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Wikipedia, A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more "weight" or influence on the result than other elements in the same set. They occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus" and "metacalculus". Discrete weightsGeneral definitionIn 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 apply this weight to various concepts. If the function is a realvalued function, then the unweighted sum of f on A is defined as
but given a weight function , the weighted sum is defined as
One common application of weighted sums arises in numerical integration. If B is a finite subset of A, one can replace the unweighted cardinality B of B by the weighted cardinality If A is a finite nonempty 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. StatisticsWeighted means are commonly used in statistics to compensate for the presence of bias. For a quantity f measured multiple independent times f_{i} with variance , the best estimate of the signal is obtained by averaging all the measurements with weight , and the resulting variance is smaller than each of the independent measurements . The Maximum likelihood method weights the difference between fit and data using the same weights w_{i} . MechanicsThe 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 . Continuous weightsIn 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 nonnegative measurable function. In this context, the weight function w(x) is sometimes referred to as a density. General definitionIf is a realvalued 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. Weighted volumeIf E is a subset of Ω, then the volume vol(E) of E can be generalized to the weighted volume
Weighted averageIf Ω has finite nonzero weighted volume, then we can replace the unweighted average by the weighted average Inner productIf and are two functions, one can generalize the unweighted inner product to a weighted inner product See the entry on Orthogonality for more details. See also
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Published  July 2009


