Multivariate normal distribution

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The multivariate normal (Gaussian) distribution [1].

The multivariate normal distribution (also called the multivariate Gaussian distribution) is the most important, useful, and widely studied distribution in multivariate statistics because of its:

  • Mathematical simplicity: It is relatively easy to work with, so it is easy to obtain multivariate methods.
  • Central Limit Theorem (multivariate version): If we have a collection of independent and identically distributed random vectors, then the sample mean vector will be approximately multivariate normally distributed for large samples.
  • Applicability to natural phenomena: Many natural phenomena may also be modeled using this distribution, just as in the univariate case [2].

The bivariate normal distribution (two variables) is easiest to understand because of its notational simplicity; in comparison, distributions with three or more variables require matrix algebra and vector notation.

Definition and properties of the multivariate normal distribution

There are several equivalent definitions for the multivariate normal distribution. One way to define it is, given a random vector X = (X1, X2, …, Xd) ∈ ℝd, X is multivariate normal if any linear combination Y = aT X = a1X1 + a2X2 + … + adXd with a ∈ ℝ [3], where T is the transpose of a.

Another way to define it is [4]:

For a set of standard normal random variables X = (X1, …, Xk) and Xi, then the expectation E(X) and covariance COV(X) are

E(X) = (0, 0, · · · , 0), COV (X) = I<sub>k</sub>.

Where Ik​ is the covariance matrix of the random vector X.

Then, for a n dimensional vector µ and n × k matrix A, we have

E(µ + AX) = µ, COV (µ + AX) = AAT

which leads to the following definition:

The distribution of random vector AX is called a multivariate normal distribution with covariance matrix Σ and is denoted by N(0, Σ). And the distribution of µ+AX is called a multivariate normal distribution with mean µ and covariance matrix Σ, N(µ, Σ).

Lie Wang.

The multivariate normal distribution can also be defined by its density function. For vector µ and positive semidefinite matrix Σ, Y ∼ Nn(µ, Σ), the density function is [5]:

density function multivariate normal distribution

And the moment generating function of Y is:

multivariate normal MGF


[1] Piotrg~commonswiki via Wikimedia Commons.  Creative Commons Attribution-Share Alike 3.0 Unported license.

[2] Penn State. Lesson 4: Multivariate Normal Distribution. Retrieved October 17, 2023 from: Lesson 4: Multivariate Normal Distribution

[3] Arias, M. Notes on the Gaussian distribution.

[4] Wang, L. Multivariate Normal Distribution. Retrieved October 17, 2023 from:

[5] The multivariate normal distribution. Retrieved October 17, 2023 from:

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