Probability Space
A probability space consists of a -algebra on and a countably additive with , so for pairwise disjoint in .
Definition 1.1 (Probability Space)
Suppose is a set and is a collection of subsets of . Then we call a -algebra if
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[ the whole space is in . ]
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If , then [ closed under complements. ]
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If is a countable collection of sets in , then [ closed under countable unions. ]
Let be a -algebra on . A function
is called a probability measure if
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Countable additivity. For any countable disjoint collection of sets in , we have
We say to be the probability of the event .
We call a probability space.
Examples
Rolling a fair die is modelled by , and for every .
Picking of labelled balls at random without replacement is modelled by , so and for each outcome .
Related
Stated in
- Definition 1.1 (Probability Space)§1.1 Probability Spaces