< 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 [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 = (X_{1}, X_{2}, …, X_{d}) ∈ ℝ^{d}, X is multivariate normal if any linear combination Y = a^{T} X = a_{1}X_{1} + a_{2}X_{2} + … + a_{d}X_{d} 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 = (X_{1}, …, X_{k}) and X_{i}, then the expectation E(X) and covariance COV(X) are

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

Where *I _{k}* 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*) = *AA*^{T}

which leads to the following definition:

Lie Wang.

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).

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]:

And the moment generating function of** Y** is:

## References

[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: https://math.mit.edu/~liewang/multinormal.pdf

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