|
Size: 943
Comment: Initial commit
|
Size: 1696
Comment: Description
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 2: | Line 2: |
σ algebra uses and re-uses many common notations. See also some [[Statistics/ProbabilityNotation|probability notation]], [[Statistics/BayesianNotation|Bayesian notation]], [[Statistics/JointProbability|joint probability notation]], [[Statistics/ConditionalProbability|conditional probability notation]], [[Statistics/ExpectedValues|expected value notation]], and [[Statistics/ConditionalExpectations|conditional expectation notation]]. |
|
| Line 13: | Line 17: |
=== Subsets === |
|
| Line 16: | Line 24: |
=== 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 45: |
| The intersection of two sets is notated as ''A ⋂ B''; the union of two sets is notated as ''A ⋃ B''. | |
| Line 35: | Line 56: |
| 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. A parallel to the functional expression of probability, ''p(A)'', is '''''P''': A -> R''. |
σ Algebra Notation
σ algebra uses and re-uses many common notations.
See also some probability notation, Bayesian notation, joint probability notation, conditional probability notation, expected value notation, and conditional expectation notation.
Sets and Subsets
The maximal set, which in probability applications is the sample space, is notated as Ω.
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.
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 = ∅
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. A parallel to the functional expression of probability, p(A), is P: A -> R.