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A Here chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. It is named after the Russian mathematician Andrey Markov. Markov chains Rwndom many applications as statistical models of real-world processes, [1] [4] [5] [6] such as studying cruise control systems in motor vehiclesqueues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics.

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Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlowhich are used for simulating sampling from complex probability distributions, Random Family Is A Complex Web Of have found application in Bayesian statisticsthermodynamicsstatistical mechanicsphysicschemistryeconomicsfinancesignal processinginformation theory and artificial intelligence.

The adjective Markovian is used to describe something that is related to a Markov process. A Markov process is a stochastic process that satisfies the Markov property [1] sometimes characterized as " memorylessness ". In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made here the process's full history.

A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set often representing timebut the precise definition of a Markov chain varies. The system's state space and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v.

Note that there is no definitive agreement in Random Family Is A Complex Web Of literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a discrete-time Markov chain DTMC[1] [17] but a few authors use the term "Markov process" to refer to a continuous-time Markov chain CTMC without explicit mention. Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs.

Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term. While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space. Besides time-index and state-space parameters, there are many other variations, extensions and generalizations see Variations.

For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise. The changes of state of the system are called transitions. The process is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state or initial distribution across the state space. By convention, we assume all possible states and transitions have been Random Family Is A Complex Web Of in the definition of the process, so there is always a next state, and the process does not terminate. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps.

Formally, the steps are the integers or natural numbersand the random process is a mapping of these to states. Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future.

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Markov studied Markov processes in the early 20th century, publishing his first paper on the topic in Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest inand a branching process, introduced by Francis Galton and Henry William Watson inpreceding the work of Markov. Andrei Kolmogorov developed in a paper a large part of the early theory of continuous-time Markov processes.

Random walks based on integers and the gambler's ruin problem are examples of Markov processes. Source any position there are two possible transitions, to the next or previous integer. The transition probabilities depend only on the current position, not on the manner in which the position was reached. For example, the transition probabilities from 5 to 4 and 5 to 6 are both 0. These probabilities are independent of whether the system was previously Random Family Is A Complex Web Of 4 or 6. Another example is the dietary habits of a creature who eats only grapes, cheese, or lettuce, and whose dietary habits conform to the following rules:.

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This Random Family Is A Complex Web Of eating habits can be modeled with a Markov Familyy since its choice tomorrow depends solely on what it ate today, not what it ate yesterday or any other time in the past. One statistical property that could be calculated is the expected percentage, over a long period, of the days on which the creature will eat grapes.

A series of independent events for example, a series of coin flips satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next step depends non-trivially on the current Comples. To see why this is the case, suppose that in the first six draws, all five nickels and a quarter are drawn. However, it is possible to model this scenario as a Markov process. This new model would be represented by possible states that is, 6x6x6 states, since each of the three coin types could have zero to five coins on the table by the end of the 6 draws.]