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Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness .
Coalescent theory is a model of how alleles sampled from a population may have originated from a common ancestor.In the simplest case, coalescent theory assumes no recombination, no natural selection, and no gene flow or population structure, meaning that each variant is equally likely to have been passed from one generation to the next.
To qualify as a probability, the assignment of values must satisfy the requirement that for any collection of mutually exclusive events (events with no common results, such as the events {1,6}, {3}, and {2,4}), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.
One justification of Occam's razor is a direct result of basic probability theory. By definition, all assumptions introduce possibilities for error; if an assumption does not improve the accuracy of a theory, its only effect is to increase the probability that the overall theory is wrong.
[27] [28] Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory". [28] In mathematics, the theory of stochastic processes is an important contribution to probability theory, [29] and continues to be an active topic of research for both theory and applications. [30] [31] [32]
Random forest techniques generate a panel of decision trees. Decision trees have the advantage that you can draw them and interpret them (even with a basic understanding of mathematics and statistics). Random Forests have thus been used for clinical decision support systems. [citation needed]
One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable X {\displaystyle X} is obtained by summing a large number N {\displaystyle N} of independent random variables with variance 1, then X {\displaystyle X} has variance N {\displaystyle N} and ...