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In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional normal distribution to higher dimensions.
A random vector X ∈ R p (a p×1 "column vector") has a multivariate normal distribution with a nonsingular covariance matrix Σ precisely if Σ ∈ R p × p is a positive-definite matrix and the probability density function of X is
It is constructed from a multivariate normal distribution over by using the probability integral transform. For a given correlation matrix R ∈ [ − 1 , 1 ] d × d {\displaystyle R\in [-1,1]^{d\times d}} , the Gaussian copula with parameter matrix R {\displaystyle R} can be written as
In the event that the variables X and Y are jointly normally distributed random variables, then X + Y is still normally distributed (see Multivariate normal distribution) and the mean is the sum of the means. However, the variances are not additive due to the correlation. Indeed,
The probability density function for the random matrix X (n × p) that follows the matrix normal distribution , (,,) has the form: (,,) = ([() ()]) / | | / | | /where denotes trace and M is n × p, U is n × n and V is p × p, and the density is understood as the probability density function with respect to the standard Lebesgue measure in , i.e.: the measure corresponding to integration ...
3.3 Multivariate normal distribution. 4 Properties. Toggle Properties subsection. 4.1 Chain rule. 4.2 f-divergence. 4.3 Sufficient statistic. 4.4 Reparameterization.
There is a one-to-one correspondence between cumulative distribution functions and characteristic functions, so it is possible to find one of these functions if we know the other. The formula in the definition of characteristic function allows us to compute φ when we know the distribution function F (or density f).
In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics.