Random Processes: Filtering, Estimation, and Detection

1 Experiments and Probability

 

1.1 DEFINITION OF AN EXPERIMENT

To fully appreciate the meaning of probability and acquire a strong mathematical foundation for analytical work, it is necessary to define precisely the concept of an experiment and sample space mathematically. These definitions provide consistent methods for the assignment of elementary probabilities in paradoxical situations, and thus allow for meaningful calculation of probabilities of events other than the elementary events. Although at the beginning this approach may seem stilted, it will lead to a concrete concept of probability and an interpretation of derived probabilities.

An experiment is specified by the three tuple (S, , (.)), where S is a finite, countable, or noncountable set called the sample space, is a Borel field specifying a set of events, and (.) is a probability measure allowing calculation of probabilities of all events.

 

1.1.1 The Sample Space

The sample space S is a set of elements called outcomes ...