In mathematics, the **moments** of a function are quantitative measures related to the shape of the function's graph. If the function represents mass, then the first moment is the center of the mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, 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 in physics.

- Significance of the moments
- Mean
- Variance
- Standardized moments
- Higher moments
- Mixed moments
- Properties of moments
- Transformation of center
- The moment of a convolution of function
- Cumulants
- Sample moments
- Problem of moments
- Partial moments
- Central moments in metric spaces
- See also
- References
- Further reading
- External links

For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from 0 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 became the first person to think systematically in terms of the moments of random variables.^{ [1] }

The *n*-th raw moment (i.e., moment about zero) of a distribution is defined by^{ [2] }

where

The n-th moment of a real-valued continuous function *f*(*x*) of a real variable about a value *c* is the integral

It is possible to define moments for random variables 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 nth inverse moment about zero is and the n-th logarithmic moment about zero is

The n-th moment about zero of a probability density function *f*(*x*) is the expected value of X^{n} and is called a *raw moment* or *crude moment*.^{ [3] } The moments about its mean μ are called *central* moments; these describe the shape of the function, independently of translation.

If *f* is a probability density function, then the value of the integral above is called the n-th moment of the probability distribution. More generally, if *F* is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the n-th moment of the probability distribution is given by the Riemann–Stieltjes integral

where *X* is a random variable that has this cumulative distribution *F*, and E is the expectation operator or mean. When

the moment is said not to exist. If the n-th moment about any point exists, so does the (*n* − 1)-th moment (and thus, all lower-order moments) about every point.

The zeroth moment of any probability density function is 1, since the area under any probability density function must be equal to one.

Moment ordinal | Moment | Cumulant | |||
---|---|---|---|---|---|

Raw | Central | Standardized | Raw | Normalized | |

1 | Mean | 0 | 0 | Mean | N/A |

2 | – | Variance | 1 | Variance | 1 |

3 | – | – | Skewness | – | Skewness |

4 | – | – | (Non-excess or historical) kurtosis | – | Excess kurtosis |

5 | – | – | Hyperskewness | – | – |

6 | – | – | Hypertailedness | – | – |

7+ | – | – | – | – | – |

The first raw moment is the mean, usually denoted

The second central moment is the variance. The positive square root of the variance is the standard deviation

The *normalised*n-th central moment or standardised moment is the n-th central moment divided by σ^{n}; the normalised n-th central moment of the random variable X is

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.^{ [4] }^{ [5] }

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 γ. 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, the median will be somewhere near *μ* − *γσ*/6; the mode about *μ* − *γσ*/2.

The fourth central moment is a measure of the heaviness of the tail of the distribution, compared to the normal distribution of the same variance. 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 3*σ*^{4}.

The kurtosis κ 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.)^{ [6] }^{ [7] } 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 κ must be greater than or equal to *γ*^{2} + 1; equality only holds for binary distributions. For unbounded skew distributions not too far from normal, κ tends to be somewhere in the area of *γ*^{2} and 2*γ*^{2}.

The inequality can be proven by considering

where *T* = (*X* − *μ*)/*σ*. This is the expectation of a square, so it is non-negative for all *a*; however it is also a quadratic polynomial in *a*. Its discriminant must be non-positive, which gives the required relationship.

**High-order moments** are moments beyond 4th-order moments.

As with variance, skewness, and kurtosis, these are higher-order statistics, 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 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. 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 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** are moments involving multiple variables.

The value is called the moment of order (moments are also defined for non-integral ). The moments of the joint distribution of random variables are defined similarly. For any integers , the mathematical expectation is called a mixed moment of order , and is called a central mixed moment of order . The mixed moment 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.

Since

where is the binomial coefficient, it follows that the moments about *b* can be calculated from the moments about *a* by:

The moment of a convolution reads

where denotes the -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.

The first raw moment and the second and third *unnormalized central* moments are additive in the sense that if *X* and *Y* are independent random variables then

(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 fact, these are the first three cumulants and all cumulants share this additivity property.

For all *k*, the k-th raw moment of a population can be estimated using the k-th raw sample moment

applied to a sample *X*_{1}, ..., *X _{n}* drawn from the population.

It can be shown that the expected value of the raw sample moment is equal to the k-th raw moment of the population, if that moment exists, for any sample size n. 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

in which the previous denominator n has been replaced by the degrees of freedom *n* − 1, and in which refers to the sample mean. This estimate of the population moment is greater than the unadjusted observed sample moment by a factor of and it is referred to as the "adjusted sample variance" or sometimes simply the "sample variance".

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)^{ [8] } in connection with research on limit theorems. In order that the probability distribution of a random variable be uniquely defined by its moments it is sufficient, for example, that Carleman's condition be satisfied:

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 of which are finite, and for each integer let

where is finite. Then there is a sequence that weakly converges to a distribution function having as its moments. If the moments determine uniquely, then the sequence weakly converges to .

Partial moments are sometimes referred to as "one-sided moments." The n-th order lower and upper partial moments with respect to a reference point *r* may be expressed as

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 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, as they focus purely on upside or downside.

Let (*M*, *d*) be a metric space, and let B(*M*) be the Borel σ-algebra on *M*, the σ-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 *d*.) Let 1 ≤ *p* ≤ ∞.

The **p-th central moment** of a measure μ on the measurable space (*M*, B(*M*)) about a given point *x*_{0} ∈ *M* is defined to be

*μ* is said to have **finite p-th central moment** if the p-th central moment of μ about *x*_{0} is finite for some *x*_{0} ∈ *M*.

This terminology for measures carries over to random variables in the usual way: if (Ω, Σ, **P**) is a probability space and *X* : Ω → *M* is a random variable, then the **p-th central moment** of *X* about *x*_{0} ∈ *M* is defined to be

and *X* has **finite p-th central moment** if the p-th central moment of *X* about *x*_{0} is finite for some *x*_{0} ∈ *M*.

- Energy (signal processing)
- Factorial moment
- Generalised mean
- Image moment
- L-moment
- Method of moments (probability theory)
- Method of moments (statistics)
- Moment-generating function
- Moment measure
- Second moment method
- Standardised moment
- Stieltjes moment problem
- Taylor expansions for the moments of functions of random variables

The **Cauchy distribution**, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the **Lorentz distribution**, **Cauchy–Lorentz distribution**, **Lorentz(ian) function**, or **Breit–Wigner distribution**. The Cauchy distribution is the distribution of the x-intercept of a ray issuing from with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.

In probability theory and statistics, **kurtosis** is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes the shape of a probability distribution and there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Different measures of kurtosis may have different interpretations.

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 variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location.

In probability theory and statistics, **skewness** is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.

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 numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or .

In probability theory, the **central limit theorem** (**CLT**) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.

In probability theory and statistics, the **multivariate normal distribution**, **multivariate Gaussian distribution**, or **joint normal distribution** is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be *k*-variate normally distributed if every linear combination of its *k* components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.

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 compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions.

In probability theory and statistics, the **Bernoulli distribution**, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability . Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/yes/true/one with probability *p* and failure/no/false/zero with probability *q*. It can be used to represent a coin toss where 1 and 0 would represent "heads" and "tails", respectively, and *p* would be the probability of the coin landing on heads. In particular, unfair coins would have

In probability theory and statistics, a **standardized moment** of a probability distribution is a moment that is normalized. The normalization is typically a division by an expression of the standard deviation which renders the moment scale invariant. This has the advantage that such normalized moments differ only in other properties than variability, facilitating e.g. comparison of shape of different probability distributions.

In probability theory and statistics, the **beta distribution** is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by *alpha* (*α*) and *beta* (*β*), that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.

In probability theory and statistics, the **cumulants**κ_{n} 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.

In probability theory and statistics, the **logistic distribution** is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails. The logistic distribution is a special case of the Tukey lambda distribution.

In probability and statistics, a **mixture distribution** is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors, in which case the mixture distribution is a multivariate distribution.

The **Skellam distribution** is the discrete probability distribution of the difference of two statistically independent random variables and each Poisson-distributed with respective expected values and . It is useful in describing the statistics of the difference of two images with simple photon noise, as well as describing the point spread distribution in sports where all scored points are equal, such as baseball, hockey and soccer.

In probability theory and statistics, the **characteristic function** of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.

In mathematics, **probabilistic metric spaces** are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers **R**_{ ≥ 0}, but in distribution functions.

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 that a random variable fluctuates far from its mean, by using its moments.

In probability theory and statistics, the **Hermite distribution**, named after Charles Hermite, is a discrete probability distribution used to model *count data* with more than one parameter. This distribution is flexible in terms of its ability to allow a moderate over-dispersion in the data.

In probability theory, the **first-order second-moment (FOSM) method**, also referenced as **mean value first-order second-moment (MVFOSM) method**, is a probabilistic method to determine the stochastic moments of a function with random input variables. The name is based on the derivation, which uses a *first-order* Taylor series and the first and *second moments* of the input variables.

- Text was copied from Moment at the Encyclopedia of Mathematics, which is released under a Creative Commons Attribution-Share Alike 3.0 (Unported) (CC-BY-SA 3.0) license and the GNU Free Documentation License.

- ↑ George Mackey (July 1980). "HARMONIC ANALYSIS AS THE EXPLOITATION OF SYMMETRY - A HISTORICAL SURVEY".
*Bulletin of the American Mathematical Society*. New Series.**3**(1): 549. - ↑ Papoulis, A. (1984).
*Probability, Random Variables, and Stochastic Processes, 2nd ed*. New York: McGraw Hill. pp. 145–149. - ↑ "Archived copy". Archived from the original on 2009-05-28. Retrieved 2009-06-24.
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*Electrical Engineering: Know It All*. Newnes. p. 884. ISBN 978-0-08-094966-6. - ↑ Ha H. Nguyen; Ed Shwedyk (2009).
*A First Course in Digital Communications*. Cambridge University Press. p. 87. ISBN 978-0-521-87613-1. - ↑ Casella, George; Berger, Roger L. (2002).
*Statistical Inference*(2 ed.). Pacific Grove: Duxbury. ISBN 0-534-24312-6. - ↑ Ballanda, Kevin P.; MacGillivray, H. L. (1988). "Kurtosis: A Critical Review".
*The American Statistician*. American Statistical Association.**42**(2): 111–119. doi:10.2307/2684482. JSTOR 2684482. - ↑ Feller, W. (1957-1971).
*An introduction to probability theory and its applications.*New York: John Wiley & Sons. 419 p.

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*Probability Theory and Statistical Inference*. New York: Cambridge University Press. pp. 109–130. ISBN 0-521-42408-9. - Walker, Helen M. (1929).
*Studies in the history of statistical method, with special reference to certain educational problems*. Baltimore, Williams & Wilkins Co. p. 71.

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*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Moments at Mathworld

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