We introduce and analyse a simple probabilistic model of genome evolution. It is based on three fundamental evolutionary events: gene duplication, loss and innovation, and it is called DLI model. The focus of the paper is around the size distribution of gene families. The formulas for equilibrium gene family sizes are derived showing that they follow a logarithmic distribution. We consider also a disjoint union of DLI models and we present the result of this study. Some empirical results for microbial genomes are presented.