With reference to the previous exercise which of the following properties are valid for weighted averages of a function on an interval . Denote the weighted average of with a weight function on by .
- Additive property: .
- Homogeneous property: for all .
- Monotone property: if on .
All of these properties are valid for weighted averages.
- Proof. We compute,
- Proof. We compute,
- Proof. Assume on , then since is nonnegative (definition of a weight function) we have for all . Next, by the monotone property of the integral we have
Then, since is nonnegative, is also nonnegative and we have
What if both functions have different weighted functions?
same question
Doesn’t change the result because you could rewrite them even if there are different weighted functions using the properties of integrals