text: In mathematics, specifically in probability theory, and yet more specifically in the theory of stochastic processes, the Chapman-Kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process.\n\nSuppose that { fi } is an indexed collection of random variables, that is, a stochastic process. Let\n\nPi1,...,in(f1 , ... ,fn)\n\nbe the joint probability density function of the values of the random variables f1 to fn. Then, the Chapman-Kolmogorov equation is\n\nPi1,...,in-1(f1 , ... ,fn-1) = ∫±∞Pi1,...,in(f1 , ... ,fn)dfn\n\nie a straightforward marginalization over the nuisance variable.\n\n(Note that we have not yet assumed anything about the temporal (or any other) ordering of the random variables -- the above equation applies equally to the marginalization of any of them).