Formal way of defining a weighted average

Question

I am looking for a nice and formal way to define moving averages :

$y = \sum_{i=0}^N w_ix_i=<\mathbf{w},\mathbf{x}>$

and especially the fact that weights have to equal one, i.e. $\sum_{i=0}^N w_i=1$.

This property seems quite obvious to me, as you somehow want the output to have the same "level" as the input values. However, I don't know how to describe it a more mathematical way ..

Can somebody help me out ? Thank you

Answer 1

If $y=\sum_{i=1}^{n}w_ix_i$ is a weighted average of the $x_i$ with $\sum_{i=1}^{n}w_i=1$ and each of the $w_i\geq0$, then $y$ is a convex combination of the $x_i$.