Probability Models And Axioms
Sample Space
- Sample space is a list (set) of possible outcomes, \(\Omega\)
- Event: a subset of the sample space. Probability is assigned to events.
Probability Axioms
- Nonnegativity: \(P(A) \ge 0\)
- Normalization: \(P(\Omega) = 1\)
- (Finite) additivity: (to be strengthened later)
If \(A \cap B = \varnothing\), then \(P(A \cup B) = P(A) + P(B)\)
\(A \cap B\) read as \(A\) intersects \(B\), \(A \cup B\) read as \(A\) unions \(B\)
- Countable Additivity Axiom:
If \(A_1, A_2, A_3, \dots\) is an infinite sequence of disjoint events, then \(P(A_1 \cup A_2 \cup A_3 \cup \cdots) = P(A_1) + P(A_2) + P(A_3) + \dots\)