Bioequivalence and Bioavailability Forum

Hi Helmut,

I am sorry that I am once again not able to tell what I am looking for . I am not talking about simulations and not about transformations either.

Two functions, generally, and the key word is really generally, when are they (or their results) to be considered independent?

We can think of f=mean(x) and g=median(x). I guess we can easily do a mental picture of plotting means versus medians, often seeing the strong relationship. Visually appealing. Independence?

OK then let us say f=range(x) and g=sine function of the range (x).
Or an F statistics with a variance in both f=numerator and g=denominator in an unbalanced anova.
Or Cmax and AUCt (which I guess are correlated and dependent(?), but the example is not great in my perspective since the two functions are not applied to a random sample but to a time series).
There is no end to the possible examples.

And so forth. Without debating to much about the specific cases, how do we generally approach it to define two (outcomes of) functions as being independent? Which mathematical/algebraic/statistical/whatever properties of functions render them mutually independent? When I understand it, I think or hope I will understand the nature of independence.
For inspiration: Are estimates of any two statistical moments independent? If yes, why? Is it only the first and second? Why? Is it generally so? Why? Etc. I am looking for the general clarity.