The most commonly used regression model in general insurance pricing is the compound Poisson model with gamma claim sizes, which belongs to the family of Tweedie's distributions. 

There are two different parametrizations for this model: the Poisson-gamma parametrization and Tweedie's compound Poisson parametrization. Insurance industry typically prefers the Poisson-gamma parametrization. We review the family of Tweedie's distributions and we derive both parametrizations of the Poisson-gamma model. 

These derivations gives us new results that help to lower computational costs for Tweedie's compound Poisson parameter estimation within generalized linear models, and we provide evidence supporting the industry preference for the Poisson-gamma parametrization.