A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a "weight" than others. They occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be constructed in both discrete and continuous settings. Statistics is a type of data analysis which practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In mathematics, a measure is a function that assigns a number, e. ...

Discrete weights

In the discrete setting, a weight function is a positive function defined on a discrete set A, which is typically finite or countable. The weight function w(a): = 1 corresponds to the unweighted situation in which all elements have equal weight. One can then use apply this weight to various concepts as follows: The word discrete comes from the Latin word discretus which means separate. ... In mathematics, a set can be thought of as any well-defined collection of distinct things considered as a whole. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics the term countable set is used to describe the size of a set, e. ...

1. If is a real-valued function, then the unweighted sum of f on A is ; but if one introduces a weight function , then one can also form the weighted sum . One common application of weighted sums arises in numerical integration.
2. If B is a finite subset of A, one can replace the unweighted cardinality |B| of B by the weighted cardinality
3. If A is a finite non-empty set, one can replace the unweighted mean or average by the weighted mean or weighted average . In this case only the relative weights are relevant.
   
   Weighted means are commonly used in statistics to compensate for the presence of bias.

The terminology weight function arises from mechanics: if one has a collection of n objects on a lever, with weights (where weight is now interpreted in the physical sense) and locations , then the lever will be in balance if the fulcrum of the lever is at the center of mass , which is also the weighted average of the positions x_i. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... Addition is one of the basic operations of arithmetic. ... In numerical analysis, the term numerical integration is used to describe a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe numerical algorithms for solving differential equations. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. ... In mathematics, there are numerous methods for calculating the average or central tendency of a list of n numbers. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ... Statistics is a type of data analysis which practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... It has been suggested that this article or section be merged with bias (disambiguation). ... Mechanics refers to: a craft relating to machinery (from the Latin mechanicus, from the Greek mechanikos, meaning one skilled in machines), or a range of disciplines in science and engineering. ... The principle of the lever tells us that the above is in static equilibrium, with all forces balancing, if F1D1 = F2D2. ... Weight is the interaction of matter with a gravitational field. ... Fulcrum is the NATO reporting name of the MiG-29, a Soviet fighter aircraft. ... The center of mass or center of inertia of an object is a point at which the objects mass can be assumed, for many purposes, to be concentrated. ...

Continuous weights

In the continuous setting, a weight is a positive measure such as w(x) dx on some domain Ω, which is typically a subset of an Euclidean space , for instance Ω could be an interval [a,b]. Here dx is Lebesgue measure and is a non-negative measurable function. In this context, the weight function w(x) is sometimes referred to as a density. In mathematics, a measure is a function that assigns a number, e. ... In mathematics, the domain of a function is the set of all input values to the function. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ...

1. If is a real-valued function, then the unweighted integral can be generalized to the weighted integral . Note that one may need to require f to be absolutely integrable with respect to the weight w(x) dx in order for this integral to be finite.
2. If E is a subset of Ω, then the volume vol(E) of E can be generalized to the weighted volume .
3. If Ω has finite non-zero weighted volume, then we can replace the unweighted average by the weighted average
4. If and are two functions, one can generalize the unweighted inner product to a weighted inner product . See the entry on Orthogonality for more details.