Probability Distributions and Computational Complexity
The complexity of probability distributions 
comes up particularly in the study of distributional problems like 
DistNP in Levin's theory of average case complexity theory.

A distribution is P-computable if
its cumulative density function can be evaluated in polynomial time.

A distribution is P-samplable if 
we can sample from them  in polynomial time.

If a distribution is P-computable then it is P-sampable.
The reverse is not true if certain one-way functions exist.

You can extend the definitions to other complexity classes.

Oded Goldreich has a nice introductory notes on the topic 
that you may want to check.