See also: Moment (physics)

The concept of moment in mathematics evolved from the concept of moment in physics. In Physics, the moment of force (often just moment, though there are other quantities of that name such as Moment of inertia) is a Pseudovector Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Physics, the moment of force (often just moment, though there are other quantities of that name such as Moment of inertia) is a Pseudovector Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. The nth moment of a real-valued function f(x) of a real variable about a value c is

It is possible to define moments for random variables in a more general fashion than moments for real values. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way See Moments in metric spaces.

The moments about zero are usually referred to simply as the moments of a function. Usually, except in the special context of the problem of moments, the function will be a probability density function. In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability The n^th moment (about zero) of a probability density function f(x) is the expected value of Xⁿ. The moments about its mean μ are called central moments; these describe the shape of the function, independently of translation. In Probability theory and Statistics, the k th moment about the Mean (or k th central moment In Euclidean geometry, a translation is moving every point a constant distance in a specified direction

If f is a probability density function, then the value integral above is called the nth moment of the probability distribution. In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable More generally, if F is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the nth moment of the probability distribution is given by the Riemann-Stieltjes integral

where X is a random variable that has this distribution and E the expectation operator. In Probability theory and Statistics, the cumulative distribution function (CDF, also probability distribution function or just distribution function In Mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way

When

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

Significance of the moments

Increasing each of the first four moments in turn whilst keeping the others constant, for a discrete uniform distribution with four values. WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one ---> In Probability

The first moment about zero, if it exists, is the expectation of X, i. e. the mean of the probability distribution of X, designated μ. In higher orders, the central moments are more interesting than the moments about zero.

The nth central moment of the probability distribution of a random variable X is

The first central moment is thus 0. In Probability theory and Statistics, the k th moment about the Mean (or k th central moment

Variance

The second central moment is the variance, the positive square root of which is the standard deviation, σ. In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values

Normalized moments

The normalised nth central moment or standardized moment is the nth central moment divided by σⁿ; the nth moment of t = (x − μ)/σ. In Probability theory and Statistics, the k th standardized moment of a Probability distribution is \frac{\mu_k}{\sigma^k}\! where These normalised central moments are dimensionless quantities, which represent the distribution independently of any linear change of scale. In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units

Skewness

The third central moment is a 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. In Probability theory and Statistics, skewness is a measure of the asymmetry of the Probability distribution of a real -valued A distribution that is skewed to the left (the tail of the distribution is heavier on the left) will have a negative skewness. A distribution that is skewed to the right (the tail of the distribution is heavier on the right), will have a positive skewness.

For distributions that are not too different from the normal (or "Gaussian") distribution, the median will be somewhere near μ − γσ/6; the mode about μ − γσ/2. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution In Statistics, the mode is the value that occurs the most frequently in a Data set or a Probability distribution.

Kurtosis

The fourth central moment is a measure of whether the distribution is tall and skinny or short and squat, 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 positive (except for a point distribution); the fourth central moment of a normal distribution is 3σ⁴. WikipediaWikiProject Probability#Standards for a discussionof standards used for probability distribution articles such as this one

The kurtosis κ is defined to be the normalized fourth central moment minus 3. In Probability theory and Statistics, kurtosis (from the Greek word κυρτός kyrtos or kurtos, meaning bulging is a measure of the "peakedness" (Equivalently, as in the next section, it is the fourth cumulant divided by the square of the variance. In Probability theory and Statistics, a Random variable X has an Expected value μ = E ( X) and a Variance σ2 ) Some authorities do not subtract three, but it is usually more convenient to have the normal distribution at the origin of coordinates. If a distribution has a peak at the mean and long tails, the fourth moment will be high and the kurtosis positive; and conversely; thus, bounded distributions tend to have low kurtosis.

The kurtosis can be positive without limit, but κ must be greater than or equal to γ² − 2; equality only holds for binary distributions. In Probability theory and Statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete Probability For unbounded skew distributions not too far from normal, κ tends to be somewhere in the area of γ² and 2γ².

The inequality can be proven by considering

where T = (X − μ)/σ. This is the expectation of a square, so it is non-negative whatever a is; on the other hand, it's also a quadratic equation in a. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. Its discriminant must be non-positive, which gives the required relationship. In Algebra, the discriminant of a Polynomial with real or complex Coefficients is a certain expression in the coefficients of the

Cumulants

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

and

and

(These can also hold for variables that satisfy weaker conditions than independence. In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other The first always holds; if the second holds, the variables are called uncorrelated). In Probability theory and Statistics, correlation, (often measured as a correlation coefficient) indicates the strength and direction of a linear

This is true because these moments are the first three cumulants; the fourth cumulant is the kurtosis times σ⁴. In Probability theory and Statistics, a Random variable X has an Expected value μ = E ( X) and a Variance σ2 In Probability theory and Statistics, a Random variable X has an Expected value μ = E ( X) and a Variance σ2

All the cumulants are polynomials in the moments; so are the factorial moments. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Probability theory, the n th factorial moment of a Probability distribution, also called the n th factorial moment of any Random variable The central moments are polynomials in the moments about zero, and conversely.

Sample moments

The moments of a population can be estimated using the sample k-th moment

applied to a sample X₁,X₂,. . . , X_n drawn from the population.

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

Problem of moments

The problem of moments seeks characterizations of sequences { μ′_n : n = 1, 2, 3, . In Mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure &mu to the sequences of moments . . } that are sequences of moments of some function f.

Partial moments

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

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. The upside potential ratio is a measure of a return of an investment asset relative to the Minimal acceptable return.

Moments in metric spaces

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. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, the Borel algebra (or Borel &sigma-algebra) on a Topological space X is a &sigma-algebra of Subsets of In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in (For technical reasons, it is also convenient to assume that M is a separable space with respect to the metric d. In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. ) Let 1 ≤ p ≤ +∞.

The p^th moment of a measure μ on the measurable space (M, B(M)) about a given point x₀ in M is defined to be

μ is said to have finite p^th moment if the p^th moment of μ about x₀ is finite for some x₀ ∈ M. In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty

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 moment of X about x₀ ∈ M is defined to be

and X has finite p^th moment if the p^th moment of X about x₀ is finite for some x₀ ∈ M. A probability space, in Probability theory, is the conventional Mathematical model of Randomness.

See also

• Binomial distribution
• Cumulant
• Hamburger moment problem
• Hausdorff moment problem
• Method of moments
• Moment about the mean
• Moment-generating function
• Normal distribution
• Standardized moment
• Stieltjes moment problem
• Taylor expansions for the moments of functions of random variables

External links

• Mathworld Website