The frequency spectrum of a mutation, and its age, in a general diffusion model

General formulae are derived for the probability density and expected age of a mutation of frequency x in a population, and similarly for a mutation with b copies in a sample of n genes. A general formula is derived for the frequency spectrum of a mutation in a sample. Variable population size models are included. Results are derived in two frameworks: diffusion process models for the frequency of the mutation; and birth and death process models. The coalescent structure within the mutant gene group and the non-mutant group is considered.