Multivariate Normal Distribution

Properties

The probability density function for a univariate normal distribution is , where –∞< x <∞; μ represents the mean of the distribution and can take on any real number value; and σ represents the standard deviation of the distribution, which can take on any positive value. This formula is extended to the multivariate distributions as . Many online sites, including Weisstein's MathWorld, cover many of the mathematical properties of this distribution.

For the bivariate distribution, p = 2, and in general, p is the number of variables. μ is a vector of size p × 1 that contains the means, and Σ is the variance covariance matrix of size p × p. In this matrix, the main diagonal elements are the variances of the p component variables, and the off diagonal elements are the covariances. For example, if we let σij represent the element in the ith row and jth column of this matrix, then σ11 would represent the variance of the first variable, σ12 is the covariance between the first and second variables, and the correlation between these two variables would be or the covariance divided by the product of the two standard deviations.

Some properties of the multivariate normal distribution are very important to consider when performing multivariate analyses. When considering a random vector X that follows a multivariate normal distribution, the following properties will hold:

1.
All linear combinations of the components of X are normally distributed.
2.
All possible subsets of the components of X will be normally distributed.
3.
Zero covariance between two components implies that those components are independent.
4.
The conditional distributions of the components are normal.

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It is important to realize that even though each univariate component of X follows the univariate normal distribution, the univariate normality of each component of some other random vector X∗ does not necessarily imply that X∗ is multivariate normal. In other words, the univariate normality of each variable is necessary but not sufficient to establish multivariate normality. Thus, a strategy of assessing multivariate normality that merely assesses each component variable separately will not be successful in detecting all deviations from multivariate normality. The relationship between the variables, numerically expressed in the variance covariance matrix, must be considered for the problem of testing multivariate normality.

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