Appendix A - Probability theoretical methods

Summary

In this appendix we will briefly describe probability theory and methods that will be used throughout the book. For a more complete treatment of probability theory, see any standard textbook on probability theory, such as [Fel68].

Probabilities and distributions

The main concept in probability theory is that of a probability space or an ensemble. The probability space consists of different objects or events, each of which has a measure – probability – assigned to it.

When each event is assigned a number, this number is termed a random variable. The random variable can be either discrete, in which case the measure is termed a probability distribution, or continuous, in which case the measure is termed “probability density.” The single events then have an extremely small probability of occurring and only the probability of a range of values, obtained through integration assuming that the probability density is finite.

When a finite number of events exists, say N events, the most common case is to assume they are equiprobable. Therefore, the probability of finding some property is P = M/N, where M is the number of events having the property. If P = 1 − f(N), where lim_{N → ∞}f (N) = 0, (i.e., the probability of finding the property approaches 1, when N becomes large) the property is said to hold almost always (or a.a.). In this book, when a property is said to hold, it actually means that it almost always holds.