Conditional Probability and Expectation
We begin the chapter by exploring some relations between measurements, that is, measurable functions, and events, that is, members of a σ-field. In particular, we explore the relation between knowledge of the value of a particular measurement or class of measurements and knowledge of an outcome of a particular event or class of events. Mathematically these are relations between classes of functions and σ-fields. Such relations will be useful in developing properties of certain special functions such as limiting sample averages arising in the study of ergodic properties of information sources. In addition, they are fundamental to the development and interpretation of conditional probability and conditional expectation, that is, probabilities and expectations when we are given partial knowledge about the outcome of an experiment.
KeywordsProbability Measure Conditional Probability Conditional Expectation Output Event Descriptive Definition
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