Bayesian credibility under a bivariate prior on the frequency and the severity of claims

•In credibility theory, we consider a model where the risk profiles of claim frequency and claim severity are dependent.•A bivariate conjugate prior is proposed for the frequency and the severity that belong to the exponential family.•Our proposed bivariate conjugate prior contains various well known distributions as special cases such as mixed Erlang.•Bivariate posterior, Bayesian premium, and predictive distributions involving claim number and aggregate loss are derived.•Numerical examples are provided to illustrate the effect of dependence, including the use of a real data set.