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Probability space (changes)

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Definition Definitions

A probability space consists of the following data:

  • The sample space SS , which is the set of possible outcomes (of an experiment) 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 also hence set differences). Implied here is the fact that the empty set and whole sample space are events inAA.

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