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Approximation of powers of some binomials
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that ( 1 + x ) α ≈ 1 + α x . {\displaystyle
Binomial_approximation
Probability distribution
hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used. If
Binomial_distribution
Something roughly the same as something else
An approximation is anything that is intentionally similar but not exactly equal to something else. The word approximation is derived from Latin approximatus
Approximation
Statistical confidence interval for success counts
with a normal distribution. The normal approximation depends on the de Moivre–Laplace theorem (the original, binomial-only version of the central limit theorem)
Binomial proportion confidence interval
Binomial_proportion_confidence_interval
Mathematical series
In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer: where α {\displaystyle
Binomial_series
Algebraic expansion of powers of a binomial
Mathematics portal Binomial approximation Binomial distribution Binomial inverse theorem Binomial coefficient Stirling's approximation Tannery's theorem Polynomials
Binomial_theorem
Approximation of a function by its tangent line at a point
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are
Linear_approximation
Region of space between a transmitting and receiving antenna
{\displaystyle P} are much larger than the radius and applying the binomial approximation for the square root, 1 + x ≈ 1 + x / 2 {\displaystyle {\sqrt {1+x}}\approx
Fresnel_zone
Number of subsets of a given size
mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is
Binomial_coefficient
Probability distribution
doi:10.1214/19-EJP380. Ehm, Werner (1991-01-01). "Binomial approximation to the Poisson binomial distribution". Statistics & Probability Letters. 11
Poisson_binomial_distribution
Mathematical approximation of a function
\end{aligned}}} When only the linear term is retained, this simplifies to the binomial approximation. The usual trigonometric functions and their inverses have the following
Taylor_series
Numerical method for the valuation of financial options
volatility, the binomial tree is a discrete time approximation to the geometric Brownian motion used in the Black–Scholes–Merton model. The binomial model assumes
Binomial options pricing model
Binomial_options_pricing_model
Type of potential energy
constant over h, then this expression can be simplified using the binomial approximation 11+h/R≈1−hR to ΔU≈GMmR[1−(1−hR)]ΔU≈GMmhR2ΔU≈m(GMR2)h since the acceleration
Gravitational_energy
Speed of sound wave through elastic medium
{\sqrt {1+{\frac {\theta }{273.15}}}}\\\end{aligned}}} Finally, the binomial approximation (assuming θ is very close to 0) of the remaining square root yields
Speed_of_sound
Energy of a moving physical body
approximated well by the classical kinetic energy. To see this, apply the binomial approximation or take the first two terms of the Taylor expansion in powers of
Kinetic_energy
Rule in statistics
parameter p of a binomial distribution that give Pr(X = 0) ≤ 0.05. The rule can then be derived either from the Poisson approximation to the binomial distribution
Rule_of_three_(statistics)
Test of statistical significance
samples these approximations break down, and there is no alternative to the binomial test. The most usual (and easiest) approximation is through the
Binomial_test
Discrete probability distribution
approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. The Poisson distribution is a good approximation of
Poisson_distribution
Inequality about exponentiations of ''1+x''
{r-1}{r}}\log 1={\frac {1}{r}}\log(1+rx)} which is the desired inequality. Binomial approximation Brannan, D. A. (2006). A First Course in Mathematical Analysis.
Bernoulli's_inequality
Probability distribution
In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that
Negative binomial distribution
Negative_binomial_distribution
Transformation of a mathematical sequence
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely
Binomial_transform
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the
Taylor's_theorem
Method in probability theory
1007/BF00533704. S2CID 121725342. Ehm, W. (1991). "Binomial approximation to the Poisson binomial distribution". Statistics & Probability Letters. 11
Stein's_method
Discrete probability distribution
[math.PR]. unpublished note The Hypergeometric Distribution and Binomial Approximation to a Hypergeometric Random Variable by Chris Boucher, Wolfram Demonstrations
Hypergeometric_distribution
filters) Binomial series Binomial theorem Binomial transform Binomial type Carlson's theorem Catalan number Fuss–Catalan number Central binomial coefficient
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Sequence of numbers ((2n) choose (n))
In mathematics the nth central binomial coefficient is the particular binomial coefficient ( 2 n n ) = ( 2 n ) ! ( n ! ) 2 for all n ≥ 0. {\displaystyle
Central_binomial_coefficient
Triangular array of the binomial coefficients
mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics
Pascal's_triangle
Family of polynomials
mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian numbers, Gaussian polynomials, or q-binomial coefficients) are q-analogs
Gaussian_binomial_coefficient
Discrete probability distribution
distribution as n→∞. This result generalises the classical Poisson approximation of the binomial distribution. Let X1,…,Xn be Bernoulli random variables with
Conway–Maxwell–binomial distribution
Conway–Maxwell–binomial_distribution
A binomial QMF – properly an orthonormal binomial quadrature mirror filter – is an orthogonal wavelet developed in 1990. The binomial QMF bank with perfect
Binomial_QMF
Type of polynomial sequence
which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities pn(x+y)=∑k=0n(nk)pk(x)pn−k(y)
Binomial_type
perimeter of an ellipse. Throughout history, a large number of closed-form approximations and expressions in terms of integrals or series have been given for
Perimeter_of_an_ellipse
Product of numbers from 1 to n
the late 18th and early 19th centuries. Stirling's approximation provides an accurate approximation to the factorial of large numbers, showing that it
Factorial
Statistical rule of thumb
selection method. Sturges's rule comes from the binomial distribution which is used as a discrete approximation to the normal distribution. If the function
Sturges's_rule
Probability Theory
theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions. The theorem was named
Poisson_limit_theorem
Discrete analog of a derivative
differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference
Finite_difference
Regression analysis technique
In statistics, binomial regression is a regression analysis technique in which the response (often referred to as Y) has a binomial distribution: it is
Binomial_regression
French mathematician (1667–1754)
unpublished result of 1733, which is the first statement of an approximation to the binomial distribution in terms of what we now call the normal or Gaussian
Abraham_de_Moivre
Addition of several numbers or other values
expresses the sum of the binomial distribution ∑i=0ni(ni)=n(2n−1), the value at a = b = 1 of the derivative with respect to a of the binomial theorem ∑i=0n(ni)i+1=2n+1−1n+1
Summation
Approximation in mathematics
Press. ISBN 0-534-24264-2. Feller, W. (1945). "On the normal approximation to the binomial distribution". The Annals of Mathematical Statistics. 16 (4):
Continuity_correction
Generalization of the binomial theorem to other polynomials
of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. For any positive integer m and any non-negative
Multinomial_theorem
Probability distribution
discrete-to-continuum approximation and where infinitely divisible and decomposable distributions are involved, such as Binomial random variables, associated
Normal_distribution
Orthogonal wavelets
a scaling sequence of an orthogonal discrete wavelet transform with approximation order A, a ( Z ) = 2 1 − A ( 1 + Z ) A p ( Z ) , {\displaystyle
Daubechies_wavelet
Stars sorted by absolute magnitude
determining d implies an error ~2× as large (thus 20%) in luminosity (see binomial approximation). Stellar distances are only directly measured accurately out to
List_of_most_luminous_stars
Near-field diffraction
the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation
Fresnel_diffraction
Approximation method in statistics
mild-conditions are satisfied (e.g. for normal, exponential, Poisson and binomial distributions), standardized least-squares estimates and maximum-likelihood
Least_squares
Fundamental theorem in probability theory and statistics
this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem. Let (Xn)n≥1
Central_limit_theorem
Family of iterative methods
Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive
Stochastic_approximation
Conjecture in combinatorial number theory
prime numbers appear two times; 6 appears three times, as do all central binomial coefficients except for 1 and 2; (it is in principle not excluded that
Singmaster's_conjecture
Sigmoid shape special function
the desired interval of approximation. Another approximation is given by Sergei Winitzki using his "global Padé approximations": erf ( x ) ≈ sgn x
Error_function
Probability distribution and special case of gamma distribution
that the exact binomial test is always more powerful than the normal approximation. Lancaster shows the connections among the binomial, normal, and chi-squared
Chi-squared_distribution
Expectation or average of the falling factorial of a random variable
involve Stirling numbers of the second kind. If a random variable X has a binomial distribution with success probability p ∈ [0,1] and number of trials n
Factorial_moment
Numerical method for calculating the gamma function
In mathematics, the Lanczos approximation is a method for computing the gamma function numerically, published by Cornelius Lanczos in 1964. It is a practical
Lanczos_approximation
Convergence in distribution of binomial to normal distribution
theorem, states that the normal distribution may be used as an approximation to the binomial distribution under certain conditions. In particular, the theorem
De_Moivre–Laplace_theorem
Algorithms for calculating square roots
these algorithms typically construct a series of increasingly accurate approximations. Most square root computation methods are iterative: after choosing
Square_root_algorithms
Mathematical function
Similar in spirit to the Lanczos approximation of the Γ {\displaystyle \Gamma } -function is Spouge's approximation. Another alternative is to use the
Digamma_function
Method in statistics
increases, since it would help reduce the variance, and thus the Taylor approximation would be applied to a smaller range of the function g at the point of
Delta_method
Probability distribution
trial. A number of papers compare the robustness of different approximations for the binomial ratio.[citation needed] In the ratio of Poisson variables R
Ratio_distribution
Probability of shared birthdays
{364}{365}}\right)^{253}\approx 0.500477.} Applying the Poisson approximation for the binomial on the group of 23 people, Poi ( ( 23 2 ) 365 ) = Poi (
Birthday_problem
Probability distribution
47, No. 1/2, June 1960, pp. 173–175 Pratt, John W. “A Normal Approximation for Binomial, F, Beta, and Other Common, Related Tail Probabilities, II.” Journal
Beta_distribution
Type of polynomial used in Numerical Analysis
by the Binomial distribution. The expectation of this approximation technique is polynomial, as it is the expectation of a function of a binomial RV. The
Bernstein_polynomial
Estimation problem in physics or engineering
or engineering education, designed to teach dimensional analysis or approximation of extreme scientific calculations. Fermi problems are usually back-of-the-envelope
Fermi_problem
Extension of the factorial function
{\displaystyle n+1} times to get an approximation for Γ ( z ) {\displaystyle \Gamma (z)} , and furthermore that this approximation becomes exact as n increases
Gamma_function
Statistical significance test
(e.g., p-value) can be calculated exactly, rather than relying on an approximation that becomes exact in the limit as the sample size grows to infinity
Fisher's_exact_test
Mathematical identity involving sums of binomial coefficients
Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following: ∑
Abel's_binomial_theorem
Bayesian statistical inference method
this difference in perspective, empirical Bayes may be viewed as an approximation to a fully Bayesian treatment of a hierarchical model wherein the parameters
Empirical_Bayes_method
filter approximation, followed by small support difference operators, may be both faster and more accurate than computing recursive approximations of each
Scale_space_implementation
Mathematical functions
{\displaystyle (x)_{n}} with yet another meaning, namely to denote the binomial coefficient ( x n ) {\displaystyle {\tbinom {x}{n}}} . In this article
Falling_and_rising_factorials
Mathematical theorem on convolved binomial coefficients
identity (or Vandermonde's convolution) is the following identity for binomial coefficients: (m+nr)=∑k=0r(mk)(nr−k) for any nonnegative integers r, m
Vandermonde's_identity
Number computed as a product of powers
· Subfactorial · Superfactorial · Primorial · Lanczos approximation · Stirling's approximation Structures & arrays Lozanić's triangle · Sierpinski triangle
Hyperfactorial
Compound probability distribution
In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable X {\displaystyle X} equal to
Beta negative binomial distribution
Beta_negative_binomial_distribution
Square matrix in which each ascending skew-diagonal from left to right is constant
use the spectral norm (operator 2-norm) to measure the error of our approximation. This suggests singular value decomposition as a possible technique
Hankel_matrix
Statistical method
not quite correct, and introduces some error. To reduce the error in approximation, Frank Yates, an English statistician, suggested a correction for continuity
Yates's correction for continuity
Yates's_correction_for_continuity
Statistical model for count data
log-linear model, especially when used to model contingency tables. Negative binomial regression is a popular generalization of Poisson regression because it
Poisson_regression
Graphical aid for deriving some concepts in combinatorics
distinguishable bins. The solution to this particular problem is given by the binomial coefficient (n+k−1k−1), which is the number of subsets of size k − 1 that
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Evaluates how likely it is that any difference between data sets arose by chance
\left({\frac {O_{1}-np}{\sqrt {np(1-p)}}}\right)^{2}.} By the normal approximation to a binomial, this is the squared of one standard normal variate, and hence
Pearson's_chi-squared_test
Pair of polynomial sequences
minimizes Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum
Chebyshev_polynomials
Mathematical result on arithmetic properties of binomial coefficients
arithmetic properties of binomial coefficients. It was discovered by Henry W. Gould in 1972. The greatest common divisors of the binomial coefficients forming
Star_of_David_theorem
Arrangement of trinomial coefficients
triangle, which contains the binomial coefficients that appear in the binomial expansion and the binomial distribution. The binomial and trinomial coefficients
Pascal's_pyramid
Smooth approximation to the maximum function
or multivariable softplus) function is a smooth maximum – a smooth approximation to the maximum function, mainly used by machine learning algorithms
LogSumExp
Probability theorem
the independence requirement. Le Cam, L. (1960). "An Approximation Theorem for the Poisson Binomial Distribution". Pacific Journal of Mathematics. 10 (4):
Le_Cam's_theorem
Motion extremely close to the speed of light
are usually above 106 (v practically indistinguishable from c), the approximation is essentially exact. The opposite case (v ≪ c) is a so-called classical
Ultrarelativistic_limit
Wave phenomenon
proportion of the wavelength. Numerical approximations may be used, including the Fresnel and Fraunhofer approximations. Because diffraction is the result
Diffraction_from_slits
Product of consecutive factorial numbers
· Subfactorial · Superfactorial · Primorial · Lanczos approximation · Stirling's approximation Structures & arrays Lozanić's triangle · Sierpinski triangle
Superfactorial
Class of statistical models
This is appropriate when the response variable can vary, to a good approximation, indefinitely in either direction, or more generally for any quantity
Generalized_linear_model
Statistical concept
transformation Anscombe, F. J. (1948), "The transformation of Poisson, binomial and negative-binomial data", Biometrika, vol. 35, no. 3–4, [Oxford University Press
Anscombe_transform
Statistical hypothesis test
as χ2 distribution with k − 1 degrees of freedom, the error in this approximation would not affect practical decisions. This conclusion caused some controversy
Chi-squared_test
Test of normality in frequentist statistics
of calculating m and a lognormal approximation of W up to n = 2000, which could be used with an existing approximation of V, but the quadratic limitation
Shapiro–Wilk_test
Matrix of partial derivatives of a vector-valued function
distance between x and p does as x approaches p. This approximation specializes to the approximation of a scalar function of a single variable by its Taylor
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Formula in mathematics
tetrahedron. The trinomial expansion can be calculated by applying the binomial expansion twice, setting d = b + c {\displaystyle d=b+c} , which leads
Trinomial_expansion
Solved prime-number problem
mathematical publications. The basic idea is to show that the central binomial coefficients must have a prime factor within the interval (n,2n) in order
Proof_of_Bertrand's_postulate
Term in probability theory
the Conway–Maxwell–Poisson distribution. Only the Poisson, binomial and negative binomial distributions satisfy the full form of this relationship. These
(a,b,0) class of distributions
(a,b,0)_class_of_distributions
Recurrence relations of binomial coefficients in Pascal's triangle
+X^{n}={\frac {X^{r}-X^{n+1}}{1-X}}={\frac {X^{n+1}-X^{r}}{x}}} . Further, by the binomial theorem, we also find that X r + k = ( 1 + x ) r + k = ∑ i = 0 r + k (
Hockey-stick_identity
Number, approximately 3.14
widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer
Pi
{\displaystyle p} pairs of identical cards from the two sets, which is the binomial coefficient ( n p ) {\displaystyle {n \choose p}} . The remaining k − 2
Trinomial_triangle
Branch of discrete mathematics
astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist
Combinatorics
Number of partitions of an integer
which p ( n ) ≡ r ( mod m ) {\displaystyle p(n)\equiv r{\pmod {m}}} . Approximation formulas exist that are faster to calculate than the exact formula given
Partition function (number theory)
Partition_function_(number_theory)
Probability distribution of only one random variable
examples of discrete univariate distributions include the binomial, geometric, negative binomial, and Poisson distributions. At least 750 univariate discrete
Univariate_distribution
Mathematical function
special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral B ( z 1 , z 2 ) = ∫ 0 1 t z
Beta_function
Theorem of matrix ranks
(where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated
Woodbury_matrix_identity
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BINOMIAL APPROXIMATION
BINOMIAL APPROXIMATION
BINOMIAL APPROXIMATION
BINOMIAL APPROXIMATION
BINOMIAL APPROXIMATION
BINOMIAL APPROXIMATION
BINOMIAL APPROXIMATION
BINOMIAL APPROXIMATION
BINOMIAL APPROXIMATION
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