In mathematics, specifically in probability theory and in particular the theory of Markovian stochastic processes, the **Chapman–Kolmogorov equation** is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process. The equation was arrived at independently by both the British mathematician Sydney Chapman and the Russian mathematician Andrey Kolmogorov.

Suppose that { *f*_{i} } is an indexed collection of random variables, that is, a stochastic process. Let

be the joint probability density function of the values of the random variables *f*_{1} to *f _{n}*. Then, the Chapman–Kolmogorov equation is

i.e. a straightforward marginalization over the nuisance variable.

(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.)

Read more about Chapman–Kolmogorov Equation: Application To Time Dilated Markov Chains

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