Rescaling densities

Feb 2016
1,849
657
.
Suppose I have some collection of discrete densities $\rho_i(n)$ defined on domains $D_i$ of differing lengths (assume one dimension). Is there some standard trick for rescaling each densities' domain so that they all have some fixed domain $D$?

I guess you could modify the argument to each $\rho$ by multiplying it by the domain width you want and dividing by the domain width you've got but this isn't going to preserve your area... unless you normalised it again but that seems strange. Any ideas?
 

skipjack

Forum Staff
Dec 2006
21,379
2,410
What does "some fixed domain" mean and why is it desirable?
 
Feb 2016
1,849
657
.
Just a reference on which all $\rho_i$ are defined on. e.g. $D_i = [-i, i], i > 1$ and $D = [-1,1]$ or something. I often see this in literature where you might have some normal looking distribution on an interval $[-x, x]$ with non-zero centre but then the distribution is shifted to lie on say $[-1,1]$. But I don't see how exactly this is done in general since you would be required to renormalise (I guess it doesn't matter?).

The other part of the question is about how I would go about finding a transformation which relates the family of densities obtain one density from the other: $\rho_{i+1}(n) = f( \rho_i(n))$. Any standard examples?
 

romsek

Math Team
Sep 2015
2,872
1,607
USA
Can't you just scale everything by the reciprocal of the integral of the shifted density over the desired domain?