Примеры вероятностных процессов и цепей Маркова

Examples of the probabilistic construct
This article contains examples of Markov chains and Markov processes in action. All examples are in the countable state space. For an overview of Markov chains in general state space, see Markov chains on a measurable state space.

A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain. This is in contrast to card games such as blackjack, where the cards represent a 'memory' of the past moves. To see the difference, consider the probability for a certain event in the game. In the above mentioned dice games, the only thing that matters is the current state of the board. The next state of the board depends on the current state, and the next roll of the dice. It does not depend on how things got to their current state. In a game such as blackjack, a player can gain an advantage by remembering which cards have already been shown (and hence which cards are no longer in the deck), so the next state (or hand) of the game is not independent of the past states.

Suppose that one starts with $10, and one wagers $1 on an unending, fair, coin toss indefinitely, or until all of the money is lost. If represents the number of dollars one has after n tosses, with , then the sequence is a Markov process. If one knows that one has $12 now, then it would be expected that with even odds, one will either have $11 or $13 after the next toss. This guess is not improved by the added knowledge that one started with $10, then went up to $11, down to $10, up to $11, and then to $12. The fact that the guess is not improved by the knowledge of earlier tosses showcases the Markov property, the memoryless property of a stochastic process.

This example came from Markov himself. Markov chose 20,000 letters from Pushkin’s Eugene Onegin, classified them into vowels and consonants, and counted the transition probabilities. The stationary distribution is 43.2 percent vowels and 56.8 percent consonants, which is close to the actual count in the book.

If one pops one hundred kernels of popcorn in an oven, each kernel popping at an independent exponentially distributed time, then this would be a continuous time Markov process. If denotes the number of kernels which have popped up to time t, the problem can be defined as finding the number of kernels that will pop in some later time. The only thing one needs to know is the number of kernels that have popped prior to the time "t". It is not necessary to know when they popped, so knowing for previous times "t" is not relevant. The process described here is an approximation of a Poisson point process – Poisson processes are also Markov processes.