In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics, the cumulants of a
probability distribution are a set of quantities that provide an alternative to the ''
moments'' of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.
The first cumulant is the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
, the second cumulant is the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
, and the third cumulant is the same as the third
central moment
In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
. 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
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...
, the -th-order cumulant of their sum is equal to the sum of their -th-order cumulants. As well, the third and higher-order cumulants of a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
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 are defined using the cumulant-generating function , which is the
natural logarithm of the
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
:
:
The cumulants are obtained from a power series expansion of the cumulant generating function:
:
This expansion is a
Maclaurin series
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ...
, so the -th cumulant can be obtained by differentiating the above expansion times and evaluating the result at zero:
:
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 prefer to define the cumulant-generating function as the natural logarithm of the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
, which is sometimes also called the ''second'' characteristic function,
:
An advantage of —in some sense the function evaluated for purely imaginary arguments—is that is well defined for all real values of even when is not well defined for all real values of , such as can occur when there is "too much" probability that has a large magnitude. Although the function will be well defined, it will nonetheless mimic in terms of the length of its
Maclaurin series
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ...
, which may not extend beyond (or, rarely, even to) linear order in the argument , and in particular the number of cumulants that are well defined will not change. Nevertheless, even when does not have a long Maclaurin series, it can be used directly in analyzing and, particularly, adding random variables. Both the
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
(also called the Lorentzian) and more generally,
stable distribution
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be sta ...
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
-th cumulant
of (the distribution of) a random variable
enjoys the following properties:
* If
and
is constant (i.e. not random) then
i.e. the cumulant is translation-invariant. (If
then we have
* If
is constant (i.e. not random) then
i.e. the
-th cumulant is homogeneous of degree
.
* If random variables
are independent then
::
: i.e. the cumulant is cumulative– hence the name.
The cumulative property follows quickly by considering the cumulant-generating function:
:
so that each cumulant of a sum of independent random variables is the sum of the corresponding cumulants of the
addend
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
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 can be approximated through an
Edgeworth series The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants. The se ...
.
The 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.
*
mean
*
the variance, or second central moment.
*
the third central moment.
*
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.
*
Cumulants of some discrete probability distributions
* The constant random variables . The cumulant generating function is . The first cumulant is and the other cumulants are zero, .
* The
Bernoulli distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabi ...
s, (number of successes in one trial with probability of success). The cumulant generating function is . The first cumulants are and . The cumulants satisfy a recursion formula
::
* The
geometric distribution
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:
* The probability distribution of the number ''X'' of Bernoulli trials needed to get one success, supported on the set \;
* ...
s, (number of failures before one success with probability of success on each trial). The cumulant generating function is . The first cumulants are , and . Substituting gives and .
* The
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
s. The cumulant generating function is . All cumulants are equal to the parameter: .
* The
binomial distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
s, (number of successes in
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
trials with probability of success on each trial). The special case is a Bernoulli distribution. Every cumulant is just times the corresponding cumulant of the corresponding Bernoulli distribution. The cumulant generating function is . The first cumulants are and . Substituting gives and . The limiting case is a Poisson distribution.
* The
negative binomial distribution
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-r ...
s, (number of failures before successes with probability of success on each trial). The special case is a geometric distribution. Every cumulant is just times the corresponding cumulant of the corresponding geometric distribution. The derivative of the cumulant generating function is . The first cumulants are , and . Substituting gives and . Comparing these formulas to those of the binomial distributions explains the name 'negative binomial distribution'. The
limiting case is a Poisson distribution.
Introducing the
variance-to-mean ratio
In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a prob ...
:
the above probability distributions get a unified formula for the derivative of the cumulant generating function:
:
The second derivative is
:
confirming that the first cumulant is and the second cumulant is .
The constant random variables have .
The binomial distributions have so that .
The Poisson distributions have .
The negative binomial distributions have so that .
Note the analogy to the classification of
conic sections
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...
by
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
: circles , ellipses , parabolas , hyperbolas .
Cumulants of some continuous probability distributions
* For the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
with
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
and
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
, the cumulant generating function is . The first and second derivatives of the cumulant generating function are and . The cumulants are , , and . The special case is a constant random variable .
* The cumulants of the
uniform distribution on the interval are , where is the
th Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
.
* The cumulants of the
exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
with parameter are .
Some properties of the cumulant generating function
The cumulant generating function , if it exists, is
infinitely differentiable
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
and
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
, and passes through the origin. Its first derivative ranges monotonically in the open interval from the
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
to the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of the support of the probability distribution, and its second derivative is strictly positive everywhere it is defined, except for the
degenerate distribution
In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter ...
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
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
, that is, (''see
Big O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
'')
:
where
is the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
. The cumulant-generating function will have
vertical asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
(s) at the negative
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of such , if such a supremum exists, and at the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of such , if such a supremum exists, otherwise it will be defined for all real numbers.
If the
support of a random variable has finite upper or lower bounds, then its cumulant-generating function , if it exists, approaches
asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
(s) whose slope is equal to the supremum and/or infimum of the support,
:
respectively, lying above both these lines everywhere. (The
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s
:
yield the
-intercepts of these asymptotes, since .)
For a shift of the distribution by ,
For a degenerate point mass at , the cgf is the straight line
, and more generally,
if and only if and are independent and their cgfs exist; (
subindependence In probability theory and statistics, subindependence is a weak form of independence.
Two random variables ''X'' and ''Y'' are said to be subindependent if the characteristic function of their sum is equal to the product of their marginal character ...
and the existence of second moments sufficing to imply independence.)
The
natural exponential family In probability and statistics, a natural exponential family (NEF) is a class of probability distributions that is a special case of an exponential family (EF).
Definition
Univariate case
The natural exponential families (NEF) are a subset of ...
of a distribution may be realized by shifting or translating , and adjusting it vertically so that it always passes through the origin: if is the pdf with cgf
and
is its natural exponential family, then
and
If is finite for a range then if then is analytic and infinitely differentiable for . Moreover for real and is strictly convex, and is strictly increasing.
Further properties of cumulants
A negative result
Given the results for the cumulants of the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, it might be hoped to find families of distributions for which
for some , with the lower-order cumulants (orders 3 to ) 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.
Cumulants and moments
The
moment generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
is given by:
:
So the cumulant generating function is the logarithm of the moment generating function
:
The first cumulant is the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
; the second and third cumulants are respectively the second and third
central moment
In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
s (the second central moment is the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
); 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 -th derivative of
at ,
:
Likewise, the cumulants can be recovered in terms of moments by evaluating the -th derivative of
at ,
:
The explicit expression for the -th moment in terms of the first cumulants, and vice versa, can be obtained by using
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after , although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French ...
for higher derivatives of composite functions. In general, we have
:
:
where
are incomplete (or partial)
Bell polynomials
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's fo ...
.
In the like manner, if the mean is given by
, the central moment generating function is given by
:
and the -th central moment is obtained in terms of cumulants as
:
Also, for , the -th cumulant in terms of the central moments is
:
The -th
moment is an -th-degree polynomial in the first cumulants. The first few expressions are:
:
The "prime" distinguishes the moments from the
central moments . To express the ''central'' moments as functions of the cumulants, just drop from these polynomials all terms in which appears as a factor:
:
Similarly, the -th cumulant is an -th-degree polynomial in the first non-central moments. The first few expressions are:
:
To express the cumulants for as functions of the central moments, drop from these polynomials all terms in which μ'
1 appears as a factor:
:
:
:
:
:
To express the cumulants for as functions of the
standardized central moments , also set in the polynomials:
:
:
:
:
The cumulants can be related to the moments by
differentiating the relationship with respect to , giving , which conveniently contains no exponentials or logarithms. Equating the coefficient of on the left and right sides and using gives the following formulas for :
:
These allow either
or
to be computed from the other using knowledge of the lower-order cumulants and moments. The corresponding formulas for the central moments
for
are formed from these formulas by setting
and replacing each
with
for
.
Cumulants and set-partitions
These polynomials have a remarkable
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
interpretation: the coefficients count certain
partitions of sets. A general form of these polynomials is
:
where
* runs through the list of all partitions of a set of size ;
*"" means is one of the "blocks" into which the set is partitioned; and
* is the size of the set .
Thus each
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer exponent ...
is a constant times a product of cumulants in which the sum of the indices is (e.g., in the term , 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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
corresponds to each term. The ''coefficient'' in each term is the number of partitions of a set of members that collapse to that partition of the integer 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
Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, pro ...
, where links to
invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
,
symmetric function
In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\l ...
s, and binomial sequences are studied via
umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blis ...
.
Joint cumulants
The joint cumulant of several random variables is defined by a similar cumulant generating function
:
A consequence is that
:
where runs through the list of all partitions of , runs through the list of all blocks of the partition , and is the number of parts in the partition. For example,
:
is the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
, and
:
If any of these random variables are identical, e.g. if , then the same formulae apply, e.g.
:
although for such repeated variables there are more concise formulae. For zero-mean random vectors,
:
:
The joint cumulant of just one random variable is its expected value, and that of two random variables is their
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
. If some of the random variables are independent of all of the others, then any cumulant involving two (or more) independent random variables is zero. If all random variables are the same, then the joint cumulant is the -th ordinary cumulant.
The combinatorial meaning of the expression of moments in terms of cumulants is easier to understand than that of cumulants in terms of moments:
:
For example:
:
Another important property of joint cumulants is multilinearity:
:
Just as the second cumulant is the variance, the joint cumulant of just two random variables is the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
. The familiar identity
:
generalizes to cumulants:
:
Conditional cumulants and the law of total cumulance
The
law of total expectation
The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X is a random variable whose expected v ...
and the
law of total variance In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if X and Y are random variables on the same probability space, and ...
generalize naturally to conditional cumulants. The case , expressed in the language of (central)
moments rather than that of cumulants, says
:
In general,
:
where
* the sum is over all
partitions
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of the set of indices, and
*
1, ...,
b are all of the "blocks" of the partition ; the expression indicates that the joint cumulant of the random variables whose indices are in that block of the partition.
Relation to statistical physics
In
statistical physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations ...
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 have a fluctuating internal energy , which can be considered a random variable drawn from a distribution
. The
partition function of the system is
:
where
= and is
Boltzmann's constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
and the notation
has been used rather than