Can you tell me some real life application of weighted Euclidean inner product?

Suppose that some physical experiment can produce any of \(n\) possible numerical values \(x_1\), \(x_2\), …, \(x_n\).

We perform \(m\) repetitions of the experiment and yield these values with various frequencies; that is, \(x_1\) occurs \(f_1\) times, \(x_2\) occurs \(f_2\) times, and so forth. So: \(f_1\) + \(f_2\) + … + \(f_n\) = \(m\).

Thus, the arithmetic average, or mean \(\overline{x}\), is:

\(\overline{x}=\frac{f_1x_1+f_2x_2+...+f_nx_n}{f_1+f_2+...+f_n}=\frac{1}{m}\left(f_1x_1+f_2x_2+...+f_nx_n\right)\)

If we let \(f=\left(f_1,f_2,...,f_n\right),\ x=\left(x_1,x_2,...,x_n\right),\ w_1=w_2=...=w_n=\frac{1}{m}\)

Then this equation can be expressed as the weighed inner product:

\(\overline{x}=\left\langle f,\ x\right\rangle =w_1f_1x_1+w_2f_2x_2+...+w_nf_nx_n\)