Concepts inStochastic lambda calculus and monads of probability distributions
Lambda calculus
The lambda calculus (also written as ¿-calculus) is a formal system in mathematical logic for expressing computation by way of variable binding and substitution. It was first formulated by Alonzo Church as a way to formalize mathematics through the notion of functions, in contrast to the field of set theory. Although not very successful in that respect, the lambda calculus found early successes in the area of computability theory, such as a negative answer to Hilbert's Entscheidungsproblem.
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Stochastic
Stochastic (from the Greek ¿¿¿¿¿¿ for aim or guess) is an adjective that refers to systems whose behavior is intrinsically non-deterministic, sporadic and categorically NOT intermittent. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E.
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Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values. For a more precise definition one needs to distinguish between discrete and continuous random variables. In the discrete case, one can easily assign a probability to each possible value: when throwing a die, each of the six values 1 to 6 has the probability 1/6.
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Formal language
In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols. The alphabet of a formal language is the set of symbols, letters, or tokens from which the strings of the language may be formed; frequently it is required to be finite. The strings formed from this alphabet are called words, and the words that belong to a particular formal language are sometimes called well-formed words or well-formed formulas.
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Operational semantics
In computer science, operational semantics is a way to give meaning to computer programs in a mathematically rigorous way. Operational semantics are classified into two categories: structural operational semantics (or small-step semantics) formally describe how the individual steps of a computation take place in a computer-based system. By opposition natural semantics (or big-step semantics) describe how the overall results of the executions are obtained.
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Expected value
In probability theory, the expected value (or expectation, or mathematical expectation, or mean, or the first moment) of a random variable is the weighted average of all possible values that this random variable can take on. The weights used in computing this average correspond to the probabilities in case of a discrete random variable, or densities in case of a continuous random variable.
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Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero-valued, or the closure of that set . This concept is used very widely in mathematical analysis. In the form of functions with support that is bounded, it also plays a major part in various types of mathematical duality theories.
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