Appendix A review of probability theory

This appendix provides a review of the basic probability theory used in this book. In addition, more specialized topics in probability theory, will appear in the relevant chapters as they are needed. It is expected that most readers will be at least somewhat familiar with this material, so the pace is fairly rapid, with few examples or derivations.

A.1 Sample spaces and probability measures

We model the results of a random experiment by a set Ω, known as a sample space, where each point of Ω corresponds to a possible outcome. For example, if we throw a pair of dice, the sample space could consist of the 36 ordered pairs (a, b) where a and b take values from 1 to 6. A combination of outcomes, known as an event, is represented by a subset of Ω. Such an event occurs if any of the outcomes in the subset occur. In the above example, the event that the total on the dice is 10, would be represented by the set {(4, 6), (5, 5), (6, 4)}. From a given collection of events, familiar set operations can be applied to build other events. The union of events, denoted with the symbol ∪, gives us the event that occurs if any one of the given events occur. The intersection of events, denoted with the symbol ∩, gives us the event that occurs if all of the given events occurs. The complement of an event A, denoted by Ac, gives us the event that occurs if A does not occur. Two events are said to be mutually exclusive if the occurrence of one means that the ...

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