A formal and generic discussion of probability distributions and their relations is introduced, both in the continuum and in a lattice formulation. Focus is placed on the time dependence, and different mechanisms for time evolution are considered. The time evolution processes may have memory effects of past events or a loss of any memory, such as in a Markov process. In the case of a memoryless process, it is shown that the time dependence of a probability density satisfies the integral Chapman-Kolmogorov equation, or a differential equation, designated master equation. In some limit it leads to the Fokker-Planck equation. A Markov process is considered, taking as example a one-dimensional random walk, and the Fokker-Planck equation is interpreted as a diffusion equation. Considering a lattice, a connection between a Langevin equation and the Fokker-Planck equation is established taking as examples conserved and not conserved variables.