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Different texts (and even different parts of this article) adopt slightly different definitions for the negative binomial distribution. They can be distinguished by whether the support starts at k = 0 or at k = r, whether p denotes the probability of a success or of a failure, and whether r represents success or failure, [1] so identifying the specific parametrization used is crucial in any ...
The main reason for studying these numbers is to obtain their factorizations.Aside from algebraic factors, which are obtained by factoring the underlying polynomial (binomial) that was used to define the number, such as difference of two squares and sum of two cubes, there are other prime factors (called primitive prime factors, because for a given they do not factorize with <, except for a ...
The multiplicative formula allows the definition of binomial coefficients to be extended [4] by replacing n by an arbitrary number α (negative, real, complex) or even an element of any commutative ring in which all positive integers are invertible: = _! = () (+) ().
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation). Abel's binomial theorem; Alternating factorial; Antichain; Beta function; Bhargava factorial; Binomial coefficient. Pascal's triangle; Binomial distribution; Binomial proportion confidence interval; Binomial-QMF (Daubechies wavelet filters ...
Thus many identities on binomial coefficients carry over to the falling and rising factorials. The rising and falling factorials are well defined in any unital ring, and therefore can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function.
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 , where a and b are numbers, and m and n are distinct non-negative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable.
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).