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Concept
Weight function
Summary
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average. Weight functions 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 "meta-calculus".
In the discrete setting, a weight function is a positive function defined on a discrete set , which is typically finite or countable. The weight function 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 real-valued function, then the unweighted sum of on is defined as
but given a weight function , the weighted sum or conical combination 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 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. For a quantity measured multiple independent times 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 .
The expected value of a random variable is the weighted average of the possible values it might take on, with the weights being the respective probabilities.
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