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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
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 indexed
Binomial_coefficient
Number of partitions of an integer
of p ( N , M , n ) {\displaystyle p(N,M,n)} is the following Gaussian binomial coefficient: ∑ n = 0 ∞ p ( N , M , n ) q n = ( N + M M ) q = ( 1 − q N +
Partition function (number theory)
Partition_function_(number_theory)
Decomposition of an integer as a sum of positive integers
partition yields a partition of n − M into at most M parts. The Gaussian binomial coefficient is defined as: ( k + ℓ ℓ ) q = ( k + ℓ k ) q = ∏ j = 1 k + ℓ
Integer_partition
prime Factoriangular number Gamma distribution Gamma function Gaussian binomial coefficient Gould's sequence Hyperfactorial Hypergeometric distribution
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Generalization of the one-dimensional normal distribution to higher dimensions
theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional
Multivariate normal distribution
Multivariate_normal_distribution
Probability distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued
Normal_distribution
Triangular array of the binomial coefficients
triangle Bernoulli's triangle Binomial expansion Cellular automata Euler triangle Floyd's triangle Gaussian binomial coefficient Hockey-stick identity Leibniz
Pascal's_triangle
Mathematical function
processing, one uses a discrete Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussian. In geostatistics they have
Gaussian_function
Statistical concept
actually a set of parameters. For example, if the mixture components are Gaussian distributions, there will be a mean and variance for each component. If
Mixture_model
Measure of linear correlation
statistics, the Pearson correlation coefficient (PCC), also known as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC), or simply the unqualified
Pearson correlation coefficient
Pearson_correlation_coefficient
Graphical aid for deriving some concepts in combinatorics
the coefficient of x n {\displaystyle x^{n}} is found using binomial series to be ( n − 1 k − 1 ) {\displaystyle {\tbinom {n-1}{k-1}}} . Gaussian binomial
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
lemma in relation to polynomials Gaussian binomial coefficient, also called Gaussian polynomial or Gaussian coefficient Gauss transformation, also called
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Multiplicative factor in a mathematical expression
algorithms such as Gaussian elimination and Cramer's rule to find solutions to the system. The leading entry (sometimes leading coefficient[citation needed])
Coefficient
Statistical distribution for dependence between random variables
applying the Gaussian copula to credit derivatives to be one of the causes of the 2008 financial crisis; see David X. Li § CDOs and Gaussian copula. Despite
Copula_(statistics)
Filter in electronics and signal processing
processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would
Gaussian_filter
Sigmoid shape special function
Tanash, I.M.; Riihonen, T. (2020). "Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data
Error_function
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 and
Combinatorics
Type of mathematical generalization
to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients: ( n k ) q = [ n
Q-analog
Statistical modeling method
dependent variable y {\displaystyle y} is a random variable that follows a Gaussian distribution, where the standard deviation is fixed and the mean is a linear
Linear_regression
Type of multi-scale signal representation
Among the suggestions that have been given, the binomial kernels arising from the binomial coefficients stand out as a particularly useful and theoretically
Pyramid_(image_processing)
product, evaluates to 1. The fibonomial coefficients can be expressed in terms of the Gaussian binomial coefficients and the golden ratio φ = 1 + 5 2 {\displaystyle
Fibonomial_coefficient
Descriptive statistic
provides methods for the estimation of ICC and repeatabilities for Gaussian, binomial and Poisson distributed data in a mixed-model framework. Notably,
Intraclass_correlation
Concept in combinatorics (part of mathematics)
q-factorials, one can move on to define the q-binomial coefficients, also known as the Gaussian binomial coefficients, as [ n k ] q = [ n ] ! q [ n − k ] ! q
Q-Pochhammer_symbol
Statistical relationship
product-moment correlation coefficient, most commonly called 'Pearson's correlation coefficient' or simply 'the correlation coefficient' (as it is the most common
Correlation
q-derivative q-difference polynomial Quantum calculus LLT polynomial q-binomial coefficient q-Pochhammer symbol q-Vandermonde identity q-Bessel polynomials q-Charlier
List_of_q-analogs
takes value 1 with probability 1/2 and value −1 with probability 1/2. The binomial distribution, which describes the number of successes in a series of independent
List of probability distributions
List_of_probability_distributions
Upper bound on intersecting set families
{n-1}{r-1}}_{q},} where the subscript q marks the notation for the Gaussian binomial coefficient, the number of subspaces of a given dimension within a vector
Erdős–Ko–Rado_theorem
Device invented by Francis Galton
the number of paths to the kth bin on the bottom is given by the binomial coefficient ( n k ) {\displaystyle {n \choose k}} . Note that the leftmost bin
Galton_board
Fundamental theorem in probability theory and statistics
histogram converges toward a Gaussian curve as n approaches infinity; this relation is known as de Moivre–Laplace theorem. The binomial distribution article details
Central_limit_theorem
Technique for proving sets have equal size
powerful insights into each or both of the sets. The symmetry of the binomial coefficients states that ( n k ) = ( n n − k ) . {\displaystyle {n \choose k}={n
Bijective_proof
Branch of finite geometry
algebraic dimension d in vector space V(n, q) is given by the Gaussian binomial coefficient, [ n d ] q = ( q n − 1 ) ( q n − q ) ⋯ ( q n − q d − 1 ) ( q
Galois_geometry
Geometric system with a finite number of points
{q^{n+1-i}-1}{q^{i+1}-1}},} which is a Gaussian binomial coefficient, a q analogue of a binomial coefficient. Dimension 0 (no lines): The space is a
Finite_geometry
Family of power series in mathematics
sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the
Generalized hypergeometric function
Generalized_hypergeometric_function
Algorithm to smooth data points
equations can be found, in the form of a single set of "convolution coefficients" that can be applied to all data sub-sets, to give estimates of the smoothed
Savitzky–Golay_filter
Theorem on the largest antichain of sets
largest size of an r-chain-free family is the sum of the r largest binomial coefficients ( n i ) {\displaystyle {\binom {n}{i}}} . The case r = 1 is Sperner's
Sperner's_theorem
Mathematical approximation of a function
_{n=0}^{\infty }{\binom {\alpha }{n}}x^{n}} whose coefficients are the generalized binomial coefficients ( α n ) = ∏ k = 1 n α − k + 1 k = α ( α − 1 ) ⋯
Taylor_series
Exponentially modified Gaussian distribution Exponentiated Weibull distribution Exposure variable Extended Kalman filter Extended negative binomial distribution
List_of_statistics_articles
Transform in numerical harmonic analysis
{\displaystyle h} . The outputs give the detail coefficients (from the high-pass filter) and approximation coefficients (from the low-pass). It is important that
Discrete_wavelet_transform
Measure of relationship two or more financial variables over time
of outcome X. Hence, we derive the joint default dependence coefficient of the binomial events 1 { τ X ≤ T } {\displaystyle 1_{\{\tau _{X}\leq T\}}}
Financial_correlation
Mathematical function for the probability a given outcome occurs in an experiment
correlation coefficient) Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution and binomial distribution
Probability_distribution
Statistical property
chi-squared statistic or the standard error for a particular regression coefficient (as used in, say, confidence intervals). Suppose a statistically independent
Standard_error
Sufficient condition for polynomial irreducibility
all of whose non-leading coefficients are divisible by p by properties of binomial coefficients, and whose constant coefficient is equal to p, and therefore
Eisenstein's_criterion
Discrete probability distribution
Poisson distribution. The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial is p
Poisson_distribution
Collection of subsets covering all t-element subsets
( n k ) q {\displaystyle {\tbinom {n}{k}}_{q}} denotes the Gaussian binomial coefficient. Similarly, the affine geometry A G ( m , q ) {\displaystyle
Covering_design
Random motion of particles suspended in a fluid
higher. Unlike the random walk, it is scale invariant. A d-dimensional Gaussian free field has been described as "a d-dimensional-time analog of Brownian
Brownian_motion
Array of nonnegative integers in combinatorics
1994 in his paper Plane Partitions V: The TSSCPP Conjecture. Gaussian binomial coefficients Voxel Richard P. Stanley, Enumerative Combinatorics, Volume
Plane_partition
(Mathematical) decomposition into a product
coefficients of the expanded forms of ( a + b ) n {\displaystyle (a+b)^{n}} and ( a − b ) n {\displaystyle (a-b)^{n}} are the binomial coefficients,
Factorization
Measure of the asymmetry of random variables
skewness. For instance, a mixed distribution consisting of very thin Gaussians centred at −99, 0.5, and 2 with weights 0.01, 0.66, and 0.33 has a skewness
Skewness
Type of mathematical expression
of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, a two-term polynomial is called a binomial, and a three-term polynomial
Polynomial
Statistics function
Tanash, I.M.; Riihonen, T. (2020). "Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data
Q-function
Statistical model for a binary dependent variable
function, which is equivalent to placing a zero-mean Gaussian prior distribution on the coefficients, but other regularizers are also possible.) Whether
Logistic_regression
Academic mathematician from New Zealand
These are formed from the Gaussian binomial coefficients in an analogous way to the expansion of powers by the binomial theorem. However, she was unable
Gloria_Olive
Statistical measure of the magnitude of a phenomenon
sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, and the risk of a particular event
Effect_size
Statistical method
regression method. A Gaussian process (GP) is a collection of random variables, any finite number of which have a joint Gaussian (normal) distribution
Bootstrapping_(statistics)
Mathematical sequences in combinatorics
though the bracket notation conflicts with a common notation for Gaussian coefficients. The mathematical motivation for this type of notation, as well
Stirling_number
Statistics models class
exponential family distribution is specified for Y (for example normal, binomial or Poisson distributions) along with a link function g (for example the
Generalized_additive_model
Iterative method for finding maximum likelihood estimates in statistical models
the next E step. It can be used, for example, to estimate a mixture of gaussians, or to solve the multiple linear regression problem. The EM algorithm
Expectation–maximization algorithm
Expectation–maximization_algorithm
Metric for fit of statistical models
regression validation, the following topics relate to goodness of fit: Coefficient of determination (the R-squared measure of goodness of fit); Lack-of-fit
Goodness_of_fit
Polynomial sequence
theory in connection with nonlinear operations on Gaussian noise. random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon
Hermite_polynomials
Cryptographic protocol designed to resist quantum computer attacks
several choices in developing the algorithm: Binomial Sampling: Although sampling to high-quality discrete Gaussian distribution is important in post-quantum
NewHope
Specialized form of regression analysis, in statistics
Daemi, Atefeh, Hariprasad Kodamana, and Biao Huang. "Gaussian process modelling with Gaussian mixture likelihood." Journal of Process Control 81 (2019):
Robust_regression
Set of statistical processes for estimating the relationships among variables
distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R.A. Fisher in his works of 1922 and 1925
Regression_analysis
Concept in probability theory and statistics
relationship between two variables of interest, using their correlation coefficient will give misleading results if there is another confounding variable
Partial_correlation
Restricted model of non-universal quantum computation
indistinguishable photons distributed among N modes is given by the binomial coefficient ( M + N − 1 M ) {\displaystyle {\tbinom {M+N-1}{M}}} . Suppose the
Boson_sampling
Projective plane Property B Prüfer sequence q-analog q-binomial theorem—see Gaussian binomial coefficient q-derivative q-series q-theta function q-Vandermonde
Index of combinatorics articles
Index_of_combinatorics_articles
Application of a function to each point in a data set
regression or a logit transformation are more appropriate for binomial or non-binomial proportions, respectively, especially due to decreased type-II
Data transformation (statistics)
Data_transformation_(statistics)
Covariance and correlation
Caution must be applied when using cross correlation function which assumes Gaussian variance for nonlinear systems. In certain circumstances, which depend
Cross-correlation
Generalization of the binomial distribution
probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each
Multinomial_distribution
Optical device which splits a beam of light in two
) {\displaystyle {\tbinom {n}{j}}} is a binomial coefficient and it is to be understood that the coefficient is zero if j ∉ { 0 , n } {\displaystyle j\notin
Beam_splitter
French mathematician (1667–1754)
statement of an approximation to the binomial distribution in terms of what we now call the normal or Gaussian function. This was the first method of
Abraham_de_Moivre
Polynomial sequence
ratios of factorials in the radial part as products of binomials shows that the coefficients are integer numbers: R n m ( ρ ) = ∑ k = 0 n − m 2 ( − 1
Zernike_polynomials
Mathematical technique used in data compression and analysis
produces as many coefficients as there are pixels in the image (i.e., there is no compression yet since it is only a transform). These coefficients can then be
Wavelet_transform
Muirhead's inequality Newton's inequalities Stein–Strömberg theorem Binomial coefficient bounds Factorial bounds XYZ inequality Fisher's inequality Ingleton's
List_of_inequalities
Geometry of the location of polynomial roots
degree, less than the natural logarithm of the degree. If the coefficients are Gaussian distributed with a mean of zero and variance of σ then the mean
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Algorithm checking for prime numbers
it can easily be proved using the binomial theorem together with the following property of the binomial coefficient: ( n k ) ≡ 0 ( mod n ) {\displaystyle
AKS_primality_test
Category of regression analysis
splines smoothing splines neural networks In Gaussian process regression, also known as Kriging, a Gaussian prior is assumed for the regression curve. The
Nonparametric_regression
Set of quantities in probability theory
and binomial sequences are studied via umbral calculus. The joint cumulant κ of several random variables X1, ..., Xn is defined as the coefficient κ1,
Cumulant
Statistical transformation
z-transformation) of a Pearson correlation coefficient is its inverse hyperbolic tangent (artanh). When the sample correlation coefficient r is near 1 or -1, its distribution
Fisher_transformation
Mathematical functions
complex functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods { ( 1 + i ) ϖ , ( 1 − i ) ϖ } , {\displaystyle
Lemniscate_elliptic_functions
Probability distribution
distribution, and the generalized inverse Gaussian distribution. Among the discrete distributions, the negative binomial distribution is sometimes considered
Gamma_distribution
Infinite matrices with Pascal's triangle as elements
a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in
Pascal_matrix
Study of the inheritance of continuously variable traits
This is the inbreeding coefficient of the example progenies bulk, provided it is unbiased with respect to the full binomial distribution. An example
Quantitative_genetics
Empirical law on the variance of species in a habitat
with the distribution of prime numbers from the eigenvalue deviations of Gaussian orthogonal and unitary ensembles of random matrix theory The first use
Taylor's_law
German polymath and scholar (1777–1855)
in which he solved the main problem by introducing q-analogs of binomial coefficients and manipulating them by several original identities that seem to
Carl_Friedrich_Gauss
Class of statistical models
attendance would typically be modelled with a Bernoulli distribution (or binomial distribution, depending on exactly how the problem is phrased) and a log-odds
Generalized_linear_model
Function for integral Fourier-like transform
amounts to recovery of a signal in iid Gaussian noise. As p {\displaystyle p} is sparse, one method is to apply a Gaussian mixture model for p {\displaystyle
Wavelet
Problem-solving method in electrostatics
continue for a total of n steps, and end at position k is given by the binomial coefficient ( n ( n + k ) / 2 ) {\displaystyle {\binom {n}{(n+k)/2}}} assuming
Method_of_images
Extension of the factorial function
of binomial coefficients motivates why the properties of the gamma function when extended to negative numbers are natural. A binomial coefficient gives
Gamma_function
Special type of prime number
1{\pmod {p^{4}}},} where the expression in left-hand side denotes a binomial coefficient. In comparison, Wolstenholme's theorem states that for every prime
Wolstenholme_prime
Random process of binary (boolean) random variables
of such strings that contain k occurrences of H is given by the binomial coefficient N ( k , n ) = ( n k ) = n ! k ! ( n − k ) ! {\displaystyle N(k,n)={n
Bernoulli_process
Probability distribution
are also ratio distributions: the Student's t-distribution arises from a Gaussian random variable divided by an independent chi-distributed random variable
Ratio_distribution
Mathematical operation that predicts future values of a discrete-time signal
the predictor coefficients a i {\displaystyle a_{i}} are given by the corresponding row of the triangle of binomial transform coefficients. This estimate
Linear_prediction
these are the only known Wilson primes. Primes p for which the binomial coefficient ( 2 p − 1 p − 1 ) ≡ 1 ( mod p 4 ) . {\displaystyle {{2p-1} \choose
List_of_prime_numbers
Probability of shared birthdays
where ! is the factorial operator, (365 n) is the binomial coefficient and kPr denotes permutation. The equation expresses the fact that
Birthday_problem
Approximation of a function by its tangent line at a point
the Fréchet derivative of f {\displaystyle f} at a {\displaystyle a} . Gaussian optics is a technique in geometrical optics that describes the behaviour
Linear_approximation
Process forming a path from many random steps
{1}/{\sqrt {n}}} corresponds to the spacing of the scaling grid) one finds the gaussian density f ( x ) = 1 2 π e − x 2 {\textstyle f(x)={\frac {1}{2{\sqrt {\pi
Random_walk
Method for solving quadratic equations
computations involving quadratic polynomials, for example in calculus evaluating Gaussian integrals with a linear term in the exponent, and finding Laplace transforms
Completing_the_square
Integrals not expressible in closed-form from elementary functions
(logarithmic integral) e − x 2 {\displaystyle e^{-x^{2}}} (error function, Gaussian integral) sin ( x 2 ) {\displaystyle \sin(x^{2})} and cos ( x 2 ) {\displaystyle
Nonelementary_integral
Function defined by a hypergeometric series
In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes
Hypergeometric_function
GAUSSIAN BINOMIAL-COEFFICIENT
GAUSSIAN BINOMIAL-COEFFICIENT
Male
Russian
Variant spelling of Russian Gennadiy, GENNADY means "noble."
Male
Russian
Variant spelling of Russian Afanasiy, AFANASY means "immortal."
Male
Russian
Variant spelling of Russian Aleksey, ALEXEY means "defender."
Male
Russian
(РоÑÑ) Russian pet form of Czech/Russian Rostislav, ROSTYA means "usurp-glory."
Female
Russian
(Людмила) Russian feminine form of Czech/Russian Ludmil, LUDMILA means "people's favor."Â
Male
Russian
(Russian ИÑидор): Russian form of Greek Isidoros, ISIDOR means "gift of Isis."
Male
Russian
Variant spelling of Russian Gennadiy, GENNADI means "noble."
Male
Russian
Variant spelling of Russian Arseniy, ARSENI means "virile."
Male
Russian
Variant spelling of Russian Vikentiy, VIKENTI means "conquering."
Male
Russian
Variant spelling of Russian Afanasiy, AFANASII means "immortal."
Male
Russian
Variant spelling of Russian Vasiliy, VASILI means "king."
Male
Russian
Variant spelling of Russian Arseniy, ARSENIY means "virile."
Male
Russian
Variant spelling of Russian Vasiliy, VASSILY means "king."
Male
Russian
Variant spelling of Russian Afanasiy, AFANASEI means "immortal."
Male
Russian
Variant spelling of Russian Irinei, IRINEY means "peaceful."
Female
Russian
(Russian Ева): Armenian and Russian form of Greek Eva, YEVA means "life."Â
Male
Russian
(Паша) Russian pet form of Czech/Russian Pavel, PASHA means "small."
Male
Russian
Variant spelling of Russian Faddei, FADEI means "courageous."
Boy/Male
Australian, French, German, Irish
Curly-headed
Male
Russian
Variant spelling of Russian Vasiliy, VASILY means "king."
GAUSSIAN BINOMIAL-COEFFICIENT
GAUSSIAN BINOMIAL-COEFFICIENT
Girl/Female
Biblical
The house of corn, or of fish.
Boy/Male
Gujarati, Hindu, Indian, Kannada
Life of the World; Worldly Life
Boy/Male
Buddhist, Hindu, Indian
Voice of the Dharma
Boy/Male
Hindu, Indian, Marathi
Goddess
Boy/Male
Indian, Sindhi
Rose; Flower; Similar to Gulab
Boy/Male
Greek
Follower/gift of Artemis (Greek goddess of the hunt and counterpart of the Roman Diana). Famous...
Boy/Male
British, English
Blend of Ray and Shawn
Boy/Male
Hindu, Indian, Tamil
Flower and Treasure
Boy/Male
Hindu, Indian, Marathi
Fertile; Truthful
Girl/Female
Arabic, Muslim, Pashtun
Silken
GAUSSIAN BINOMIAL-COEFFICIENT
GAUSSIAN BINOMIAL-COEFFICIENT
GAUSSIAN BINOMIAL-COEFFICIENT
GAUSSIAN BINOMIAL-COEFFICIENT
GAUSSIAN BINOMIAL-COEFFICIENT
n.
An expression consisting of two terms connected by the sign plus (+) or minus (-); as, a + b, or 7 - 3.
a.
Of or pertaining to Prussia.
a.
Of or pertaining to two names; binomial.
n. & a.
Trinomial.
n.
A Russian village community.
n.
A monomial.
n.
Prussian leather.
n.
A quantity consisting of three terms, connected by the sign + or -; as, x + y + z, or ax + 2b - c2.
n.
A rule or principle expressed in algebraic language; as, the binominal formula.
a.
Having two names; -- used of the system by which every animal and plant receives two names, the one indicating the genus, the other the species, to which it belongs.
a.
Binominal.
n.
A native or inhabitant of Prussia.
n.
One who, not being a Russian, favors Russian policy and aggrandizement.
a.
Consisting of two terms; pertaining to binomials; as, a binomial root.
n.
A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality.
a.
Consisting of but a single term or expression.
n.
A numerical coefficient in any particular case of the binomial theorem.
a.
Of or pertaining to Lithuania (formerly a principality united with Poland, but now Russian and Prussian territory).
a.
Consisting of three terms; of or pertaining to trinomials; as, a trinomial root.
n.
A native or inhabitant of Russia; the language of Russia.