What does moment of functions tells you?

Question

What does the nth moment of a function tell us about that function?.

Like in the case of distributions, 1st moment tells about the expected value as we are taking the weighted average.

But what does the 2nd moment of the function tells us? On its own. i.e. apart from used in computing variance.

So, If for PDF, $E[X]$ is telling me avearge value of X. But what does $E[X^2]$ is saying about function? and in similar lines $E[X^3], E[X^4], ... E[X^n]$

Are you familiar with skewness and kurtosis. Of course, you are asking about noncentral and nonstandardized moments, so these two moments aren't exactly what you asked about. — Eric Towers, May 01 '21 at 04:53
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — José Carlos Santos, May 01 '21 at 06:13
@EricTowers I little familiar with skewness but not kurtosis. and Thanks for the link to moments. I don't know why I never read that general formula and always considered moments with c=0. Reading that general formula made things somewhat clear. — Amarjeet Kapoor, May 01 '21 at 07:06
@EricTowers Just one more question. So, If I consider n-th moment with $c = 0 $. Does this mean for every larger value $n$ we are just giving more importance to values that are far from the point around which we are finding a moment? or there is some different reason or significance of higher moments. — Amarjeet Kapoor, May 01 '21 at 07:12
I tend to think of the first two or three (central, standardized) moments describing the "middle" of a (unimodal) distribution and subsequent moments describing (odd $n$) asymmetry or (even $n$) heaviness of more extreme (far from $0$) events. If the distribution is not unimodal, moments do not have simple interpretations. — Eric Towers, May 01 '21 at 07:16
@JoséCarlosSantos Thanks, I will go through the links you shared. — Amarjeet Kapoor, May 01 '21 at 07:16

What does moment of functions tells you?

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