ABSTRACT
This chapter is a short review of the basics of probability that are needed to discuss random sets in subsequent chapters. For more details, see the Appendix.
Gathering information for decision making is frequent and essential in human activities. Rather than being complete, information is in general uncertain in many respects. Throughout this course we will encounter different types of uncertainty, but first, let us start out with a familiar type of uncertainty, namely randomness. After all, the analysis of randomness will serve as a guideline for studying other types of uncertainty. Below is a simple example of information gathering. Suppose that we are interested in the annual income of individuals in the
population of Las Cruces, say, in 2004. Suppose that, for some reasons (e.g., costs and time), we are unable to conduct a census (i.e., a complete enumeration) throughout the entire population, and hence we can only rely on the information obtained from a small part of that population, i.e., from a sample of that population. Suppose that the physical population of individuals is identified as a finite set U = {u1, . . . , uN}, where N is the population size. A sample in U is a subset of U . Our variable of interest is θ, the annual income. We use θ(uk) to denote the annual income of the individual uk. Thus, θ is a map from U to R, the set of real numbers. To obtain partial knowledge about θ, we are going to conduct a sampling survey, i.e., to select a sample A from U . Then, from the knowledge of the map θ on A, i.e., the values θ(u), u ∈ A, we wish to “guess” or estimate θ, or some function of it, e.g., the population total
T (θ) = ∑ u∈U
θ(u)
by ∑ u∈A
θ(u).
