Probability
Dependencies:
Read sections 1.1, 1.2, 1.3 and 1.4 from [prob-and-rand-proc] for an intuitive understanding and to know the definitions of 'event', 'sample space', 'probability' and 'probability measure'. You may optionally read chapter 0 too.
A probability space is a triple
is the sample space, also called the set of all outcomes. is a -algebra over . is called the set of all events. is a measure over such that . is called a probability measure.
One can usually let
Additional properties
These simple properties can be proven using the definition of probability space above:
- Let
be a countable set of events. Then . - Let
be an event. Define . Then (since and are disjoint and ). . (because and are disjoint). .
References
prob-and-rand-proc
Probability and Random Processes
ORF 309 / MAT 380 Lecture Notes, Princeton University, 2016-02-22
Probability and Random Processes
ORF 309 / MAT 380 Lecture Notes, Princeton University, 2016-02-22
Dependency for:
- Linearity of expectation
- Random variable
- Independence of random variables (incomplete)
- Conditional expectation
- Expected value of a random variable
- Conditional variance
- Conditional probability (incomplete)
- Independence of composite events
Info:
- Depth: 3
- Number of transitive dependencies: 5
Transitive dependencies:
- /sets-and-relations/countable-set
- /sets-and-relations/de-morgan-laws
- σ-algebra
- Measure
- σ-algebra is closed under countable intersections