Modeling and Analysis of Compositional Data

Chapter 6Random compositions

6.1 Sample space

Random variables, random functions, stochastic processes, and other more complex random objects, are mathematical models of experiments or observational procedures which appear in practice. For simplicity, all these cases are here called random variables. Mathematical models for random variables are made of, at least, three elements: a probability space, a sample space, and a function from the probability space to the sample space that is identified with the random variable. The probability space is an abstract representation of the possible conditions in which the experiment is carried out. The sample space contains the possible outputs of the experiment. The experiment itself is a function that assigns the output of the experiment to each experimental condition in the probability space. The mathematical details are summarized in Appendix B. In practice, attention is centered in the sample space and its structure. Some examples follow.

Example 6.1 (Wind speed measurement)

Consider an experiment that consists in measuring the horizontal wind speed at a given point. Assuming that the experiment is absolutely accurate and that the wind speed is unbounded, all positive real numbers are possible outputs of the experiment. The sample space of this experiment must contain the positive real numbers, . However, other impossible outputs ...