# Formal way of defining a weighted average

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

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I'm not sure if this helps you or not, but you're taking a convex combination of the $x_i$. – Max Oct 30 '12 at 19:39
Yes, I think that pretty much what I was looking for. :) – Johan Oct 30 '12 at 19:58
... and the weights $w_i$ are a vector in the unit simplex. – AKE Nov 28 '12 at 23:28

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 $x_i$.