname{var} (X\mid Y)).</math>

In general,<ref>{{cite journal | last1 = Brillinger | first1 = D.R. | year = 1969 | title = The Calculation of Cumulants via Conditioning | journal = Annals of the Institute of Statistical Mathematics | volume = 21 | pages = 215–218 | doi=10.1007/bf02532246| s2cid = 122673823 }}</ref>
<math display=block>\kappa(X_1,\dots,X_n)=\sum_\pi \kappa(\kappa(X_{\pi_1}\mid Y), \dots, \kappa(X_{\pi_b}\mid Y))</math>
where
* the sum is over all [[partition of a set|partitions]]&nbsp;{{pi}} of the set {{math|{{mset|1, ..., ''n''}}}} of indices, and
* {{pi}}<sub>1</sub>,&nbsp;...,&nbsp;{{pi}}<sub>b</sub> are all of the "blocks" of the partition {{pi}}; the expression {{math|''κ''(''X''<sub>{{pi}}<sub>''m''</sub></sub>)}} indicates that the joint cumulant of the random variables whose indices are in that block of the partition.

=== Conditional cumulants and the conditional expectation ===
For certain settings, a derivative identity can be established between the conditional cumulant and the conditional expectation. For example, suppose that {{math|1=''Y'' = ''X'' + ''Z''}} where {{math|1=''Z''}} is standard normal independent of {{math|1=''X''}}, then for any {{math|1=''X''}} it holds that<ref>{{cite journal | doi=10.1109/TIT.2022.3216012 | title=Conditional Mean Estimation in Gaussian Noise: A Meta Derivative Identity with Applications | date=2023 | last1=Dytso | first1=Alex | last2=Poor | first2=H. Vincent | last3=Shamai Shitz | first3=Shlomo | journal=IEEE Transactions on Information Theory | volume=69 | issue=3 | pages=1883–1898 | s2cid=253308274 }}</ref> 
<math display=block>\kappa_{n+1}(X\mid Y=y) =  \frac{ \mathrm{d}^n}{ \mathrm{d} y^n}\operatorname E(X\mid Y = y), \, n \in \mathbb{N}, \, y \in \mathbb{R}.</math>
The results can also be extended to the exponential family.<ref>{{cite journal | doi=10.1109/TIT.2023.3249163 | title=Meta Derivative Identity for the Conditional Expectation | date=2023 | last1=Dytso | first1=Alex | last2=Cardone | first2=Martina | last3=Zieder | first3=Ian | journal=IEEE Transactions on Information Theory | volume=69 | issue=7 | pages=4284–4302 | s2cid=257247930 }}</ref>

== Relation to statistical physics ==
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.

A system in equilibrium with a thermal bath at temperature {{math|''T''}}  have a fluctuating internal energy {{math|''E''}}, which can be considered a random variable drawn from a distribution <math display=inline> E\sim p(E)</math>.  The [[partition function (statistical mechanics)|partition function]] of the system is
<math display="block">Z(\beta) = \sum_i e^{-\beta E_i} ,</math>
where [[Thermodynamic beta|{{math|''β''}}]] =&nbsp;{{math|1/(''kT'')}} and {{math|''k''}} is the [[Boltzmann constant]] and the notation <math display=inline>\langle A \rangle</math> has been used rather than <math display=inline>\operatorname{E}[A]</math> for the expectation value to avoid confusion with the energy, {{math|''E''}}. Hence the first and second cumulant for the energy {{math|''E''}} give the average energy and heat capacity.
<math display=block>
\begin{align}
\langle E \rangle_c & = \frac{\partial \log Z}{\partial (-\beta)} = \langle E \rangle \\[6pt]
\langle E^2 \rangle_c & = \frac{\partial\langle E\rangle_c}{\partial (-\beta)} = k T^2 \frac{\partial \langle E\rangle}{\partial T} = kT^2C
\end{align}
</math>

The [[Helmholtz free energy]] expressed in terms of
<math display=block>F(\beta) = -\beta^{-1}\log Z(\beta) \, </math>
further connects thermodynamic quantities with cumulant generating function for the energy. Thermodynamics properties that are derivatives of the free energy, such as its [[internal energy]], [[entropy]], and [[specific heat]] capacity, all can be readily expressed in terms of these cumulants. Other free energy can be a function of other variables such as the magnetic field or chemical potential <math display=inline>\mu</math>, e.g.
<math display=block>\Omega=-\beta^{-1}\log(\langle \exp(-\beta E -\beta\mu N) \rangle),\,</math>
where {{math|''N''}} is the number of particles and <math display=inline>\Omega</math> is the grand potential.   Again 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 {{math|''E''}} and {{math|''N''}}.<!--The relation between the cumulant and statistical physics is not explicitly stated here, so this material seems out of place. [Other editor:] NB: Just added this relation. Deserves expanding-->

== History ==
The history of cumulants is discussed by [[Anders Hald]].<ref>
[[Anders Hald|Hald, A.]] (2000) "The early history of the cumulants and the [[Gram–Charlier series]]" ''International Statistical Review'', 68 (2): 137–153. (Reprinted in {{Cite book|editor-link=Steffen Lauritzen|editor-first=Steffen L.|editor-last=Lauritzen|title=Thiele: Pioneer in Statistics|publisher= Oxford U. P.|year=2002|isbn=978-0-19-850972-1|title-link=Thorvald N. Thiele}})</ref><ref>
{{Cite book|first1=Anders|last1=Hald|title=A History of Mathematical Statistics from 1750 to 1930 |author-link=Anders Hald|year=1998 |publisher=Wiley |location=New York |isbn=978-0-471-17912-2}}</ref>

Cumulants were first introduced by [[Thorvald N. Thiele]], in 1889, who called them ''semi-invariants''.<ref>H. Cramér (1946) Mathematical Methods of Statistics, Princeton University Press, Section 15.10, p. 186.</ref>  They were first called ''cumulants'' in a 1932 paper by [[Ronald Fisher]] and [[John Wishart (statistician)|John Wishart]].<ref>[[Ronald Fisher|Fisher, R.A.]], [[John Wishart (statistician)|John Wishart, J.]] (1932) [http://plms.oxfordjournals.org/content/s2-33/1/195.full.pdf+html ''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.&nbsp;195–208 {{doi| 10.1112/plms/s2-33.1.195}}</ref> Fisher was publicly reminded of Thiele's work by Neyman, who also notes previous published citations of Thiele brought to Fisher's attention.<ref>Neyman, J. (1956): ‘Note on an Article by Sir Ronald Fisher,’ ''Journal of the Royal Statistical Society'', Series B (Methodological), 18, pp. 288–94.</ref>   [[Stephen Stigler]] has said{{Citation needed|date=January 2011}} that the name ''cumulant'' was suggested to Fisher in a letter from [[Harold Hotelling]].  In a paper published in 1929, Fisher had called them ''cumulative moment functions''.<ref>{{cite journal|last1=Fisher|first1=R. A.|title=Moments and Product Moments of Sampling Distributions|journal=Proceedings of the London Mathematical Society|date=1929|volume=30|pages=199–238|doi=10.1112/plms/s2-30.1.199|url=https://digital.library.adelaide.edu.au/dspace/bitstream/2440/15200/1/74pt2.pdf|hdl=2440/15200|hdl-access=free}}<!--|access-date=7 August 2015--></ref> 

The partition function in statistical physics was introduced by [[Josiah Willard Gibbs]] in 1901.{{Citation needed|date=January 2011}} The free energy is often called Gibbs free energy. In [[statistical mechanics]], cumulants are also known as [[Ursell function]]s relating to a publication in 1927.{{Citation needed|date=January 2011}}

== Cumulants in generalized settings ==

=== Formal cumulants ===
More generally, the cumulants of a sequence {{math|1={ ''m''<sub>''n''</sub> : ''n'' = 1, 2, 3, ... } }}, not necessarily the moments of any probability distribution, are, by definition,
<math display=block>1+\sum_{n=1}^\infty \frac{m_n t^n}{n!} = \exp \left( \sum_{n=1}^\infty \frac{\kappa_n t^n}{n!} \right) ,</math>
where the values of {{math|''κ''<sub>''n''</sub>}} for {{math|1=''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.

=== Bell numbers ===
In [[combinatorics]], the {{math|''n''}}th [[Bell number]] is the number of partitions of a set of size {{math|''n''}}.  All of the [[Bell number#Generating function|cumulants of the sequence of Bell numbers are equal to 1]].  The Bell numbers are the [[Moment-generating function#Examples|moments of the Poisson distribution with expected value 1]].

=== Cumulants of a polynomial sequence of binomial type ===
For any sequence {{math|{{mset| ''κ''<sub>''n''</sub> : ''n'' {{=}} 1, 2, 3, ... }}}} of [[scalar (mathematics)|scalars]] in a [[field (mathematics)|field]] of characteristic zero, being considered formal cumulants, there is a corresponding sequence {{math|{{mset| ''μ''′ : ''n'' {{=}} 1, 2, 3, ... }}}} of formal moments, given by the polynomials above.{{clarify|reason=what polynomials|date=January 2011}}{{Citation needed|date=January 2011}} For those polynomials, construct a [[polynomial sequence]] in the following way.  Out of the polynomial
<math display=block>
\begin{align}
\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{align}
</math>
make a new polynomial in these plus one additional variable {{math|''x''}}:
<math display=block>
\begin{align}
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{align}
</math>
and then generalize the pattern.  The pattern is that the numbers of blocks in the aforementioned partitions are the exponents on {{math|''x''}}.  Each coefficient is a polynomial in the cumulants; these are the [[Bell polynomials]], named after [[Eric Temple Bell]].{{Citation needed|date=January 2011}}

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|date=January 2011}}

=== Free cumulants ===
In the above moment-cumulant formula\
<math display=block>\operatorname E(X_1\cdots X_n)=\sum_\pi\prod_{B\,\in\,\pi}\kappa(X_i : i\in B)</math>
for joint cumulants, one sums over ''all'' partitions of the set {{math|{{mset| 1, ..., ''n'' }}}}.  If instead, one sums only over the [[noncrossing partition]]s, then, by solving these formulae for the <math display=inline>\kappa</math> in terms of the moments, one gets '''free cumulants''' rather than conventional cumulants treated above. These free cumulants were introduced by Roland Speicher and play a central role in [[free probability]] theory.<ref>{{cite journal |last=Speicher |first=Roland |year=1994 |title=Multiplicative functions on the lattice of non-crossing partitions and free convolution |journal=[[Mathematische Annalen]] |volume=298 |issue=4 |pages=611–628 |doi=10.1007/BF01459754 |s2cid=123022311 }}</ref><ref name="Novak-Śniady">{{Cite journal|last1=Novak|first1=Jonathan|last2=Śniady|first2=Piotr|year=2011|title=What Is a Free Cumulant?|journal=[[Notices of the American Mathematical Society]]|volume=58|issue=2|pages=300–301|issn=0002-9920}}</ref>  In that theory, rather than considering [[statistical independence|independence]] of [[random variable]]s, defined in terms of [[Tensor product of algebras|tensor products of algebras]] of random variables, one considers instead [[free independence]] of random variables, defined in terms of [[free product]]s of algebras.<ref name="Novak-Śniady"/>

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.<ref name="Novak-Śniady"/> 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.

== See also ==
* [[Entropic value at risk]]
* [[Multiset#Cumulant generating function|Cumulant generating function from a multiset]]
* [[Cornish–Fisher expansion]]
* [[Edgeworth expansion]]
* [[Polykay]]
* [[k-statistic]], a minimum-variance [[unbiased estimator]] of a cumulant
* [[Ursell function]]
* [[Total position spread]] tensor as an application of cumulants to analyse the electronic [[wave function]] in [[quantum chemistry]].

== References ==
{{reflist}}

== External links ==
* {{MathWorld | urlname=Cumulant | title=Cumulant}}
* [http://jeff560.tripod.com/c.html cumulant] on the [http://jeff560.tripod.com/mathword.html Earliest known uses of some of the words of mathematics]

{{Theory of probability distributions}}

[[Category:Moments (mathematics)]]