# Wolfram Language & System 10.4 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# DiscreteMarkovProcess

DiscreteMarkovProcess[i0,m]
represents a discrete-time, finite-state Markov process with transition matrix m and initial state .

DiscreteMarkovProcess[p0,m]
represents a Markov process with initial state probability vector .

represents a Markov process with transition matrix from the graph g.

## DetailsDetails

• DiscreteMarkovProcess is also known as a discrete-time Markov chain.
• DiscreteMarkovProcess is a discrete-time and discrete-state random process.
• The states of DiscreteMarkovProcess are integers between 1 and , where is the length of transition matrix m.
• The transition matrix m specifies conditional transition probabilities mi,jProbability[x[k+1]jx[k]i], where is the state of the process at time k. »
• A discrete Markov process can be seen as a random walk on a graph, where the probability of transitioning from state to state is specified by .
• EstimatedProcess[data,DiscreteMarkovProcess[n]] indicates that a process with n states should be estimated.
• The transition matrix in the case of a graph g is constructed to give equal probability of transitioning to each incident vertex.
• DiscreteMarkovProcess allows m to be an × matrix with non-negative elements and rows that sum to 1, is an integer between 1 and , and is a vector of length of non-negative elements that sum to 1.
• DiscreteMarkovProcess can be used with such functions as MarkovProcessProperties, PDF, Probability, and RandomFunction.

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

Define a discrete Markov process:

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Simulate it:

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Find the PDF for the state at time :

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Find the long-run proportion of time the process is in state 2:

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