Unit 1: Probability models and axioms
1. Lec. 1: Probability models and axioms
1.1. Sample Space
- Definition: the set of all of possible outcomes is called the sample space of the experiment,. Denoted by
- A subset of the sample space is called event. Probability is assigned to events.
- sample space can be finite, infinite, continuous, etc.
- The construction of a sample space is a description of the possible outcomes of a probabilistic experiment.
- discrete/finite example:
- Two rolls of a tetrahedral die.
- roll twice, get 16 outcomes. (the order sometimes matters)
- continuous example:
- (x, y) such that : a square box.
1.2. Probability axioms
- (Finite) additivity:
- denote empty set.
- means A and B are disjoint events.
- A, B and C are disjoint events.
- and similarly for k disjoint events.
1.3. Discrete Models
- Example: Coin tosses. Consider an experiment involving a single coin toss. There are two possible outcomes, heads (H) and tails (T). The sample space is , and the events are
Discrete Probability Law
- the probability of any event is the sum of the probabilities of its elements:
Discrete Uniform Law
- Assume consists of n equals likely elements
- Then the probability of any event A is given by
1.4. Continuous Models
- Example: (x, y) such that
- Because it's a right triangle with length 1/2 of each side.
- Because it's the area of a single point.
1.5. Countable additivity
Probability calculation: discrete but infinite sample space
- Keep tossing a coin and the outcome is the number of tosses until we observe heads for the first time.
- Legitimate check:
- To support this calculation, we'll need:
Countable Additivity Axiom
- If is an infinite sequence of disjoint events, then
- Notice the events should be sequence, like the example above.
- Example (x, y) such that can not be arranged in a sequence. and the elements are NOT countable.
- Additivity holds only for "countable" sequences of events.
Countable and uncountable sets
- Countable: can be put in 1-1 correspondence with positive integers
- positive integers:
- pairs of positive integers
- rational numbers q with 0 < q < 1
- the interval [0, 1]
- the reals, the plane...
1.6. Interpretations of probability theory
- A framework for analyzing phenomena with uncertain outcomes
- Rules for consistent reasoning
- Used for predictions and decisions
- tetrahedral [,tetrə'hi:drəl, -'he-] adj. 四面体的；有四面的
- axiom ['æksiəm] n. [数] 公理；格言；自明之理
- discrete [dis'kri:t] adj. 离散的，不连续的
- legitimate [li'dʒitimət, li'dʒitimeit] adj. 合法的；正当的；合理的；正统的 vt. 使合法；认为正当（等于legitimize）
- additivity [,ædi'tivəti] n. 添加；相加性
- unit square [数] 单位平方形