# Rescaling densities

#### Joppy

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
What does "some fixed domain" mean and why is it desirable?

#### Joppy

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
Can't you just scale everything by the reciprocal of the integral of the shifted density over the desired domain?