what is the posterior predictive distribution of a negative binomial, with a conjugate prior beta distribution placed over the probability? $r$ is assumed to be fixed. Definition is as on Wikipedia, where $p$ is probability of success (not failure), $r$ is the number of failures till the experiment stops, and the support starts at 0.
what about the posterior predictive distribution of a pareto distribution, with a conjugate prior gamma distribution placed over the shape $k$? Minimum $x_m$ is assumed to be fixed.