Editing Weight function (section)

=== 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.