Calculating expectation for a custom distribution in Mathematica

Question

This question builds on the great answers I got on an earlier question:

Can one extend the functionality of PDF, CDF, FindDistributionParameters etc in Mathematica?<

Answer 1

The following page contains some tips on enabling custom distributions (i.e. written from scratch without TransformedDisribution or ProbabilityDistribution) for use in CopulaDistribution, RandomVariate, etc: https://mathematica.stackexchange.com/questions/20067/efficient-generation-of-random-variates-from-a-copula-distribution/26169#26169