Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Hi @Kerighan I agree that this doesn't (fully) answer your question, but I think I'll leave my answer here as it might be useful to other users... As for approximating moments: I suppose that you could approximate them using numerical integration for fixed $\alpha,\beta,\gamma,\delta$. I don't have a good idea if you want an answer for every $\alpha,\beta,\gamma,\delta$