Editing Weight function (section)

== Discrete weights ==
=== General definition ===
In the discrete setting, a weight function <math>w \colon A \to \R^+</math> is a positive function defined on a [[discrete mathematics|discrete]] [[Set (mathematics)|set]] <math>A</math>, which is typically [[finite set|finite]] or [[countable]].  The weight function <math>w(a) := 1</math> corresponds to the ''unweighted'' situation in which all elements have equal weight.  One can then apply this weight to various concepts.

If the function <math>f\colon A \to \R</math> is a [[real number|real]]-valued [[mathematical function|function]], then the ''unweighted [[summation|sum]] of <math>f</math> on <math>A</math>'' is defined as

:<math>\sum_{a \in A} f(a);</math>

but given a ''weight function'' <math>w\colon A \to \R^+</math>, the '''weighted sum''' or [[conical combination]] is defined as

:<math>\sum_{a \in A} f(a) w(a).</math>

One common application of weighted sums arises in [[numerical integration]].

If ''B'' is a [[finite set|finite]] subset of ''A'', one can replace the unweighted [[cardinality]] |''B''| of ''B'' by the ''weighted cardinality'' 

:<math>\sum_{a \in B} w(a).</math>

If ''A'' is a [[finite set|finite]] non-empty set, one can replace the unweighted [[mean]] or [[average]] 

:<math>\frac{1}{|A|} \sum_{a \in A} f(a)</math>

by the [[weighted mean]] or [[weighted average]] 

:<math> \frac{\sum_{a \in A} f(a) w(a)}{\sum_{a \in A} w(a)}.</math>

In this case only the ''relative'' weights are relevant.

=== Statistics ===
Weighted means are commonly used in [[statistics]] to compensate for the presence of [[Bias_(statistics)|bias]].  For a quantity <math>f</math> measured multiple independent times <math>f_i</math> with [[variance]] <math>\sigma^2_i</math>, the best estimate of the signal is obtained  by averaging all the measurements with weight {{nowrap|<math display="inline">w_i = 1 / {\sigma_i^2}</math>,}} and the resulting variance is smaller than each of the independent measurements {{nowrap|<math display="inline"> \sigma^2 = 1 / \sum_i w_i</math>.}} The [[maximum likelihood]] method weights the difference between fit and data using the same weights {{nowrap|<math>w_i</math>.}}

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 [[probability|probabilities]]. More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable.

In [[linear regression|regressions]] in which the [[dependent variable]] is assumed to be affected by both current and lagged (past) values of the [[independent variable]], a [[distributed lag]] function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a [[moving average model]] specifies an evolving variable as a weighted average of current and various lagged values of a random variable.

=== Mechanics ===
The terminology ''weight function'' arises from [[mechanics]]: if one has a collection of <math>n</math> objects on a [[lever]], with weights <math>w_1, \ldots, w_n</math> (where [[weight]] is now interpreted in the physical sense) and locations {{nowrap|<math>\boldsymbol{x}_1,\dotsc,\boldsymbol{x}_n</math>,}} then the lever will be in balance if the [[Lever|fulcrum]] of the lever is at the [[center of mass]] 

:<math>\frac{\sum_{i=1}^n w_i \boldsymbol{x}_i}{\sum_{i=1}^n w_i},</math>

which is also the weighted average of the positions {{nowrap|<math>\boldsymbol{x}_i</math>.}}