1. Probability
Definitions of probability
Classical definition: $\frac{m}{n}$ assumes finite outcomes Frequentist definition: $ ^{lim} _{n \to \infty} \frac{m}{n} $ Bayesian definition: Subjective belief Axiomatic definition (Kolmogorov)
- Axiom 1: $P(A) \geq 0$
- Axiom 2: $P(S) = 1$
- Axiom 3: $P(A \cup B) = P(A) + P(B)$ for any two mutually exclusive events A and B defined over S
- Axiom 4: If $A_i \cap A_j = \null$ for every $i \neq j$, $P(U^\infty _{i=1} A_i) = \Sigma ^\infty _{i=1} P(A_i)$
- Only when S has an infinite number of members
Terminology
Experiment: A process by which an observation is made
Event: One outcome of an experiment
Sample space: a set of elements that include all possible outcomes of an experiment
Sample point: an element in a sample space
Discrete sample space: a sample that contains a finite number of sample points
Continuous sample space: a sample space that contains an infinite and uncountable number of sample points
Probability: any number assigned to each event that satisfy the axioms