# Weight function

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.

## Discrete weights

In the discrete setting, a weight function [itex]w: A \to {\Bbb R}^+[itex] is a positive function defined on a discrete set A, which is typically finite or countable. The weight function [itex]w(a) := 1[itex] corresponds to the unweighted situation in which all elements have equal weight. One can then use apply this weight to various concepts as follows:

1. If [itex]f: A \to {\Bbb R}[itex] is a real-valued function, then the unweighted sum of f on A is [itex]\sum_{a \in A} f(a)[itex]; but if one introduces a weight function [itex]w: A \to {\Bbb R}^+[itex], then one can also form the weighted sum [itex]\sum_{a \in A} f(a) w(a)[itex]. 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 [itex]\sum_{a \in B} w(a)[itex]
3. If A is a finite non-empty set, one can replace the unweighted mean or average [itex]\frac{1}{|A|} \sum_{a \in A} f(a)[itex] by the weighted mean or weighted average [itex] \frac{\sum_{a \in A} f(a) w(a)}{\sum_{a \in A} w(a)}.[itex] 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 [itex]w_1, \ldots, w_n[itex] (where weight is now interpreted in the physical sense) and locations [itex]x_1,\ldots,x_n[itex], then the lever will be in balance if the fulcrum of the lever is at the center of mass [itex]\frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}[itex], which is also the weighted average of the positions [itex]x_i[itex].

## Continuous weights

In the continuous setting, a weight is a positive measure such as w(x) dx on some domain [itex]\Omega[itex], which is typically a subset of an Euclidean space [itex]{\Bbb R}^n[itex], for instance [itex]\Omega[itex] could be an interval [itex][a,b][itex]. Here dx is Lebesgue measure and [itex]w: \Omega \to \R^+[itex] is a non-negative measurable function. In this context, the weight function w(x) is sometimes referred to as a density.

1. If [itex]f: \Omega \to {\Bbb R}[itex] is a real-valued function, then the unweighted integral [itex]\int_\Omega f(x)\ dx[itex] can be generalized to the weighted integral [itex]\int_\Omega f(x)\ w(x) dx[itex]. 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 [itex]\Omega[itex], then the volume vol(E) of E can be generalized to the weighted volume [itex] \int_E w(x)\ dx[itex].
3. If [itex]\Omega[itex] has finite non-zero weighted volume, then we can replace the unweighted average [itex]\frac{1}{vol(\Omega)} \int_\Omega f(x)\ dx[itex] by the weighted average [itex] \frac{\int_\Omega f(x)\ w(x) dx}{\int_\Omega w(x)\ dx}.[itex]
4. If [itex]f: \Omega \to {\Bbb R}[itex] and [itex]g: \Omega \to {\Bbb R}[itex] are two functions, one can generalize the unweighted inner product [itex]\langle f, g \rangle := \int_\Omega f(x) g(x)\ dx[itex] to a weighted inner product [itex]\langle f, g \rangle := \int_\Omega f(x) g(x)\ w(x) dx[itex]. See the entry on Orthogonality for more details.

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