Defining and evaluating new continuous distributions based on Mathematica's distributions

Question

A number of functions operate with Mathematica's set of distributions e.g.

Through[{Mean, StandardDeviation, Variance}[UniformDistribution[{umin,umax}]]]

which returns the correct result

    {(umax + umin)/2, (umax - umin)/(2 Sqrt[3]), 1/12 (umax - umin)^2}

However, when I add two continuous distributions, thus creating a kind of annular distribution, it will not be evaluated.

Through[{Mean, StandardDeviation, Variance}[
  UniformDistribution[{umin, umax}] - 
   UniformDistribution[{\[Epsilon] umin, \[Epsilon] umax}]]]

I think I had a very good look at the Mathematica documentation but I simply could not find any indication of how to define "my own distribution".

Please note, that I know how to handle such functions, i.e., how to calculate the Mean ... The issue of this question is, are there ways in Mathematica, I have missed, that allow me to use Mathematica's functionality on distributions with my distributions I have defined using Mathematica's distributions.

Answer 1

The way to combine distributions would be to use TransformedDistribution. Your code could be written as follows:

Through[{Mean, StandardDeviation, Variance}[
  TransformedDistribution[
   x - y, 
   {
    x \[Distributed] UniformDistribution[{umin, umax}], 
    y \[Distributed] UniformDistribution[{\[Epsilon] umin, \[Epsilon] umax}]
   }
  ]]]

If you want to define distributions that work like the built-in distributions, have a look at this workshop.