Naming convention for "Functional Variance"?

[quasar987]

TL;DR Summary

naming convention for "functional variance"

For a continuous function f:[a,b]-->R there is a well-known notion of "functional mean"
$$
\mu[f] = \frac{1}{b-a}\int_a^bf(x)\,dx.
$$
I am wondering if there is a name for the corresponding notion of "functional variance", i.e. the quantity
$$
\frac{1}{b-a}\int_a^b(f(x)-\mu[f])^2\,dx.
$$
?? Thanks. Also, does this quantity play a role somewhere in the literature? I'm writing a paper that involves this thing but google is very useless at answering my question since it just spits out pages upon pages about the variance of a continuous random variable.

 

[pbuk]

For a continuous function f:[a,b]-->R there is a well-known notion of "functional mean"

Is there? I have never heard of the term "functional mean" and it is not used in the Wikipedia page you linked.

 

[pbuk]

However we define the mean of a (suitable) function ## f ## in the interval ## [a, b] ## as
$$
\mu[f] = \frac{1}{b-a}\int_a^bf(x)\,dx.
$$
And we can similarly define the variance of a (suitable) function ## f ## in the interval ## [a, b] ## as
$$
\frac{1}{b-a}\int_a^b(f(x)-\mu[f])^2\,dx.
$$

 

[quasar987]

Is there? I have never heard of the term "functional mean" and it is not used in the Wikipedia page you linked.

I'm not saying the term "functional mean" is well known, but that the concept of the mean of a function is well known. And I am curious if the corresponding concept of "functional variance" (aka variance of a function) appears somewhere in the literature.

 

[fresh_42]

I was curious and calculated @pbuk's formulas for ##f(x)=x^2## on ##[0,1]## and ##[-1.1]##. This resulted in ##\mu[f]=1/3\, , \,\sigma [x^2]=4/45\approx 9\%## in the first case and ##\mu[f]=0\, , \,\sigma [x^2]=0## in the second.

I can see why the mean is meaningful and a valuable quantity in proofs for approximations, but what should that ##9\%## be good for?

 

[pbuk]

I was curious and calculated @pbuk's formulas for ##f(x)=x^2## on ##[0,1]## and ##[-1.1]##. This resulted in ##\mu[f]=1/3\, , \,\sigma [x^2]=4/45\approx 9\%## in the first case and ##\mu[f]=0\, , \,\sigma [x^2]=0## in the second.

I think you should check your results for ##[-1,1]##.

I can see why the mean is meaningful and a valuable quantity in proofs for approximations, but what should that ##9\%## be good for?

Yes, I also wonder that. @quasar987?

 

[quasar987]

I was curious and calculated @pbuk's formulas for ##f(x)=x^2## on ##[0,1]## and ##[-1.1]##. This resulted in ##\mu[f]=1/3\, , \,\sigma [x^2]=4/45\approx 9\%## in the first case and ##\mu[f]=0\, , \,\sigma [x^2]=0## in the second.

I can see why the mean is meaningful and a valuable quantity in proofs for approximations, but what should that ##9\%## be good for?

Are you saying the area under the curve of the positive function $f(x)=x^2$ is 0?

I assign to the variance of a function $f$ the same meaning that Var[X] hold for a random variable X; it is a measure of dispersion: small variance means f tends to sheepishly sticks close to its mean µ[f] whereas large variance means f likes to venture away from its mean.

 

[fresh_42]

I think you should check your results for ##[-1,1]##.

I hate odd numbers of sign errors. Thanks.