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Las cantidades conocidas se expresan por las primeras letras del alfa beto: a, b, c, d

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Half of the game is learning how to forget those events in the past that eat away at you and cloud your reason.

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and B both report revenue? Such an exchange is referred to as a “barter transaction.” An even more challenging revenue recognition issue evolved from a specific type of barter transaction, a round-trip transaction. As an example, <span>if Company A sells advertising services (or energy contracts, or commodities) to Company B and almost simultaneously buys an almost identical product from Company B, can Company A report revenue at the fair value of the product sold? Because the company’s revenue would be approximately equal to its expense, the net effect of the transaction would have no impact on net income or cash flow. However, the amount of reve

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are acceptable for financial reporting purposes. However, the direct method discloses more information about a company. Partly because companies want to limit information disclosed, the indirect method is more commonly used. <span>The reporting of investing and financing activities is the same for both direct and indirect methods. Only the reporting of CFO is different. Direct Method Under the direct method, the statement of cash flows reports net cash flows from operations as major clas

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a }}\mathbf {t} {\Big )}} In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. <span>One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly)

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e definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. <span>The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Contents [hide] 1 Notation and parametrization 2 Definition 3 Properties 3.1 Density function 3.1.1 Non-degenerate case 3.1.2 Degenerate case 3.2 Higher moments 3.3 Lik

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al {CN}}_{0}\|{\mathcal {CN}}_{1})=\operatorname {tr} \left({\boldsymbol {\Sigma }}_{1}^{-1}{\boldsymbol {\Sigma }}_{0}\right)-k+\ln {|{\boldsymbol {\Sigma }}_{1}| \over |{\boldsymbol {\Sigma }}_{0}|}.} Mutual information[edit source] <span>The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which P {\displaystyle P} is the full multivariate distribution and Q {\displaystyle Q} is the product of the 1-dimensional marginal distributions. In the notation of the Kullback–Leibler divergence section of this article, Σ 1

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ldsymbol {\rho }}_{0}} is the correlation matrix constructed from Σ 0 {\displaystyle {\boldsymbol {\Sigma }}_{0}} . <span>In the bivariate case the expression for the mutual information is: I ( x ; y ) = − 1 2 ln ( 1 − ρ 2 ) . {\displaystyle I(x;y)=-{1 \over 2}\ln(1-\rho ^{2}).} Cumulative distribution function[edit source] The notion of cumulative distribution function (cdf) in dimension 1 can be extended in two ways to the multidimensional case, based

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y two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. <span>Conditional distributions[edit source] If N-dimensional x is partitioned as follows x = [ x 1 x 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {x} _{1}\\\mathbf {x} _{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} and accordingly μ and Σ are partitioned as follows μ = [ μ 1 μ 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle {\boldsymbol {\mu }}={\begin{bmatrix}{\boldsymbol {\mu }}_{1}\\{\boldsymbol {\mu }}_{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} Σ = [ Σ 11 Σ 12 Σ 21 Σ 22 ] with sizes [ q × q q × ( N − q ) ( N − q ) × q ( N − q ) × ( N − q ) ] {\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{12}\\{\boldsymbol {\Sigma }}_{21}&{\boldsymbol {\Sigma }}_{22}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times q&q\times (N-q)\\(N-q)\times q&(N-q)\times (N-q)\end{bmatrix}}} then the distribution of x 1 conditional on x 2 = a is multivariate normal (x 1 | x 2 = a) ~ N(μ, Σ) where μ ¯ = μ 1 + Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\bar {\boldsymbol {\mu }}}={\boldsymbol {\mu }}_{1}+{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} and covariance matrix Σ ¯ = Σ 11 − Σ 12 Σ 22 − 1 Σ 21 . {\displaystyle {\overline {\boldsymbol {\Sigma }}}={\boldsymbol {\Sigma }}_{11}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}{\boldsymbol {\Sigma }}_{21}.} [13] This matrix is the Schur complement of Σ 22 in Σ. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops the rows and columns corresponding to the variables being conditioned upon, and then inverts back to get the conditional covariance matrix. Here Σ 22 − 1 {\displaystyle {\boldsymbol {\Sigma }}_{22}^{-1}} is the generalized inverse of Σ 22 {\displaystyle {\boldsymbol {\Sigma }}_{22}} . Note that knowing that x 2 = a alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} ; compare this with the situation of not knowing the value of a, in which case x 1 would have distribution N q ( μ 1 , Σ 11 ) {\displaystyle {\mathcal {N}}_{q}\left({\boldsymbol {\mu }}_{1},{\boldsymbol {\Sigma }}_{11}\right)} . An interesting fact derived in order to prove this result, is that the random vectors x 2 {\displaystyle \mathbf {x} _{2}} and y 1 = x 1 − Σ 12 Σ 22 − 1 x 2 {\displaystyle \mathbf {y} _{1}=\mathbf {x} _{1}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\mathbf {x} _{2}} are independent. The matrix Σ 12 Σ 22 −1 is known as the matrix of regression coefficients. Bivariate case[edit source] In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is [14]

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) {\displaystyle \operatorname {E} (X_{1}\mid X_{2}##BAD TAG##\rho E(X_{2}\mid X_{2}##BAD TAG##} and then using the properties of the expectation of a truncated normal distribution. Marginal distributions[edit source] <span>To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra. [16] Example Let X = [X 1 , X 2 , X 3 ] be multivariate normal random variables with mean vector μ = [μ 1 , μ 2 , μ 3 ] and covariance matrix Σ (standard parametrization for multivariate

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{\displaystyle {\boldsymbol {\Sigma }}'={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{13}\\{\boldsymbol {\Sigma }}_{31}&{\boldsymbol {\Sigma }}_{33}\end{bmatrix}}} . Affine transformation[edit source] <span>If Y = c + BX is an affine transformation of X ∼ N ( μ , Σ ) , {\displaystyle \mathbf {X} \ \sim {\mathcal {N}}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}),} where c is an M × 1 {\displaystyle M\times 1} vector of constants and B is a constant M × N {\displaystyle M\times N} matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T i.e., Y ∼ N ( c + B μ , B Σ B T ) {\displaystyle \mathbf {Y} \sim {\mathcal {N}}\left(\mathbf {c} +\mathbf {B} {\boldsymbol {\mu }},\mathbf {B} {\boldsymbol {\Sigma }}\mathbf {B} ^{\rm {T}}\right)} . In particular, any subset of the X i has a marginal distribution that is also multivariate normal. To see this, consider the following example: to extract the subset (X 1 , X 2 , X 4 )

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implies that the variance of the dot product must be positive. An affine transformation of X such as 2X is not the same as the sum of two independent realisations of X. Geometric interpretation[edit source] See also: Confidence region <span>The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. The directions of the principal axes of the ellipsoids are given by the eigenvec

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urs of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. <span>The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ. The squared relative lengths of the principal axes are given by the corresponding eigenvalues. If Σ = UΛU T = UΛ 1/2 (UΛ 1/2 ) T is an eigendecomposition where the columns of U are unit eigenvectors and Λ is a diagonal matrix of the eigenvalues, then we have

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{\mu }}+\mathbf {U} {\mathcal {N}}(0,{\boldsymbol {\Lambda }}).} Moreover, U can be chosen to be a rotation matrix, as inverting an axis does not have any effect on N(0, Λ), but inverting a column changes the sign of U's determinant. <span>The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ 1/2 , rotated by U and translated by μ. Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λ i yields a non-singular multivariate normal distribution. If any Λ i is zero and U is square, the re

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mathbf {y} _{1}=\mathbf {x} _{1}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\mathbf {x} _{2}} are independent. The matrix Σ 12 Σ 22 −1 is known as the matrix of regression coefficients. Bivariate case[edit source] <span>In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is [14] X 1 ∣ X 2 = x 2 ∼ N ( μ 1 + σ 1 σ 2 ρ ( x 2 − μ 2 ) , ( 1 − ρ 2 ) σ 1 2 ) . {\displaystyle X_{1}\mid X_{2}=x_{2}\ \sim \ {\mathcal {N}}\left(\mu _{1}+{\frac {\sigma _{1}}{\sigma _{2}}}\rho (x_{2}-\mu _{2}),\,(1-\rho ^{2})\sigma _{1}^{2}\right).} where ρ {\displaystyle \rho } is the correlation coefficient between X 1 and X 2 . Bivariate conditional expectation[edit source] In the general case[edit source] (

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In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is where is the correlation coefficient between X 1 and X 2 .

mathbf {y} _{1}=\mathbf {x} _{1}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\mathbf {x} _{2}} are independent. The matrix Σ 12 Σ 22 −1 is known as the matrix of regression coefficients. Bivariate case[edit source] <span>In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is [14] X 1 ∣ X 2 = x 2 ∼ N ( μ 1 + σ 1 σ 2 ρ ( x 2 − μ 2 ) , ( 1 − ρ 2 ) σ 1 2 ) . {\displaystyle X_{1}\mid X_{2}=x_{2}\ \sim \ {\mathcal {N}}\left(\mu _{1}+{\frac {\sigma _{1}}{\sigma _{2}}}\rho (x_{2}-\mu _{2}),\,(1-\rho ^{2})\sigma _{1}^{2}\right).} where ρ {\displaystyle \rho } is the correlation coefficient between X 1 and X 2 . Bivariate conditional expectation[edit source] In the general case[edit source] (

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The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ 1/2 , rotated by U and translated by μ.

{\mu }}+\mathbf {U} {\mathcal {N}}(0,{\boldsymbol {\Lambda }}).} Moreover, U can be chosen to be a rotation matrix, as inverting an axis does not have any effect on N(0, Λ), but inverting a column changes the sign of U's determinant. <span>The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ 1/2 , rotated by U and translated by μ. Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λ i yields a non-singular multivariate normal distribution. If any Λ i is zero and U is square, the re

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The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ. The squared relative lengths of the principal axes are given by the corresponding eigenvalues.

urs of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. <span>The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ. The squared relative lengths of the principal axes are given by the corresponding eigenvalues. If Σ = UΛU T = UΛ 1/2 (UΛ 1/2 ) T is an eigendecomposition where the columns of U are unit eigenvectors and Λ is a diagonal matrix of the eigenvalues, then we have

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The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ. The squared relative lengths of the principal axes are given by the corresponding eigenvalues.

urs of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. <span>The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ. The squared relative lengths of the principal axes are given by the corresponding eigenvalues. If Σ = UΛU T = UΛ 1/2 (UΛ 1/2 ) T is an eigendecomposition where the columns of U are unit eigenvectors and Λ is a diagonal matrix of the eigenvalues, then we have

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The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean.

implies that the variance of the dot product must be positive. An affine transformation of X such as 2X is not the same as the sum of two independent realisations of X. Geometric interpretation[edit source] See also: Confidence region <span>The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. The directions of the principal axes of the ellipsoids are given by the eigenvec

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If Y = c + BX is an affine transformation of where c is an vector of constants and B is a constant matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T . Corollaries: sums of Gaussian are Gaussian, marginals of Gaussian are Gaussian.

{\displaystyle {\boldsymbol {\Sigma }}'={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{13}\\{\boldsymbol {\Sigma }}_{31}&{\boldsymbol {\Sigma }}_{33}\end{bmatrix}}} . Affine transformation[edit source] <span>If Y = c + BX is an affine transformation of X ∼ N ( μ , Σ ) , {\displaystyle \mathbf {X} \ \sim {\mathcal {N}}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}),} where c is an M × 1 {\displaystyle M\times 1} vector of constants and B is a constant M × N {\displaystyle M\times N} matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T i.e., Y ∼ N ( c + B μ , B Σ B T ) {\displaystyle \mathbf {Y} \sim {\mathcal {N}}\left(\mathbf {c} +\mathbf {B} {\boldsymbol {\mu }},\mathbf {B} {\boldsymbol {\Sigma }}\mathbf {B} ^{\rm {T}}\right)} . In particular, any subset of the X i has a marginal distribution that is also multivariate normal. To see this, consider the following example: to extract the subset (X 1 , X 2 , X 4 )

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><head> If Y = c + BX is an affine transformation of where c is an vector of constants and B is a constant matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T . Corollaries: sums of Gaussian are Gaussian, marginals of Gaussian are Gaussian. <html>

{\displaystyle {\boldsymbol {\Sigma }}'={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{13}\\{\boldsymbol {\Sigma }}_{31}&{\boldsymbol {\Sigma }}_{33}\end{bmatrix}}} . Affine transformation[edit source] <span>If Y = c + BX is an affine transformation of X ∼ N ( μ , Σ ) , {\displaystyle \mathbf {X} \ \sim {\mathcal {N}}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}),} where c is an M × 1 {\displaystyle M\times 1} vector of constants and B is a constant M × N {\displaystyle M\times N} matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T i.e., Y ∼ N ( c + B μ , B Σ B T ) {\displaystyle \mathbf {Y} \sim {\mathcal {N}}\left(\mathbf {c} +\mathbf {B} {\boldsymbol {\mu }},\mathbf {B} {\boldsymbol {\Sigma }}\mathbf {B} ^{\rm {T}}\right)} . In particular, any subset of the X i has a marginal distribution that is also multivariate normal. To see this, consider the following example: to extract the subset (X 1 , X 2 , X 4 )

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To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions an

) {\displaystyle \operatorname {E} (X_{1}\mid X_{2}##BAD TAG##\rho E(X_{2}\mid X_{2}##BAD TAG##} and then using the properties of the expectation of a truncated normal distribution. Marginal distributions[edit source] <span>To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra. [16] Example Let X = [X 1 , X 2 , X 3 ] be multivariate normal random variables with mean vector μ = [μ 1 , μ 2 , μ 3 ] and covariance matrix Σ (standard parametrization for multivariate

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Conditional distributions If N-dimensional x is partitioned as follows and accordingly μ and Σ are partitioned as follows then the distribution of x 1 conditional on x 2 = a is multivariate normal (x 1 | x 2 = a) ~ N( μ , Σ ) where and covariance matrix This matrix is the Schur complement of Σ 22 in Σ. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops t

y two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. <span>Conditional distributions[edit source] If N-dimensional x is partitioned as follows x = [ x 1 x 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {x} _{1}\\\mathbf {x} _{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} and accordingly μ and Σ are partitioned as follows μ = [ μ 1 μ 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle {\boldsymbol {\mu }}={\begin{bmatrix}{\boldsymbol {\mu }}_{1}\\{\boldsymbol {\mu }}_{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} Σ = [ Σ 11 Σ 12 Σ 21 Σ 22 ] with sizes [ q × q q × ( N − q ) ( N − q ) × q ( N − q ) × ( N − q ) ] {\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{12}\\{\boldsymbol {\Sigma }}_{21}&{\boldsymbol {\Sigma }}_{22}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times q&q\times (N-q)\\(N-q)\times q&(N-q)\times (N-q)\end{bmatrix}}} then the distribution of x 1 conditional on x 2 = a is multivariate normal (x 1 | x 2 = a) ~ N(μ, Σ) where μ ¯ = μ 1 + Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\bar {\boldsymbol {\mu }}}={\boldsymbol {\mu }}_{1}+{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} and covariance matrix Σ ¯ = Σ 11 − Σ 12 Σ 22 − 1 Σ 21 . {\displaystyle {\overline {\boldsymbol {\Sigma }}}={\boldsymbol {\Sigma }}_{11}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}{\boldsymbol {\Sigma }}_{21}.} [13] This matrix is the Schur complement of Σ 22 in Σ. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops the rows and columns corresponding to the variables being conditioned upon, and then inverts back to get the conditional covariance matrix. Here Σ 22 − 1 {\displaystyle {\boldsymbol {\Sigma }}_{22}^{-1}} is the generalized inverse of Σ 22 {\displaystyle {\boldsymbol {\Sigma }}_{22}} . Note that knowing that x 2 = a alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} ; compare this with the situation of not knowing the value of a, in which case x 1 would have distribution N q ( μ 1 , Σ 11 ) {\displaystyle {\mathcal {N}}_{q}\left({\boldsymbol {\mu }}_{1},{\boldsymbol {\Sigma }}_{11}\right)} . An interesting fact derived in order to prove this result, is that the random vectors x 2 {\displaystyle \mathbf {x} _{2}} and y 1 = x 1 − Σ 12 Σ 22 − 1 x 2 {\displaystyle \mathbf {y} _{1}=\mathbf {x} _{1}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\mathbf {x} _{2}} are independent. The matrix Σ 12 Σ 22 −1 is known as the matrix of regression coefficients. Bivariate case[edit source] In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is [14]

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the distribution of x 1 conditional on x 2 = a is multivariate normal (x 1 | x 2 = a) ~ N( μ , Σ ) where and covariance matrix

y two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. <span>Conditional distributions[edit source] If N-dimensional x is partitioned as follows x = [ x 1 x 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {x} _{1}\\\mathbf {x} _{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} and accordingly μ and Σ are partitioned as follows μ = [ μ 1 μ 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle {\boldsymbol {\mu }}={\begin{bmatrix}{\boldsymbol {\mu }}_{1}\\{\boldsymbol {\mu }}_{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} Σ = [ Σ 11 Σ 12 Σ 21 Σ 22 ] with sizes [ q × q q × ( N − q ) ( N − q ) × q ( N − q ) × ( N − q ) ] {\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{12}\\{\boldsymbol {\Sigma }}_{21}&{\boldsymbol {\Sigma }}_{22}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times q&q\times (N-q)\\(N-q)\times q&(N-q)\times (N-q)\end{bmatrix}}} then the distribution of x 1 conditional on x 2 = a is multivariate normal (x 1 | x 2 = a) ~ N(μ, Σ) where μ ¯ = μ 1 + Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\bar {\boldsymbol {\mu }}}={\boldsymbol {\mu }}_{1}+{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} and covariance matrix Σ ¯ = Σ 11 − Σ 12 Σ 22 − 1 Σ 21 . {\displaystyle {\overline {\boldsymbol {\Sigma }}}={\boldsymbol {\Sigma }}_{11}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}{\boldsymbol {\Sigma }}_{21}.} [13] This matrix is the Schur complement of Σ 22 in Σ. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops the rows and columns corresponding to the variables being conditioned upon, and then inverts back to get the conditional covariance matrix. Here Σ 22 − 1 {\displaystyle {\boldsymbol {\Sigma }}_{22}^{-1}} is the generalized inverse of Σ 22 {\displaystyle {\boldsymbol {\Sigma }}_{22}} . Note that knowing that x 2 = a alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} ; compare this with the situation of not knowing the value of a, in which case x 1 would have distribution N q ( μ 1 , Σ 11 ) {\displaystyle {\mathcal {N}}_{q}\left({\boldsymbol {\mu }}_{1},{\boldsymbol {\Sigma }}_{11}\right)} . An interesting fact derived in order to prove this result, is that the random vectors x 2 {\displaystyle \mathbf {x} _{2}} and y 1 = x 1 − Σ 12 Σ 22 − 1 x 2 {\displaystyle \mathbf {y} _{1}=\mathbf {x} _{1}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\mathbf {x} _{2}} are independent. The matrix Σ 12 Σ 22 −1 is known as the matrix of regression coefficients. Bivariate case[edit source] In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is [14]

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In the bivariate case the expression for the mutual information is:

ldsymbol {\rho }}_{0}} is the correlation matrix constructed from Σ 0 {\displaystyle {\boldsymbol {\Sigma }}_{0}} . <span>In the bivariate case the expression for the mutual information is: I ( x ; y ) = − 1 2 ln ( 1 − ρ 2 ) . {\displaystyle I(x;y)=-{1 \over 2}\ln(1-\rho ^{2}).} Cumulative distribution function[edit source] The notion of cumulative distribution function (cdf) in dimension 1 can be extended in two ways to the multidimensional case, based

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The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which is the full multivariate distribution and is the product of the 1-dimensional marginal distributions

al {CN}}_{0}\|{\mathcal {CN}}_{1})=\operatorname {tr} \left({\boldsymbol {\Sigma }}_{1}^{-1}{\boldsymbol {\Sigma }}_{0}\right)-k+\ln {|{\boldsymbol {\Sigma }}_{1}| \over |{\boldsymbol {\Sigma }}_{0}|}.} Mutual information[edit source] <span>The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which P {\displaystyle P} is the full multivariate distribution and Q {\displaystyle Q} is the product of the 1-dimensional marginal distributions. In the notation of the Kullback–Leibler divergence section of this article, Σ 1

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The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value

e definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. <span>The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Contents [hide] 1 Notation and parametrization 2 Definition 3 Properties 3.1 Density function 3.1.1 Non-degenerate case 3.1.2 Degenerate case 3.2 Higher moments 3.3 Lik

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One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution.

a }}\mathbf {t} {\Big )}} In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. <span>One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly)

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operatorname {sgn}(\rho ){\frac {\sigma _{Y}}{\sigma _{X}}}(x-\mu _{X})+\mu _{Y}.} This is because this expression, with sgn(ρ) replaced by ρ, is the best linear unbiased prediction of Y given a value of X. [4] Degenerate case[edit] <span>If the covariance matrix Σ {\displaystyle {\boldsymbol {\Sigma }}} is not full rank, then the multivariate normal distribution is degenerate and does not have a density. More precisely, it does not have a density with respect to k-dimensional Lebesgue measure (which is the usual measure assumed in calculus-level probability courses). Only random vectors

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If the covariance matrix is not full rank, then the multivariate normal distribution is degenerate and does not have a density.

operatorname {sgn}(\rho ){\frac {\sigma _{Y}}{\sigma _{X}}}(x-\mu _{X})+\mu _{Y}.} This is because this expression, with sgn(ρ) replaced by ρ, is the best linear unbiased prediction of Y given a value of X. [4] Degenerate case[edit] <span>If the covariance matrix Σ {\displaystyle {\boldsymbol {\Sigma }}} is not full rank, then the multivariate normal distribution is degenerate and does not have a density. More precisely, it does not have a density with respect to k-dimensional Lebesgue measure (which is the usual measure assumed in calculus-level probability courses). Only random vectors

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A staggered board consists of a board of directors whose members are grouped into classes; for example, Class 1, Class 2, Class 3, etc. Each class represents a certain percentage of the total number of board positions.

[imagelink] DEFINITION of '<span>Staggered Board' A staggered board consists of a board of directors whose members are grouped into classes; for example, Class 1, Class 2, Class 3, etc. Each class represents a certain percentage of the total number of board positions. For example, a class is commonly comprised on one-third of the total board members. During each election term only one class is open to elections, thereby staggering the board directorship. BREAKING DOWN 'Staggered Board' A staggered board is also known as a classified board because of the different "classes" involved. A

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Historical Paper data storage (1725) Drum memory (1932) Magnetic-core memory (1949) Plated wire memory (1957) Core rope memory (1960s) Thin-film memory (1962) Twistor memory (~1968) Bubble memory (~1970) Floppy disk (1971) v t e <span>Static random-access memory (static RAM or SRAM) is a type of semiconductor memory that uses bistable latching circuitry (flip-flop) to store each bit. SRAM exhibits data remanence, [1] but it is still volatile in the conventional sense that data is eventually lost when the memory is not powered. The term static differentiates SRAM

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VGT6 microcontroller as seen by an optical microscope. Characteristics[edit] Advantages: Low power consumption Simplicity – a refresh circuit is not needed Reliability Disadvantages: Price Capacity Clock rate and power[edit] <span>The power consumption of SRAM varies widely depending on how frequently it is accessed; in some instances, it can use as much power as dynamic RAM, when used at high frequencies, and some ICs can consume many watts at full bandwidth. On the other hand, static RAM used at a somewhat slower pace, such as in applications with moderately clocked microprocessors, draws very little power and can have a nearly negligible power consumption when sitting idle – in the region of a few micro-watts. Several techniques have been proposed to manage power consumption of SRAM-based memory structures. [2] SRAM exists primarily as: general purpose products with asynchronous interface, such as the ubiquitous 28-pin 8K × 8 and 32K × 8 chips (often but not always named something along t

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eral megabytes may be used in complex products such as digital cameras, cell phones, synthesizers, etc. SRAM in its dual-ported form is sometimes used for realtime digital signal processing circuits. [citation needed] In computers[edit] <span>SRAM is also used in personal computers, workstations, routers and peripheral equipment: CPU register files, internal CPU caches and external burst mode SRAM caches, hard disk buffers, router buffers, etc. LCD screens and printers also normally employ static RAM to hold the image displayed (or to be printed). Static RAM was used for the main memory of some early personal computers such as

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Hunchentoot - The Common Lisp web server formerly known as TBNL [imagelink] Hunchentoot - The Common Lisp web server formerly known as TBNL Abstract Hunchentoot is a web server written in Common Lisp and at the same time a toolkit for building dynamic websites. As a stand-alone web server, Hunchentoot is capable of HTTP/1.1 chunking (both directions), persistent connections (keep-alive), and SSL. Hunchentoot provides facilities like automatic session handling (with and without cookies), logging, customizable error handling, and easy acces

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same time a toolkit for building dynamic websites. As a stand-alone web server, Hunchentoot is capable of HTTP/1.1 chunking (both directions), persistent connections (keep-alive), and SSL. <span>Hunchentoot provides facilities like automatic session handling (with and without cookies), logging, customizable error handling, and easy access to GET and POST parameters sent by the client. It does not include functionality to programmatically gener

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The net income figure is used to prepare the statement of retained earnings.

recurring income is persistent. If an item in the unusual or infrequent component of income from continuing operations is deemed not to be persistent, then recurring (pre-tax) income from continuing operations should be adjusted. <span>The net income figure is used to prepare the statement of retained earnings. Balance Sheet A balance sheet provides a "snapshot" of a company's financial condition. Think of the balance sheet as a photo of the business at a sp

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last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

est and since show commands are so important, I'm adding their own section. These are the ones I have so far, I've tried to group them according to their role in troubleshooting. Can you guys think of any others I might need on the CCENT? <span>Show Interfaces show ip interface brief show mac address-table show ip protocols show ip route show cdp neighbors show cdp neighbors detail show cdp interface show running-config show version