Weight function (가중 함수), Weighted Average (가중 평균)

출처 : Wikipedia

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.

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 measurablefunction. In this context, the weight function w(x) is sometimes referred to as a density.

General definition

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.

Weighted volume

If E is a subset of Ω, then the volume vol(E) of E can be generalized to the weighted volume

.

Weighted average

If Ω has finite non-zero weighted volume, then we can replace the unweighted average $ \frac{1}{\text{vol} \left( \Omega \right)}\int_{\Omega}^{}f\left( x \right)dx $ by the weighted average $\frac{\int_{\Omega}^{}f\left( x \right)w\left( x \right)dx}{\int_{\Omega}^{}w\left( x \right)dx}$

Inner product

If $f:\Omega\to \mathbb{R} $ and $g:\Omega\to \mathbb{R} $ are two functions, one can generalize the unweighted inner product $\left\langle f,g \right\rangle :=\int_{\Omega}^{}f\left( x \right)g\left( x \right)dx$ to a weighted inner product $\left\langle f,g \right\rangle :=\int_{\Omega}^{}f\left( x \right)g\left( x \right)w\left( x \right)dx$