Foundation of Probability

#ECE313

Axioms of Probability(概率公理)

Terms

• $\mathrm{\Omega }$ : Denote the sample space(样本空间)，表明所有可能结果的集合
• Event(事件): An event is a subset of the sample space $\mathrm{\Omega }$
• Outcome: denote as $\omega$ is an element of sample space
• Complement of an event(事件的补集): denote as $E^c$
• Exclusive(互斥事件) & Partition of a sample space

• The number of elements of in a set is called the cardinality of the set
  

• Experiment modelled by triplet
  

Axioms

Event Axioms

The set of events, $F$ is required to satisfy the following axioms

• 利用公理三基于已定义的事件推出新的事件（将新定义的命题进行拆解）
  Property:
  

Probability Axioms

注意：将互斥事件加和得到总概率只适用于有限集或者可数无限集，不适用于不可数无限集