We introduce and analyze a simple discrete probabilistic model of genome evolution. It is based on four fundamental evolutionary events: gene duplication, loss, accumulated change and innovation. We call it the DLCI model. It is the first such model rigorously analyzed. The focus of the paper is on the size distribution of gene families. We derive formulae for the equilibrium of gene family sizes, and show that they follow a logarithmic distribution. We also consider a disjoint union of DLCI models and present empirical results for bacterial genomes.