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Probability theory
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Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. It is not possible to predict precisely results of random events, two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, a great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory and this culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of space, introduced by Richard von Mises. This became the mostly undisputed axiomatic basis for modern probability theory, most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment. The power set of the space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results, one collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the set of the sample space of die rolls. In this case, is the event that the die falls on some odd number, If the results that actually occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one, the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6 and this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, and the event has a probability of 1, discrete probability theory deals with events that occur in countable sample spaces. Modern definition, The modern definition starts with a finite or countable set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by Ω

2.
Markov chain
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In probability theory and related fields, a Markov process, named after the Russian mathematician Andrey Markov, is a stochastic process that satisfies the Markov property. e. Conditional on the present state of the system, its future, a Markov chain is a type of Markov process that has either discrete state space or discrete index set, but the precise definition of a Markov chain varies. Andrey Markov studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906, random walks on the integers and the Gamblers ruin problem are examples of Markov processes and were studied hundreds of years earlier. These two processes are Markov processes in time, while random walks on the integers and the Gamblers ruin problem are examples of Markov processes in discrete time. The algorithm known as PageRank, which was proposed for the internet search engine Google, is based on a Markov process. The adjective Markovian is used to something that is related to a Markov process. A Markov chain is a process with the Markov property. The term Markov chain refers to the sequence of variables such a process moves through. It can thus be used for describing systems that follow a chain of linked events, the systems state space and time parameter index need to be specified. In addition, there are extensions of Markov processes that are referred to as such. Moreover, the index need not necessarily be real-valued, like with the state space. Notice that the 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 space of a Markov chain does not have any generally agreed-on restrictions. However, many applications of Markov chains employ finite or countably infinite state spaces, besides time-index and state-space parameters, there are many other variations, extensions and generalizations. For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, the changes of state of the system are called transitions. The probabilities associated with state changes are called transition probabilities. The process is characterized by a space, a transition matrix describing the probabilities of particular transitions. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not terminate