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Binomial Distribution, Binomial
Binomial may refer to: In mathematics *Binomial (polynomial), a polynomial with two terms *Binomial coefficient, numbers appearing in the expansions of powers of binomials *Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition *Binomial theorem, a theorem about powers of binomials *Binomial type, a property of sequences of polynomials In probability and statistics *Binomial distribution, a type of probability distribution *Binomial process *Binomial test, a test of significance In computing science *Binomial heap, a data structure In linguistics *Binomial pair, a sequence of two or more words or phrases in the same grammatical category, having some semantic relationship and joined by some syntactic device In biology * Binomial nomenclature, a Latin two-term name for a species, such as ''Sequoia sempervirens'' In finance *Binomial options pricing model, a numerical method for the valuation of options In politics *Binomial voting system, a voting system used i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial (polynomial)
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of sparse polynomial after the monomials. Definition A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form :a x^m - bx^n \,, where and are numbers, and and are distinct nonnegative integers and is a symbol which is called an indeterminate or, for historical reasons, a variable. In the context of Laurent polynomials, a ''Laurent binomial'', often simply called a ''binomial'', is similarly defined, but the exponents and may be negative. More generally, a binomial may be written as: :a x_1^\dotsb x_i^ - b x_1^\dotsb x_i^ Examples :3x - 2x^2 :xy + yx^2 :0.9 x^3 + \pi y^2 :2 x^3 + 7 Operations on simple binomials *The binomial can be factored as the product of two other binomials: :: x^2 - y^2 = (x - y)(x + y). :This is a special case of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial QMF
A binomial QMF – properly an orthonormal binomial quadrature mirror filter – is an orthogonal wavelet developed in 1990. The binomial QMF bank with perfect reconstruction (PR) was designed by Ali Akansu, and published in 1990, using the family of binomial polynomials for subband decomposition of discrete-time signals. Akansu and his fellow authors also showed that these binomial-QMF filters are identical to the wavelet filters designed independently by Ingrid Daubechies from compactly supported orthonormal wavelet transform perspective in 1988 (Daubechies wavelet). It was an extension of Akansu's prior work on Binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ... and Hermite polynomials wherein he developed the Modified Hermite Transformation (MHT) in 1987. Lat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the exponents and are nonnegative integers with , and the coefficient of each term is a specific positive integer depending on and . For example, for , (x+y)^4 = x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4. The coefficient in the term of is known as the binomial coefficient \tbinom or \tbinom (the two have the same value). These coefficients for varying and can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where \tbinom gives the number of different combinations of elements that can be chosen from an -element set. Therefore \tbinom is often pronounced as " choose ". History Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid ment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers \left\ in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities :p_n(x+y)=\sum_^n\, p_k(x)\, p_(y). Many such sequences exist. The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences put on firm footing the vague 19th century notions of umbral calculus. Examples * In consequence of this definition the binomial theorem can be stated by saying that the sequence is of binomial type. * The sequence of " lower factorials" is defined by(x)_n=x(x-1)(x-2)\cdot\cdots\cdot(x-n+1).(In the theory of special functions, this same notation denotes upper fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: ''success'' (with probability ''p'') or ''failure'' (with probability q=1-p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., ''n'' = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size ''n'' drawn with replacement from a population of size ''N''. If the sampling is carried out without replacement, the draws are not independent and so the resulting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Process
A binomial process is a special point process in probability theory. Definition Let P be a probability distribution and n be a fixed natural number. Let X_1, X_2, \dots, X_n be i.i.d. random variables with distribution P , so X_i \sim P for all i \in \. Then the binomial process based on ''n'' and ''P'' is the random measure : \xi= \sum_^n \delta_, where \delta_=\begin1, &\textX_i\in A,\\ 0, &\text.\end Properties Name The name of a binomial process is derived from the fact that for all measurable sets A the random variable \xi(A) follows a binomial distribution with parameters P(A) and n : : \xi(A) \sim \operatorname(n,P(A)). Laplace-transform The Laplace transform of a binomial process is given by : \mathcal L_(f)= \left \int \exp(-f(x)) \mathrm P(dx) \rightn for all positive measurable functions f . Intensity measure The intensity measure \operatorname\xi of a binomial process \xi is given by : \operatorname\xi =n P. Generalizations A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Test
In statistics, the binomial test is an exact test of the statistical significance of deviations from a theoretically expected distribution of observations into two categories using sample data. Usage The binomial test is useful to test hypotheses about the probability (\pi) of success: : H_0:\pi=\pi_0 where \pi_0 is a user-defined value between 0 and 1. If in a sample of size n there are k successes, while we expect n\pi_0, the formula of the binomial distribution gives the probability of finding this value: : \Pr(X=k)=\binomp^k(1-p)^ If the null hypothesis H_0 were correct, then the expected number of successes would be n\pi_0. We find our p-value for this test by considering the probability of seeing an outcome as, or more, extreme. For a one-tailed test, this is straightforward to compute. Suppose that we want to test if \pi\pi_0 using the summation of the range from k to n instead. Calculating a p-value for a two-tailed test is slightly more complicated, since a bin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Heap
In computer science, a binomial heap is a data structure that acts as a priority queue but also allows pairs of heaps to be merged. It is important as an implementation of the mergeable heap abstract data type (also called meldable heap), which is a priority queue supporting merge operation. It is implemented as a heap similar to a binary heap but using a special tree structure that is different from the complete binary trees used by binary heaps. Binomial heaps were invented in 1978 by Jean Vuillemin. Binomial heap A binomial heap is implemented as a set of binomial trees (compare with a binary heap, which has a shape of a single binary tree), which are defined recursively as follows: * A binomial tree of order 0 is a single node * A binomial tree of order k has a root node whose children are roots of binomial trees of orders k-1, k-2, ..., 2, 1, 0 (in this order). A binomial tree of order k has 2^k nodes, and height k. The name comes from the shape: a binomial tree of ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Pair
In linguistics and stylistics, an irreversible binomial, frozen binomial, binomial freeze, binomial expression, binomial pair, or nonreversible word pair is a pair or group of words used together in fixed order as an idiomatic expression or collocation. The words have some semantic relationship and are usually connected by the words ''and'' or ''or''. They also belong to the same part of speech: nouns (''milk and honey''), adjectives (''short and sweet''), or verbs (''do or die''). The order of word elements cannot be reversed. The term "irreversible binomial" was introduced by Yakov Malkiel in 1954, though various aspects of the phenomenon had been discussed since at least 1903 under different names: a "terminological imbroglio". Ernest Gowers used the name Siamese twins (i.e., conjoined twins) in the 1965 edition of Fowler's ''Modern English Usage''. The 2015 edition reverts to the scholarly name, "irreversible binomials", as "Siamese twins" had become offensive. Many irrev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Nomenclature
In taxonomy, binomial nomenclature ("two-term naming system"), also called nomenclature ("two-name naming system") or binary nomenclature, is a formal system of naming species of living things by giving each a name composed of two parts, both of which use Latin grammatical forms, although they can be based on words from other languages. Such a name is called a binomial name (which may be shortened to just "binomial"), a binomen, name or a scientific name; more informally it is also historically called a Latin name. The first part of the name – the '' generic name'' – identifies the genus to which the species belongs, whereas the second part – the specific name or specific epithet – distinguishes the species within the genus. For example, modern humans belong to the genus ''Homo'' and within this genus to the species ''Homo sapiens''. ''Tyrannosaurus rex'' is likely the most widely known binomial. The ''formal'' introduction of this system of naming species is credit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Options Pricing Model
In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" ( lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting. The binomial model was first proposed by William Sharpe in the 1978 edition of ''Investments'' (), and formalized by Cox, Ross and Rubinstein in 1979 and by Rendleman and Bartter in that same year. For binomial trees as applied to fixed income and interest rate derivatives see . Use of the model The Binomial options pricing model approach has been widely used since it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM is based on the description of an underlying instrument over a period of time rather than a single point. As a consequence, it is used to value Am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
