Probability Theory with Markov Chains

A Markov chain is a mathematical system that transitions from one state to another based on probabilistic rules. The defining feature of a Markov chain is that the possible future states are fixed regardless of how the process arrived at its current state. In other words, the probability of transitioning to any particular state is solely determined by the current state and the amount of time elapsed. Letters, numbers, weather conditions, baseball scores, or stock performance can all be included in the state space, or set of all possible states.
Finite state machines can be used to model Markov chains, and random walks are a prime example of their utility in mathematics. They are commonly used in economics, game theory, queueing (communication) theory, genetics, and finance, and they emerge broadly in statistical and information-theoretic contexts. While Markov chains with any size of state space can be discussed, the initial theory and most applications are focused on cases with a finite (or countably infinite) number of states. Many applications of Markov chains necessitate familiarity with common matrix methods.

KEY FEATURE

Probability theory is concerned with discrete and continuous random variables, probability distributions, and stochastic processes.

The law of large numbers and the central limit theorem are two major results in probability theory that describe such behaviour.

Probability theory, as a mathematical foundation for statistics, is essential to many human activities that involve quantitative data analysis.

Probability theory methods can also be used to describe complex systems with only a partial understanding of their state, as in statistical mechanics or sequential estimation.

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