# Multivariate normal distribution

< Probability distributions < Multivariate normal distribution

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 .

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 , where T is the transpose of a.

Another way to define it is :

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 :

And the moment generating function of Y is:

## References

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

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

 Arias, M. Notes on the Gaussian distribution.

 Wang, L. Multivariate Normal Distribution. Retrieved October 17, 2023 from: https://math.mit.edu/~liewang/multinormal.pdf

 The multivariate normal distribution. Retrieved October 17, 2023 from: https://www.biostat.jhsph.edu/~iruczins/teaching/140.752/notes/ch4.pdf

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