Bivariate Distributions

Abstract Bivariate distributions play a special role in environmental science since they are the underpinnings of analyses of relationships between two random variables. These relationships include such analyses as biserial correlation between continuous and discrete random variables; measures of association in an r × c contingency table; Pearson's product–moment correlation; Kendall's tau; Spearman's rank correlation; and many others. In addition to studies of the dependency between two variables, interest may center on other analyses; for example, multivariate general (or generalized) linear modeling when two or more response variables are involved. Inferences involving two random variables rely on knowledge of their bivariate distribution. Probably the best known example is estimation and testing of the correlation coefficient in the bivariate normal distribution for two continuous random variables. There are many other distributions of importance as well, especially those relating two discrete random variables or a discrete to a continuous variable. This article describes a few of the more common distributions in environmental practice and, where appropriate, mentions the multivariate distribution of which the bivariate is a special case.