The weighted average value of a function over a region provides the average value of a function over that region weighted by some factor so that it's skewed to "important" values. The equation can be thought of as a generalization of the mean with the pdf being replaced by a weighting function (with less restrictions).

Generally given a function , a weighting function and a region the weighted average value can be calculated as

For example for three dimensions the function with a weighting function over a region can be calculated as