We consider an inhomogeneous diffusion channel (IDC) for molecular communication, where the diffusivities vary in space. We first study the statistical properties of molecular noise. These properties can be fully characterized by two IDC parameters related to a space-dependent diffusion coefficient and a constant drift velocity. We then characterize a noise power for the IDC by introducing the concept of geometric noise power since the molecular noise can be a heavy-tailed distribution. Finally, we derive the bit error rate of molecular communication with timing modulation in terms of two IDC parameters.