Multi-dimensional distribution

multivariate distribution

A probability distribution on the -algebra of Borel sets of an -dimensional Euclidean space . One usually speaks of a multivariate distribution as the distribution of a multi-dimensional random variable, or random vector, , meaning by this the joint distribution of the real random variables given on the same space of elementary events ( may be regarded as coordinate variables in the space ). A multivariate distribution is uniquely determined by its distribution function — the function

of the real variables .

As in the one-dimensional case, the most widespread multivariate distributions are the discrete and the absolutely-continuous distributions. In the discrete case a multivariate distribution is concentrated on a finite or countable set of points of such that

(see, for example, Multinomial distribution). In the absolutely-continuous case almost-everywhere (with respect to Lebesgue measure) on ,

where is the density of the multivariate distribution:

for any from the -algebra of Borel subsets of , and

The distribution of any random variable (and also, for any , the distribution of the variables ) relative to a multivariate distribution is called a marginal distribution. The marginal distributions are completely determined by the given multivariate distribution. When are independent, then

and

where and are, respectively, the marginal distribution functions and densities of the .

The mathematical expectation of any function of is defined by the integral of this function with respect to the multivariate distribution; in particular, in the absolutely-continuous case it is defined by the integral

The characteristic function of a multivariate distribution is the function of given by

where . The fundamental characteristics of a multivariate distribution are the moments (cf. Moment): the mixed moments and the central mixed moments , where is the order of the corresponding moment. The roles of the expectation and the variance for a multivariate distribution are played by and the set of second-order central mixed moments, which form the covariance matrix. If for all , , then are called pairwise uncorrelated or orthogonal (the covariance matrix is diagonal). If the rank of the covariance matrix is less than , then the multivariate distribution is called a degenerate distribution; in this case the distribution is concentrated on some linear manifold in of dimension .

For methods of investigating dependencies between see Correlation; Regression.

References

[a1]	N.L. Johnson, S. Kotz, "Discrete distributions" , Houghton Mifflin (1969)
[a2]	N.L. Johnson, S. Kotz, "Continuous multivariate distributions" , Wiley (1942)