I was thinking about a way to do a "weighted average" (that's what I call it, could be dead wrong) of a variable $x$ defined for a given range $x_1\leq x\leq x_N$, weighted by an always positive function $f(x)$.
I came up with the following formula:
$\frac{\sum_{i}^N x_i * f(x_i)}{\sum_i^N f(x_i)}$
where $i=1..N$ for any number $N$ of $x$ values. If $f(x)$ is any constant function, then the result is simply $(x_1+ \cdots + x_N)/N$.
The above equation seems rather familiar but I can't figure out what it might be, or if it is anything at all.
Is the above equation some known mathematical concept? Does it even make sense?
It's called the weighted average. :) Seriously: this is exactly how generalized weighted averages are defined. Nice work inventing it for yourself!