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Generalization of the one-dimensional normal distribution to higher dimensions From Wikipedia, the free encyclopedia
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.
Probability density function | |||
Notation | |||
---|---|---|---|
Parameters |
μ ∈ Rk — location Σ ∈ Rk × k — covariance (positive semi-definite matrix) | ||
Support | x ∈ μ + span(Σ) ⊆ Rk | ||
exists only when Σ is positive-definite | |||
Mean | μ | ||
Mode | μ | ||
Variance | Σ | ||
Entropy | |||
MGF | |||
CF | |||
Kullback–Leibler divergence | See § Kullback–Leibler divergence |
The multivariate normal distribution of a k-dimensional random vector can be written in the following notation:
or to make it explicitly known that X is k-dimensional,
with k-dimensional mean vector
such that and . The inverse of the covariance matrix is called the precision matrix, denoted by .
A real random vector is called a standard normal random vector if all of its components are independent and each is a zero-mean unit-variance normally distributed random variable, i.e. if for all .[1]: p. 454
A real random vector is called a centered normal random vector if there exists a deterministic matrix such that has the same distribution as where is a standard normal random vector with components.[1]: p. 454
A real random vector is called a normal random vector if there exists a random -vector , which is a standard normal random vector, a -vector , and a matrix , such that .[2]: p. 454 [1]: p. 455
Formally: