Continuous Random Variables: Probability Distributions and Their Applications in Hydrology

Abstract

The probability distributions covered in this chapter refer to models of continuous random variables. Emphasis is given to the models that are generally employed in frequency analysis of hydrologic continuous random variables. Following the formal description of each model, the reader will find, in most cases, a brief example of its application in hydrology. The list of models detailed in this chapter is not exhaustive, as it does not include all distributions that have possible uses in Statistical Hydrology. However, the list includes other models that are not currently employed in the frequency analysis of hydrologic continuous random variables but are key elements in setting out the foundations of statistical inference, such as the sampling distributions, as well as other generally useful models, like the uniform and beta distributions. Throughout the chapter, the focus is deliberately kept on describing the models, their main shape characteristics, and usual applications, and not on systematically providing mathematical proofs for expected values and other population descriptors. By the end of the chapter, the normal bivariate model and the principles for building copulas, for describing the dependence structure between variables, are introduced, as an illustration of multivariate probability distributions.