By formulating a hypothesis for the elasticity of the cumulative distribution function, Dagum derived a set of eleven frequency distributions some of which are commonly employed in water engineering. This system is called as the Dagum system or family. This chapter revisits this system, discusses its properties, and derives its individual frequency distributions.

The Pearson system of frequency distributions is based on a differential equation which satisfies certain mathematical conditions but its physical basis remains obscure. This chapter discusses this system and the distributions that are derived therefrom. Some of these distributions, such as gamma, Pearson type III and its logarithmic version, exponential, and normal and its logarithmic version, are frequently used in water engineering. This chapter presents the Pearson system and its underlying hypothesis and derives different member distributions of this system.

A wide spectrum of frequency distributions that are commonly used in hydrologic, hydraulic, environmental and water resources engineering can be derived by employing the principle of maximum entropy. Entropy maximization provides a general framework for deriving any probability distribution subject to appropriate constraints. This chapter discusses this framework and derives a number of distributions which satisfy different constraints.

The D’Addario system contains a number of distributions that are commonly used in environmental and water engineering. These distributions result from the integration of a probability generating function and a transformation function. This chapter discusses the D’Addario system and discusses the hypotheses that are used for deriving this system and the distribution members of the system.

Using a slightly different definition of distribution elasticity, Esteban proposed a system of distributions some of which are used in hydrologic, hydraulic, environmental, and water resources engineering. This chapter visits the Esteban system of distributions and discusses the hypotheses that are used for deriving distributions of this system.

A wide range of random variables occur in hydrometeorology, hydrology, geohydrology, hydraulics, and water quality engineering.These random variables are described by frequency distributions. The usual practice is to plot the data of the variable and then fit an appropriate distribution. Often more than one distribution can be adequately fitted.These distributions originate from one or the other type of system of distributions. This chapter outlines these systems and discusses the need for describing these distribution systems. The chapter is concluded with the organization of the book.

Previously we have revisited the different distribution systems. In this chapter, we will discuss one more system based on the genetic theory, i.e., Charlier type A and B curves, which are based on the fundamental hypothesis of elementary errors.

Employing the elasticity of the cumulative distribution function (CDF) F(x), Stoppa (1993) proposed a differential equation which can be used to derive a set of distributions which constitute the Stoppa system or family. Some of these distributions are quite general. This chapter revisits this system and derives its individual frequency distributions.

Systems of frequency distributions are derived by the use of Bessel functions and the method of expansions in terms of cumulants or moments. The resulting distributions may be useful in hydrologic, hydraulic, environmental, and water resources engineering. These methods are discussed in this chapter.

A wide variety of frequency distributions are used in hydrologic, hydraulic, environmental, and water resources engineering. Using a hypothesis that relates the probability density function to the cumulative distribution function and its complement, Burr derived a set of twelve distributions that exhibit different characteristics and some of these distributions are commonly used in water engineering. This paper revisits the Burr system of distributions and discusses the hypothesis that is used for deriving these distributions. Using this hypothesis, it then derives these distributions and discusses the theory of cumulative moments for deriving parameters of these distributions.

There are many frequency distributions whose cumulative distribution functions (CDFs) cannot be expressed in closed form. Examples of such distributions are normal, lognormal, gamma, Pearson type III, among others. If a distribution has a closed form CDF then its probability density function (PDF) can be easily obtained by differentiation but vice versa is not tractable. Using certain hypotheses on the relation between PDF and CDF based on empirical data, the CDFs of a large number of distributions can be derived. This chapter discusses the derivation of CDFs of such distributions many of which are frequently used in hydrologic, hydraulic, environmental, and water resources engineering.

A wide spectrum of frequency distributions, used in hydrologic, hydraulic, environmental, and water resources engineering, are derived using transformations of some basic frequency distributions. The basic distributions that have been used are normal, logistic, beta, Laplace, and other distributions, and the transformations used are logarithmic, power, and exponential. This chapter revisits the distributions obtained by transformation and transformations applied to basic distributions.