Classification, Parameter Estimation and State Estimation, 2nd Edition

CProbability Theory

This appendix summarizes concepts from probability theory. This summary only concerns those concepts that are part of the mathematical background required for understanding this book. Mathematical peculiarities that are not relevant here are omitted. At the end of the appendix references to detailed treatments are given.

C.1 Probability Theory and Random Variables

The axiomatic development of probability involves the definitions of three concepts. Taken together these concepts are called an experiment. The three concepts are:

1. A set Ω consisting of outcomes ω_i. A trial is the act of randomly drawing a single outcome. Hence, each trial produces one ω ∈ Ω.
2. A is a set of certain¹ subsets of Ω. Each subset α ∈ A is called an event. The event {ω_i}, which consists of a single outcome, is called an elementary event. The set Ω is called the certain event. The empty subset Ø is called the impossible event. We say that an event α occurred if the outcome ω of a trial is contained in α, that is if ω ∈ α.

3. A real function P(α) is defined on A. This function, called probability, satisfies the following axioms:

   I: P(α) ≥ 0

   II: P(Ω) = 1

   III: If α, β ∈ A and α∩β = Ø then P(α∪β) = P(α) + P(β)

Example The space of outcomes corresponding to the colours of a traffic light is Ω = {red, green, yellow}. The set A may consist of subsets like Ø, red, green, yellow, red ∪ green, red ∩ green, red ∪ green ∪ yellow, etc. Thus, P(green) is the probability that the light will be green. ...