|
Size: 943
Comment: Initial commit
|
Size: 1950
Comment: Some details and structure
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 2: | Line 2: |
σ algebra uses and re-uses many common statistics [[Statistics/ProbabilityNotation|notations]]. |
|
| Line 13: | Line 15: |
| The sample space could be a discrete set, like ''Ω = {heads, tails}''. It could be a set of discrete numbers, like ''Ω = '''N''''' (all real numbers). It could be a continuous range, like ''Ω = [0,1]''. === Subsets === |
|
| Line 16: | Line 24: |
=== Power sets === The power set of a set (''P(Ω)'') is the set of all subsets, including the empty set (''∅'') and the set itself (''Ω''). This becomes analagous to a probability function in descrete cases. === Intersections and Unions === The '''intersection''' of two sets is notated as ''A ⋂ B''; the '''union''' of two sets is notated as ''A ⋃ B''. The intersection of all subsets ''A,,i,,'' can be expressed as: {{attachment:intersection.svg}} The union of all subsets ''A,,i,,'' can be expressed as: {{attachment:union.svg}} |
|
| Line 23: | Line 53: |
| The intersection of two sets is notated as ''A ⋂ B''; the union of two sets is notated as ''A ⋃ B''. | A pair of sets are '''disjoint''' if there is no intersection, which is expressed as ''A ⋂ B = ∅'' |
| Line 25: | Line 55: |
| A pair of sets are '''disjoint''' if there is no intersection, which is expressed as ''A ⋂ B = ∅'' | ---- == Sigma Algebras == |
| Line 35: | Line 69: |
| A map translates a (sub)set into a real number. This can be expressed as '''''P''': Ω -> R''. | A map translates a (sub)set into a real number: '''''M''': A -> '''R'''''. === Probability Measures === '''Probability measures''' are the primary use of maps. A parallel to the functional expression of probability, ''p(A)'', is '''''P''': A -> [0,1]''. |
σ Algebra Notation
σ algebra uses and re-uses many common statistics notations.
Contents
Sets and Subsets
The maximal set, which in probability applications is the sample space, is notated as Ω.
The sample space could be a discrete set, like Ω = {heads, tails}. It could be a set of discrete numbers, like Ω = N (all real numbers). It could be a continuous range, like Ω = [0,1].
Subsets
Subsets are usually named with calligraphic uppercase letters, but that's not exactly practical in typed notes. Capital letters will be used instead.
A subset of Ω is expressed as A ⊆ Ω.
Power sets
The power set of a set (P(Ω)) is the set of all subsets, including the empty set (∅) and the set itself (Ω).
This becomes analagous to a probability function in descrete cases.
Intersections and Unions
The intersection of two sets is notated as A ⋂ B; the union of two sets is notated as A ⋃ B.
The intersection of all subsets Ai can be expressed as:
The union of all subsets Ai can be expressed as:
Properties
A pair of sets are disjoint if there is no intersection, which is expressed as A ⋂ B = ∅
Sigma Algebras
Maps
Maps are usually named with blackboard bold letters, but that's not exactly practical in typed notes. Bold capital letters will be used instead.
A map translates a (sub)set into a real number: M: A -> R.
Probability Measures
Probability measures are the primary use of maps.
A parallel to the functional expression of probability, p(A), is P: A -> [0,1].