Feature request: Random variable (distribution object) as result of mathematical functions of random variables (distributions)

Simple example that should have been possible:
X ~ Uniform(a,b)
Y ~ Uniform(c,d)
X,Y are independent. Then:
Z = X + Y
has some trapezoidoid (🙃) distribution over (a+c,b+d).

I would like to be able to sample Z directly. I would also like to read probability densities, interval probabilities, etc. from the Z distribution object. If a,b,c,d are trainable variables, I would like to be able to optimize them for a Maximum Likelihood Objective obtained by querying the Z distribution object with batches of targets.

This is obviously a very simple example. Ideally, I would be able to define distributions by applying (almost) arbitrary functions to multiple distribution objects, and seamlessly use the resulting distribution objects as I would use the built-in distributions. Regardless of the number of nesting levels, everything would be trainable end-to-end, to the extent the constituent functions and distributions were trainable.

I must admit I do not know how _tfp.distribution_s are trained now (although I have magically succeeded both via samples and via log-density), so I understand there is a possibility that my wishes are unrealistic. I have done my best trying to find documentation, blogs and web questions on how to combine random variables like this, with no success. Mixtures and joint distributions combine distribution probabilities and the joint sequential distribution allows building a sort of chaining graph, but to my knowledge none of the distribution classes offer a way to apply math functions to random variables represented as distribution objects and treat the resulting random variable as a distribution object.

Here is a minimal example program based on the example, trying (naïvely?) to add the variables directly:

import tensorflow as tf
import tensorflow_probability as tfp

X = tfp.distributions.Uniform(2, 3)
Y = tfp.distributions.Uniform(-1,2)

Z = X+Y

And here is the console output:

Traceback (most recent call last):
  File ".\dist_sum_test.py", line 7, in <module>
    Z = X+Y
TypeError: unsupported operand type(s) for +: 'Uniform' and 'Uniform'

Start with the minimal TensorFlow Probability example in the issue and review how mixtures, joint distributions, and existing distribution APIs represent composition. Done would require a settled scope for transformed multi-distribution objects, including sampling, density or probability queries, and trainable parameters; the issue names no files or tests.