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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 in modern mathematics ...
, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
, and the second moment is the
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
. If the function is a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
, then the first moment 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 ...
, 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 ...
, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of
moment Moment or Moments may refer to: * Present time Music * The Moments, American R&B vocal group Albums * ''Moment'' (Dark Tranquillity album), 2020 * ''Moment'' (Speed album), 1998 * ''Moments'' (Darude album) * ''Moments'' (Christine Guldbrand ...
in physics. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution ( Hausdorff moment problem). The same is not true on unbounded intervals ( Hamburger moment problem). In the mid-nineteenth century,
Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebysh ...
became the first person to think systematically in terms of the moments of random variables.


Significance of the moments

The '-th raw moment (i.e., moment about zero) of a distribution is defined by\mu'_n = \langle x^n\ranglewhere\langle f(x) \rangle = \begin \sum f(x)P(x), & \text \\ \int f(x)P(x) dx, & \text \endThe -th moment of a real-valued
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
''f''(''x'') of a real variable about a value ''c'' is the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
\mu_n = \int_^\infty (x - c)^n\,f(x)\,\mathrmx.It is possible to define moments for
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s in a more general fashion than moments for real-valued functions — see moments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with ''c'' = 0. For the second and higher moments, the central moment (moments about the mean, with ''c'' being the mean) are usually used rather than the moments about zero, because they provide clearer information about the distribution's shape. Other moments may also be defined. For example, the th inverse moment about zero is \operatorname\left ^\right/math> and the -th logarithmic moment about zero is \operatorname\left ln^n(X)\right The -th moment about zero of a probability density function ''f''(''x'') 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 ...
of and is called a ''raw moment'' or ''crude moment''. The moments about its mean are called ''central'' moments; these describe the shape of the function, independently of
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
. If ''f'' is a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
, then the value of the integral above is called the -th moment of the
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
. More generally, if ''F'' is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the -th moment of the probability distribution is given by the Riemann–Stieltjes integral\mu'_n = \operatorname \left ^n\right= \int_^\infty x^n\,\mathrmF(x)where ''X'' is a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
that has this cumulative distribution ''F'', and is the expectation operator or mean. When\operatorname\left X^n \ \right= \int_^\infty \left, x^n\\,\mathrmF(x) = \inftythe moment is said not to exist. If the -th moment about any point exists, so does the -th moment (and thus, all lower-order moments) about every point. The zeroth moment of any
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
is 1, since the area under any
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
must be equal to one.


Standardized moments

The ''normalised'' -th central moment or standardised moment is the -th central moment divided by ; the normalised -th central moment of the random variable is \frac = \frac = \frac . These normalised central moments are dimensionless quantities, which represent the distribution independently of any linear change of scale. For an electric signal, the first moment is its DC level, and the second moment is proportional to its average power.


Notable moments


Mean

The first raw moment 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 '' ar ...
, usually denoted \mu \equiv \operatorname


Variance

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 ...
. The positive square root of the variance is the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
\sigma \equiv \left(\operatorname\left x - \mu)^2\rightright)^\frac.


Skewness

The third central moment is the measure of the lopsidedness of the distribution; any symmetric distribution will have a third central moment, if defined, of zero. The normalised third central moment is called the skewness, often . A distribution that is skewed to the left (the tail of the distribution is longer on the left) will have a negative skewness. A distribution that is skewed to the right (the tail of the distribution is longer on the right), will have a positive skewness. For distributions that are not too different from 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 ...
, the
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
will be somewhere near ; the
mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...
about .


Kurtosis

The fourth central moment is a measure of the heaviness of the tail of the distribution. Since it is the expectation of a fourth power, the fourth central moment, where defined, is always nonnegative; and except for a point distribution, it is always strictly positive. The fourth central moment of a normal distribution is . The kurtosis is defined to be the standardized fourth central moment (Equivalently, as in the next section, excess kurtosis is the fourth cumulant divided by the square of the second cumulant.) If a distribution has heavy tails, the kurtosis will be high (sometimes called leptokurtic); conversely, light-tailed distributions (for example, bounded distributions such as the uniform) have low kurtosis (sometimes called platykurtic). The kurtosis can be positive without limit, but must be greater than or equal to ; equality only holds for binary distributions. For unbounded skew distributions not too far from normal, tends to be somewhere in the area of and . The inequality can be proven by considering\operatorname\left left(T^2 - aT - 1\right)^2\right/math>where . This is the expectation of a square, so it is non-negative for all ''a''; however it is also a quadratic
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
in ''a''. Its
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
must be non-positive, which gives the required relationship.


Higher moments

High-order moments are moments beyond 4th-order moments. As with variance, skewness, and kurtosis, these are
higher-order statistics In statistics, the term higher-order statistics (HOS) refers to functions which use the third or higher power of a sample, as opposed to more conventional techniques of lower-order statistics, which use constant, linear, and quadratic terms (zero ...
, involving non-linear combinations of the data, and can be used for description or estimation of further shape parameters. The higher the moment, the harder it is to estimate, in the sense that larger samples are required in order to obtain estimates of similar quality. This is due to the excess
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
consumed by the higher orders. Further, they can be subtle to interpret, often being most easily understood in terms of lower order moments – compare the higher-order derivatives of jerk and
jounce In physics, the fourth, fifth and sixth derivatives of position are defined as derivatives of the position vector with respect to time – with the first, second, and third derivatives being velocity, acceleration, and jerk, respectively. Unlike ...
in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
. For example, just as the 4th-order moment (kurtosis) can be interpreted as "relative importance of tails as compared to shoulders in contribution to dispersion" (for a given amount of dispersion, higher kurtosis corresponds to thicker tails, while lower kurtosis corresponds to broader shoulders), the 5th-order moment can be interpreted as measuring "relative importance of tails as compared to center (
mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...
and shoulders) in contribution to skewness" (for a given amount of skewness, higher 5th moment corresponds to higher skewness in the tail portions and little skewness of mode, while lower 5th moment corresponds to more skewness in shoulders).


Mixed moments

Mixed moments are moments involving multiple variables. The value E ^k/math> is called the moment of order k (moments are also defined for non-integral k). The moments of the joint distribution of random variables X_1 ... X_n are defined similarly. For any integers k_i\geq0, the mathematical expectation E \cdots^/math> is called a mixed moment of order k (where k=k_1+...+k_n), and E
X_1-E[X_1 X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
^\cdots(X_n-E[X_n])^] is called a central mixed moment of order k. The mixed moment E
X_1-E[X_1 X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
(X_2-E[X_2])] is called the covariance and is one of the basic characteristics of dependency between random variables. Some examples are covariance, coskewness and cokurtosis. While there is a unique covariance, there are multiple co-skewnesses and co-kurtoses.


Properties of moments


Transformation of center

Since (x - b)^n = (x - a + a - b)^n = \sum_^n (x - a)^i(a - b)^ where \binom is the
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
, it follows that the moments about ''b'' can be calculated from the moments about ''a'' by: E\left x - b)^n\right= \sum_^n E\left x - a)^i\righta - b)^.


The moment of a convolution of function

The moment of a convolution h(t) = (f * g)(t) = \int_^\infty f(\tau) g(t - \tau) \, d\tau reads \mu_n = \sum_^ \mu_i \mu_ /math> where \mu_n ,\cdot\,/math> denotes the n-th moment of the function given in the brackets. This identity follows by the convolution theorem for moment generating function and applying the chain rule for differentiating a product.


Cumulants

The first raw moment and the second and third ''unnormalized central'' moments are additive in the sense that if ''X'' and ''Y'' are
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 * Independe ...
random variables then \begin m_1(X + Y) &= m_1(X) + m_1(Y) \\ \operatorname(X + Y) &= \operatorname(X) + \operatorname(Y) \\ \mu_3(X + Y) &= \mu_3(X) + \mu_3(Y) \end (These can also hold for variables that satisfy weaker conditions than independence. The first always holds; if the second holds, the variables are called
uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, ther ...
). In fact, these are the first three cumulants and all cumulants share this additivity property.


Sample moments

For all ''k'', the -th raw moment of a population can be estimated using the -th raw sample moment \frac\sum_^ X^k_i applied to a sample drawn from the population. It can be shown that the expected value of the raw sample moment is equal to the -th raw moment of the population, if that moment exists, for any sample size . It is thus an unbiased estimator. This contrasts with the situation for central moments, whose computation uses up a degree of freedom by using the sample mean. So for example an unbiased estimate of the population variance (the second central moment) is given by \frac\sum_^n \left(X_i - \bar\right)^2 in which the previous denominator has been replaced by the degrees of freedom , and in which \bar X refers to the sample mean. This estimate of the population moment is greater than the unadjusted observed sample moment by a factor of \tfrac, and it is referred to as the "adjusted sample variance" or sometimes simply the "sample variance".


Problem of moments

Problems of determining a probability distribution from its sequence of moments are called ''problem of moments''. Such problems were first discussed by P.L. Chebyshev (1874)Feller, W. (1957-1971). ''An introduction to probability theory and its applications.'' New York: John Wiley & Sons. 419 p. in connection with research on limit theorems. In order that the probability distribution of a random variable X be uniquely defined by its moments \alpha_k = EX^k it is sufficient, for example, that Carleman's condition be satisfied: \sum_^\infin\frac = \infin A similar result even holds for moments of random vectors. The ''problem of moments'' seeks characterizations of sequences that are sequences of moments of some function ''f,'' all moments \alpha_k(n) of which are finite, and for each integer k\geq1 let \alpha_k(n)\rightarrow \alpha_k ,n\rightarrow \infin, where \alpha_k is finite. Then there is a sequence ' that weakly converges to a distribution function \mu having \alpha_k as its moments. If the moments determine \mu uniquely, then the sequence ' weakly converges to \mu.


Partial moments

Partial moments are sometimes referred to as "one-sided moments." The -th order lower and upper partial moments with respect to a reference point ''r'' may be expressed as \mu_n^- (r) = \int_^r (r - x)^n\,f(x)\,\mathrmx, \mu_n^+ (r) = \int_r^\infty (x - r)^n\,f(x)\,\mathrmx. If the integral function do not converge, the partial moment does not exist. Partial moments are normalized by being raised to the power 1/''n''. The
upside potential ratio The upside-potential ratio is a measure of a return of an investment asset relative to the minimal acceptable return. The measurement allows a firm or individual to choose investments which have had relatively good upside performance, per unit of d ...
may be expressed as a ratio of a first-order upper partial moment to a normalized second-order lower partial moment. They have been used in the definition of some financial metrics, such as the
Sortino ratio The Sortino ratio measures the risk-adjusted return of an investment asset, portfolio, or strategy. It is a modification of the Sharpe ratio but penalizes only those returns falling below a user-specified target or required rate of return, while ...
, as they focus purely on upside or downside.


Central moments in metric spaces

Let be a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, and let B(''M'') be the Borel -algebra on ''M'', the -algebra generated by the ''d''- open subsets of ''M''. (For technical reasons, it is also convenient to assume that ''M'' is a separable space with respect to the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
''d''.) Let . The -th central moment of a measure on the measurable space (''M'', B(''M'')) about a given point is defined to be \int_ d\left(x, x_0\right)^p \, \mathrm \mu (x). ''μ'' is said to have finite -th central moment if the -th central moment of about ''x''0 is finite for some . This terminology for measures carries over to random variables in the usual way: if is a probability space and is a random variable, then the -th central moment of ''X'' about is defined to be \int_M d \left(x, x_0\right)^p \, \mathrm \left( X_* \left(\mathbf\right) \right) (x) = \int_\Omega d \left(X(\omega), x_0\right)^p \, \mathrm \mathbf (\omega) = \operatorname (X, x_0)^p and ''X'' has finite -th central moment if the -th central moment of ''X'' about ''x''0 is finite for some .


See also

* Energy (signal processing) * Factorial moment *
Generalised mean In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). D ...
* Image moment * L-moment * Method of moments (probability theory) *
Method of moments (statistics) In statistics, the method of moments is a method of estimation of population parameters. The same principle is used to derive higher moments like skewness and kurtosis. It starts by expressing the population moments (i.e., the expected value ...
* Moment-generating function *
Moment measure In probability and statistics, a moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as ...
*
Second moment method In mathematics, the second moment method is a technique used in probability theory and analysis to show that a random variable has positive probability of being positive. More generally, the "moment method" consists of bounding the probability th ...
* Standardised moment *
Stieltjes moment problem In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (''m''0, ''m''1, ''m''2, ...) to be of the form :m_n = \int_0^\infty x^n\,d\mu(x) for some measure ''&m ...
* Taylor expansions for the moments of functions of random variables


References

* Text was copied fro
Moment
at the Encyclopedia of Mathematics, which is released under
Creative Commons Attribution-Share Alike 3.0 (Unported) (CC-BY-SA 3.0) license
and the
GNU Free Documentation License The GNU Free Documentation License (GNU FDL or simply GFDL) is a copyleft license for free documentation, designed by the Free Software Foundation (FSF) for the GNU Project. It is similar to the GNU General Public License, giving readers the r ...
.


Further reading

* *


External links

*
Moments at Mathworld
{{DEFAULTSORT:Moment (Mathematics) Moment (physics)