Markov processes

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Smith's markov process theorem

Consider(1)If the probability distribution is governed by a Markovprocess, then(2)(3)Assuming no time dependence, so ,(4)

Markov sequence

A sequence , , ... of random variates is called Markov (or Markoff) if, for any ,i.e., if the conditional distribution of assuming , , ..., equals the conditional distribution of assuming only (Papoulis 1984, pp. 528-529). The transitional densities of a Markov sequence satisfy the Chapman-Kolmogorov equation.

Markov process

A random process whose future probabilities are determined by its most recent values. A stochastic process is called Markov if for every and , we haveThis is equivalent to(Papoulis 1984, p. 535).

Markov chain

A Markov chain is collection of random variables (where the index runs through 0, 1, ...) having the property that, given the present, the future is conditionally independent of the past.In other words,If a Markov sequence of random variates take the discrete values , ..., , thenand the sequence is called a Markov chain (Papoulis 1984, p. 532).A simple random walk is an example of a Markovchain.The Season 1 episode "Man Hunt" (2005) of the television crime drama NUMB3RS features Markov chains.

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