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