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.

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.