A family of random probabilities is defined and studied. This family contains the Dirichlet process as a special case, corresponding to an inner point in the appropriate parameter space. The extension makes it possible to have random means with larger or smaller skewnesses as compared to skewnesses under the Dirichlet prior, and also in other ways amounts to additional modelling flexibility. The usefulness of such random probabilities for nonparametric Bayesian statistics is discussed. The posterior distribution is complicated, but inference can nevertheless be carried out via simulation, and some exact formulae are derived for the case of random means. The class of nonparametric priors provides an instructive example where the speed with which the posterior forgets its prior with increasing data sample size depends on special aspects of the prior, which is a different situation from that of parametric inference.