Multivariate Normal/Gaussian Distribution Family

A Multivariate Normal/Gaussian Distribution Family is a multivariate distribution family that is a member of a Gaussian distribution and can be instantiated in multivariate Gaussian functions.

• Context:
  • It can be represented as a [math]\displaystyle{ k }[/math]-dimensional Vector Probability Distribution [math]\displaystyle{ \mathcal{N}_k(\boldsymbol\mu,\, \boldsymbol\Sigma) }[/math], where
    [math]\displaystyle{ \boldsymbol\mu }[/math] is a mean vector: [math]\displaystyle{ \lt \operatorname{E}[X_1], \operatorname{E}[X_2], \ldots, \operatorname{E}[X_k]\gt }[/math],
    and [math]\displaystyle{ \boldsymbol\Sigma }[/math] is a [math]\displaystyle{ k \times k }[/math] covariance matrix: [math]\displaystyle{ [\operatorname{Cov}[X_i, X_j]], i=1,2,\ldots,k; j=1,2,\ldots,k }[/math].
  • It can range from being a Spherical Gaussian Distribution to being a Non-Spherical Gaussian Distribution.
  • It can be instantiated in a Multivariate Normal Distribution Function.
• Example(s):
  • a Bivariate Normal Distribution.
  • …
• Counter-Example(s):
  • a Univariate Gaussian.
  • a Gaussian Mixture Function.
• See: Joint Probability Distribution, Multinomial Probability Distribution, Location Parameter, Covariance Matrix, Positive-Definite Matrix, Random Vector.

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Multivariate_normal_distribution Retrieved:2017-6-14.
  • 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. One possible 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. ...
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2006b

• (Dubnicka, 2006i) ⇒ Suzanne R. Dubnicka. (2006). “Special Multivariate Distributions. Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
  • TERMINOLOGY : The bivariate normal distribution can be extended easily to the multivariate normal distribution. In particular, X = (X1, . . ., Xk) has a multivariate normal distribution if its joint pdf is given by

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