Week 1 Flashcards
What is the difference between Deterministic and Random processes?
Deterministic: Outcome can be predicted exactly in advance.
Random: Outcome not known exactly, but can describe probability distribution of outcome.
What is a random experiment?
A process that produces uncertain outcomes from a well-defined set of possible outcomes.
E.g: Tossing a coin, drawing a card.
What is a sample space?
What are the possible sizes of a sample space?
Set of all possible outcomes of a random experiment, with any individual outcome being a sample point.
Denoted by omega.
Can be finite, countably infinite or infinite.
What is an event, in terms of a sample space?
An event is a subset of the sample space.
Event A occurs if random experiment outcome is a member of A.
What is an event space?
Collection of all events, denoted by F.
What are three possible operations on events?
Complement: The sample points that do not occur in an Event A.
Union: The sample points in A, B or both.
Intersection: The sample points in A and B.
What is a partition?
A collection of sets {A1 … An} is a partition of a universal set omega iff:
- ) Non-Overlap: {A1 … An} is disjoint (no shared points between sets).
- ) Decompose: Union of all sets {A1 … An} is omega.
How is area used as a proxy for probability?
The area of omega = 1.
The area of an event = probability of that event.
What is a Probability Law?
A function P: F -> [0, 1] that maps an event A to a real number in [0, 1].
What are the Kolmogorov axioms which a Probability Law satisfies?
Non-Negativity: P(A) >= 0 for any A in F.
Unit Measure: P(omega) = 1.
Additivity of Disjoint Events: If two regions in A1, A2.. do not overlap then the area of the combined region is the sum of the area of each region.
What are the components of a Probability Space?
Triplet consisting of (Sample space, Event space, Probability law).
What are three properties of Probability Laws?
- ) If A c B then P(A) <= P(B).
- ) P(A u B) = P(A) + P(B) - P (A n B).
- ) P(A^c) = 1 - P (A).
