Compound Distribution --- Uniform Distribution with Normally Distributed Parameters

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Uniform Distribution whose parameters are distributed Normally.

Given,

$$W \sim U[X,Y]$$

$$X \sim N[\mu_{X},\sigma^2_{X}]$$

$$Y \sim N[\mu_{Y},\sigma^2_{Y}]$$

$$X,Y$$ can be assumed to be independent if it simplifies matters. But when $$X>Y$$ there is confusion since Uniform Distrbution is undefined when $$X>Y$$. Is there some sort of right truncated and left trucated extensions of the normal distribution that we can use for $$X,Y$$. Please suggest how we can resolve this issue as well.

To Determine,

$$f_{W}(w), F_{W}(w), E(W)$$

Related General Question

Starting with the above special case, it quickly becomes apparent there are many combinations possible. Hence was wondering if there were general techniques to derive the density, distribution function, expected value, higher moments, conditional expectations etc. of compound distributions and some source where certain combinations and results therein were given with detailed steps and complete proofs: Compound Distributions --- Basic Techniques and Key General Results from First Principles

Please note, This question is a cross posting on mathematics and statistics forum. Posting here to get a wider audience and possibly other applicable techniques. If this is deemed a redundant question, please let me know and I am happy to delete this to reduce the noise in the forum: https://math.stackexchange.com/questions/1610422/compound-distribution-uniform-distribution-with-normally-distributed-paramet

• $X$ and $Y$ won't be independent if you require $Y>X$. But it might perhaps make sense to reparameterize from $(X,Y)$ to $(U=Y-X,V=Y+X)$ and then you have a uniform with mean $V/2$ and spread $U$. Your constraint is then simply that $U>0$. Commented Jan 16, 2016 at 11:24
• This is a tough one to answer because you seem to be asking us how to ask the question! Could you please tell us what the actual statistical problem is that you are trying to solve?
– whuber
Commented Jan 16, 2016 at 18:23
• @whuber Thanks for your time to look into this. Please note, I have expanded this now to the general case from which this problem comes up and linked the questions. Commented Jan 17, 2016 at 3:24
• If you replace $X$ in the uniform with $min(X,Y)$ and similarly $Y$ with $max(X,Y)$. This eliminates the 0 probability measure you identified. Commented Aug 8, 2017 at 2:06
• There is an answer now at math SE Commented Feb 1, 2021 at 14:07