Multivariate Distributions

Abstract

When analysing statistical data, it is often necessary to work with models, which involve several random variables . This is obviously the case if the purpose of the analysis is to draw inference about the relationship between two or more variables in a population from their observed values in a sample drawn from the population. Also when discussing the structure of statistical models, the simultaneous distribution of several random variables is an important concept. Assume that the data consist of the n observations x₁,…,x_n. For theoretical reasons, x₁,…,x_n are regarded as the observed or realized values of n random variables X₁,…,X_n The statistical model for such data is a specification of the joint or simultaneous distribution of X₁,…,X_n For example, suppose that x₁,…,x_n form a random sample from a population or from a distribution, the mean of which is to be estimated using the sample mean ${\rm{\bar x}} = ({{\rm{x}}_1} + \ldots + {{\rm{x}}_{\rm{n}}})/{\rm{n}}$. From knowledge of the simultaneous distribution of X₁,…,X_n, it is possible to derive the distribution of the random variable ${\rm{\bar X}} = ({{\rm{X}}_1} + \ldots + {{\rm{X}}_{\rm{n}}})/{\rm{n}}$. Using this distribution, it is then possible to formulate statements concerning the precision of the estimate.