Could anyone clarify as to why central moments have integration limits from 0 to infinity rather than minus infinity to positive infinity? I am asking with particular reference to moment analysis chromatographic peaks (a distribution of a compound's concentration as a function of time). The chromatographic peaks are slightly asymmetric Gaussians which appear at a certain time "t" (>0) as an output of a detector. This characteristic time is the first moment M1. The area under the peak is taken as the zeroth moment. Most authors use the limit from 0 to infinity, whereas Wikipedia definition shows minus infinity to plus infinity as limits for central moments. The choice of minus infinity to plus infinity makes more sense, what is the logic of using 0 to infinity? Please see the attached equation or what is the assumption behind this choice.
Thanks.
Assuming that your "chromatographic peaks" are represented by the $c(t)$ in your screenshot, then both formulas are equivalent. If you integrated with boundaries $\pm \infty$, the whole domain from $-\infty$ to $0$ would just give you zero, as your peaks do not show up until $t > 0$. In general, if you had probability density functions $c$ which can also take values for $t \leq 0$, you need the formula with boundaries $\pm \infty$, because to calculate the moments, you really need all the information that your density function contains. But in your case, the formula from your screenshot does the job.