The Azimuth Project
Probability space

Contents

Definitions

A probability space consists of the following data:

  • The sample space SS, which is the set of possible outcomes (of an experiment.

  • The event algebra AA, where each event consists of a set of outcomes in SS, and the collection of events constitutes a σ\sigma-algebra – it is closed under countable sequences of union, intersection and complement operations (and hence set differences). Implied here is that the empty set and whole sample space are events in AA.

  • A measure function PP, which assigns a probability to each event in AA. PP must be additive on countable disjoint unions, and must assign 1 to the whole sample space SS.

A random variable is a function XX from the sample space S into a range space VV, which is measurable, which means: there is a σ\sigma-algebra of subsets of VV, and the inverse image of every such subset under the function XX is an event in AA.