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#probability

Question

The negative binomial distribution also arises as a [...] of Poisson distributions

Answer

continuous mixture

*Can be used to model over dispersed count observations, known as Gamma-Poisson distribution.*

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The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution.

) . {\displaystyle \operatorname {Poisson} (\lambda )=\lim _{r\to \infty }\operatorname {NB} \left(r,{\frac {\lambda }{\lambda +r}}\right).} Gamma–Poisson mixture[edit source] <span>The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution. That is, we can view the negative binomial as a Poisson(λ) distribution, where λ is itself a random variable, distributed as a gamma distribution with shape = r and scale θ = p/(1 − p) or correspondingly rate β = (1 − p)/p. To display the intuition behind this statement, consider two independent Poisson processes, “Success” and “Failure”, with intensities p and 1 − p. Together, the Success and Failure pr

#fourier-analysis

When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:

The functions *f* and *g* are orthogonal when this integral is zero: As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.

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ia (Redirected from Orthogonal function) Jump to: navigation, search In mathematics, orthogonal functions belong to a function space which is a vector space (usually over R) that has a bilinear form. <span>When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: ⟨ f , g ⟩ = ∫ f ( x ) g ( x ) d x . {\displaystyle \langle f,g\rangle =\int f(x)g(x)\,dx.} The functions f and g are orthogonal when this integral is zero: ⟨ f , g ⟩ = 0. {\displaystyle \langle f,\ g\rangle =0.} As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Suppose {f n }, n = 0, 1, 2, … is a sequence of orthogonal functions. If f n has positive support then ⟨ f n

Tags

#linear-algebra #matrix-decomposition

Question

If matrix **A** can be eigendecomposed and if none of its eigenvalues are zero, then **A** is **[...]** and its inverse is given by **[...]**

Answer

nonsingular

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amp;0\\1&3\\\end{bmatrix}}{\begin{bmatrix}-2c&0\\c&d\\\end{bmatrix}}={\begin{bmatrix}1&0\\0&3\\\end{bmatrix}},[c,d]\in \mathbb {R} } Matrix inverse via eigendecomposition[edit source] Main article: Inverse matrix <span>If matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is nonsingular and its inverse is given by A − 1 = Q Λ − 1 Q − 1 {\displaystyle \mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1}} Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate: [ Λ

Tags

#linear-algebra #matrix-decomposition

Question

**[...]** has real eigenvalues and the eigenvectors can be chosen to be orthogonal

Answer

real symmetric matrix

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{\displaystyle A=A^{*}} ), which implies that it is also complex normal, the diagonal matrix Λ has only real values, and if A is unitary, Λ takes all its values on the complex unit circle. Real symmetric matrices[edit source] <span>As a special case, for every N×N real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen such that they are orthogonal to each other. Thus a real symmetric matrix A can be decomposed as A = Q Λ Q T {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T}} where Q is an orthogonal matrix, and Λ is a diagonal matrix whose entries are the eigenvalues of A. Useful facts[edit source] Useful facts regarding eigenvalues[edit source] The product of the eigenvalues is equal to the determinant of A det

#distributions

The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the *latent* random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution").

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istribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. <span>The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Contents [hide] 1 Definition 2 Properties 3 Applications 3.1 Testing 3.2 Overdispersion modeling 3.3 Bayesian inference 3.4 Convolution 4 Computation 5 Examples 6 See als

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#distributions

Question

The compound distribution is the result of **[...]** the *latent* random variables representing the parameters of the parametrized distribution

Answer

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The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution").

istribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. <span>The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Contents [hide] 1 Definition 2 Properties 3 Applications 3.1 Testing 3.2 Overdispersion modeling 3.3 Bayesian inference 3.4 Convolution 4 Computation 5 Examples 6 See als

Tags

#distributions

Question

The compound distribution is integrated over [...] that represents the parameters of the parametrized distribution

Answer

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The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution").

istribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. <span>The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Contents [hide] 1 Definition 2 Properties 3 Applications 3.1 Testing 3.2 Overdispersion modeling 3.3 Bayesian inference 3.4 Convolution 4 Computation 5 Examples 6 See als

#mathematical-structures

In many fields of mathematics, **morphism** refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

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st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth

#mathematical-structures

In many fields of mathematics, **morphism** refers to a structure-preserving map from one mathematical structure to another.

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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphism

st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth

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#mathematical-structures

Question

Answer

morphism

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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another.

st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth

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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another.

st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth

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his article is about the mathematical concept. For other uses, see Endomorphic. [imagelink] Orthogonal projection onto a line, m, is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism. <span>In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map, f: V → V, and an endomorphism of a group, G, is a group homomorphism f: G → G. In general, we can talk about endomorphi

Question

Answer

endomorphism

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In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself.

his article is about the mathematical concept. For other uses, see Endomorphic. [imagelink] Orthogonal projection onto a line, m, is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism. <span>In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map, f: V → V, and an endomorphism of a group, G, is a group homomorphism f: G → G. In general, we can talk about endomorphi

In operator theory, a **multiplication operator** is an operator *T*_{f} defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f . That is,

for all φ in the domain of *T*_{f} , and all x in the domain of φ (which is the same as the domain of f ).

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This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009) (Learn how and when to remove this template message) <span>In operator theory, a multiplication operator is an operator T f defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is, T f φ ( x ) = f ( x ) φ ( x ) {\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of th

Multiplication operators generalize the notion of operator given by a diagonal matrix.

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{\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. <span>Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of the results of operator theory is a spectral theorem, which states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication

Question

[...] generalize the notion of operator given by a diagonal matrix.

Answer

Multiplication operators

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Multiplication operators generalize the notion of operator given by a diagonal matrix.

{\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. <span>Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of the results of operator theory is a spectral theorem, which states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication

Question

Multiplication operators generalize the notion of operator given by a [...].

Answer

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Multiplication operators generalize the notion of operator given by a diagonal matrix.

{\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. <span>Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of the results of operator theory is a spectral theorem, which states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication

Question

In operator theory, the value of a **[...]** at a function φ is given by multiplication by a fixed function f .

Answer

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In operator theory, a multiplication operator is an operator T f defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f . That is, for all φ in the d

This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009) (Learn how and when to remove this template message) <span>In operator theory, a multiplication operator is an operator T f defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is, T f φ ( x ) = f ( x ) φ ( x ) {\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of th

Question

In operator theory, the value of a **multiplication operator** at a function φ is given by [...] .

Answer

multiplication by a fixed function f

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In operator theory, a multiplication operator is an operator T f defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f . That is, for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f ).

This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009) (Learn how and when to remove this template message) <span>In operator theory, a multiplication operator is an operator T f defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is, T f φ ( x ) = f ( x ) φ ( x ) {\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of th

#matrices #spectral-theorem

In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, , which are as simple as one can hope to find.

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putations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. <span>In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical per

Tags

#matrices #spectral-theorem

Question

In general, the spectral theorem identifies a class of [...] that can be modeled by multiplication operators

Answer

linear operators

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In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators

putations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. <span>In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical per

Tags

#matrices #spectral-theorem

Question

In general, the spectral theorem identifies a class of linear operators that can be modeled by [...]

Answer

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In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators

putations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. <span>In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical per

ok, si es de ves en cuando que lo realizas esta bien, porque todos los dias es malo..

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uando"? [imagelink] 2 Votes YBANEZ Please explain to me what my good friend want to convey when she said, "a <span>ok, si es de ves en cuando que lo realizas esta bien, porque todos los dias es malo.." Thank you very much Posted Mar 11, 2010 | 22310 views | link

Question

ok, si es [...] que lo realizas esta bien, porque todos los dias es malo..

Answer

de vez en cuando

de vez en vez o de cuando en cuando son también aceptable..

de vez en vez o de cuando en cuando son también aceptable..

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ok, si es de ves en cuando que lo realizas esta bien, porque todos los dias es malo..

uando"? [imagelink] 2 Votes YBANEZ Please explain to me what my good friend want to convey when she said, "a <span>ok, si es de ves en cuando que lo realizas esta bien, porque todos los dias es malo.." Thank you very much Posted Mar 11, 2010 | 22310 views | link

Question

▸ ¿pero qué **[...]** ? ¿que haga yo tu trabajo?

are you trying to get me to do your work, or what?

are you trying to get me to do your work, or what?

Answer

pretendes

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main (disambiguation). [imagelink] Illustration showing f, a function from the pink domain X to the blue codomain Y. The yellow oval inside Y is the image of f. Both the image and the codomain are sometimes called the range of f. <span>In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. [1] Conversely, the set of values the function takes on as output is termed the image of th

Question

Answer

the set of "input" values

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In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

main (disambiguation). [imagelink] Illustration showing f, a function from the pink domain X to the blue codomain Y. The yellow oval inside Y is the image of f. Both the image and the codomain are sometimes called the range of f. <span>In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. [1] Conversely, the set of values the function takes on as output is termed the image of th

#functional-analysis

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tional · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective Constructions Restriction · Composition · λ · Inverse Generalizations Partial · Multivalued · Implicit v t e I<span>n mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space have a

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n mathematics, a function space is a set of functions between two fixed sets.

tional · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective Constructions Restriction · Composition · λ · Inverse Generalizations Partial · Multivalued · Implicit v t e I<span>n mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space have a

#inner-product-space #vector-space

A **vector space** (also called a **linear space**) is a collection of objects called **vectors**, which may be added together and multiplied ("scaled") by numbers, called *scalars*.

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company, see Vector Space Systems. [imagelink] Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w. <span>A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of v

Tags

#inner-product-space #vector-space

Question

Answer

vector space

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A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

company, see Vector Space Systems. [imagelink] Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w. <span>A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of v

#linear-algebra

In mathematics, more specifically in abstract algebra and linear algebra, a **bilinear form** on a vector space *V* is a bilinear map *V* × *V* → *K* , where *K* is the field of scalars. In other words, a bilinear form is a function *B* : *V* × *V* → *K* that is linear in each argument separately:

*B*(**u**+**v**,**w**) =*B*(**u**,**w**) +*B*(**v**,**w**) and*B*(*λ***u**,**v**) =*λB*(**u**,**v**)*B*(**u**,**v**+**w**) =*B*(**u**,**v**) +*B*(**u**,**w**) and*B*(**u**,*λ***v**) =*λB*(**u**,**v**)

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Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one

Tags

#linear-algebra

Question

a **bilinear form** is [...descriptive]

Answer

linear in each argument separately

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In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K , where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is <span>linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) <span><body><html>

Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one