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In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe" [1]) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair (,) is called a measurable space.
An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers.It is the algebra on which the Borel measure is defined. . Given a real random variable defined on a probability space, its probability distribution is by definition also a measure on the Borel a
Denotes the set of rational numbers (fractions of two integers). It is often denoted also by . Denotes the set of p-adic numbers, where p is a prime number. Denotes the set of real numbers. It is often denoted also by .
Random variables are usually written in upper case Roman letters, such as or and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable.
Sigma's original name may have been san, but due to the complicated early history of the Greek epichoric alphabets, san came to be identified as a separate letter in the Greek alphabet, represented as ΟΊ. [2] Herodotus reports that "san" was the name given by the Dorians to the same letter called "sigma" by the Ionians. [i] [3]
the set of natural numbers in set theory (although or N is more common in other areas of mathematics) an asymptotic dominant notation related to big O notation; in probability theory, a possible outcome of an experiment; the arithmetic function counting a number's distinct prime factors [68]
If (,,) is a formula with a free set variable and free number variables , then the set {(,,)} is the intersection of the sets of the form {(,,)} as ranges over the set of natural numbers. The Σ 0 0 = Π 0 0 = Δ 0 0 {\displaystyle \Sigma _{0}^{0}=\Pi _{0}^{0}=\Delta _{0}^{0}} formulas can be checked by going over all cases one by one, which is ...
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations, and any other mathematical objects and assembling them into expressions and formulas. Mathematical notation is widely used in mathematics , science , and engineering for representing complex concepts and properties in a concise ...