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Что (кто) такое jointly normal random variables - определение

GENERALIZATION OF THE ONE-DIMENSIONAL NORMAL DISTRIBUTION TO HIGHER DIMENSIONS

Multivariate gaussian distribution; Multivariate Gaussian distribution; Multivariate normal; Multivariate Gaussian; Bivariate Gaussian distribution; MVN; Bivariate normal distribution; Joint normality; Jointly normal; Jointly Gaussian; Jointly gaussian; Multivariate Gaussian random variable; Multinormal distribution; Jointly normally distributed; Bivariate normal; Gaussian discriminant analysis; Normal random vector; Multinormal; Multivariate normal random variable; Mardia's test; BHEP test; Gaussian random vector; Joint normal distribution; Multidimensional normal distribution; Friedman Rafsky Test; Multivariate Gaussian vector

$Left: Classification of seven multivariate normal classes. Coloured ellipses are 1 sd error ellipses. Black marks the boundaries between the classification regions. <math>p_e</math> is the probability of total classification error. Right: the error matrix. <math>p_{ij}</math> is the probability of classifying a sample from normal <math>i</math> as <math>j</math>. These are computed by the numerical method of ray-tracing <ref name="Das" /> ([https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions Matlab code]).$
$Top: the probability of a bivariate normal in the domain <math>x\sin y-y\cos x>1</math> (blue regions). Middle: the probability of a trivariate normal in a toroidal domain. Bottom: converging Monte-Carlo integral of the probability of a 4-variate normal in the 4d regular polyhedral domain defined by <math>\sum_{i=1}^4 \vert x_i \vert < 1</math>. These are all computed by the numerical method of ray-tracing. <ref name="Das"></ref>$
$'''a:''' Probability density of a function <math>\cos x^2</math> of a single normal variable <math>x</math> with <math>\mu=-2</math> and <math>\sigma=3</math>. '''b:''' Probability density of a function <math>x^y</math> of a normal vector <math>(x, y)</math>, with mean <math>\boldsymbol{\mu}=(1, 2)</math>, and covariance <math>\mathbf{\Sigma} = \begin{bmatrix} .01 & .016 \\ .016 & .04 \end{bmatrix}</math>. '''c:''' Heat map of the joint probability density of two functions of a normal vector <math>(x, y)</math>, with mean <math>\boldsymbol{\mu}=(-2, 5)</math>, and covariance <math>\mathbf{\Sigma} = \begin{bmatrix} 10 & -7 \\ -7 & 10 \end{bmatrix}</math>. '''d:''' Probability density of a function <math>\sum_{i=1}^4 \vert x_i \vert</math> of 4 iid standard normal variables. These are computed by the numerical method of ray-tracing. <ref name="Das" />$

Exchangeable random variables

SEQUENCE OF RANDOM VARIABLES SUCH THAT, FOR ANY FINITE PERMUTATION OF THE INDICES, THE JOINT PROBABILITY DISTRIBUTION OF THE PERMUTED SEQUENCE EQUALS THAT OF THE ORIGINAL

Exchangeable events; Interchangeable random variables; Exchangeability; Exchangeable sequence; Exchangeable random variable; Exchangeable matrix; Exchangeable correlation matrix

In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered.

https://en.wikipedia.org/wiki/Exchangeable_random_variables

Exchangeability

SEQUENCE OF RANDOM VARIABLES SUCH THAT, FOR ANY FINITE PERMUTATION OF THE INDICES, THE JOINT PROBABILITY DISTRIBUTION OF THE PERMUTED SEQUENCE EQUALS THAT OF THE ORIGINAL

Exchangeable events; Interchangeable random variables; Exchangeability; Exchangeable sequence; Exchangeable random variable; Exchangeable matrix; Exchangeable correlation matrix

·noun The quality or state of being exchangeable.

Weakly dependent random variables

Draft:Weakly dependent random variables

In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale. A (time) sequence of random variables is weakly dependent if distinct portions of the sequence have a covariance that asymptotically decreases to 0 as the blocks are further separated in time.

https://en.wikipedia.org/wiki/Weakly_dependent_random_variables

Википедия

Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. 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) correlated real-valued random variables each of which clusters around a mean value.

https://en.wikipedia.org/wiki/Multivariate normal distribution