Foundation of Probability Theory Review of Set Theory Review of Set Theory Definition 6.Union The union of A and B,A U B,is the set of all basic outcomes in S that belong to either A or B. The union of A and B occurs if and only if either A or B(or both)occurs. AUB The union of A and B is also called the B logical sum. Foundation of Probability Theory Introduction to Statistics and Econometrics May22,2019 26/248
Foundation of Probability Theory Introduction to Statistics and Econometrics May 22, 2019 26/248 Review of Set Theory Foundation of Probability Theory Review of Set Theory Definition 6. [ Union ] The union of A and B, 𝐴 ∪ 𝐵, is the set of all basic outcomes in S that belong to either A or B. The union of A and B occurs if and only if either A or B (or both) occurs. The union of A and B is also called the logical sum
Foundation of Probability Theory Review of Set Theory Review of Set Theory Definition 7.Collective Exhaustiveness Suppose A1,A2,...,An are n events in the sample space S, where n is any positive integer. If U1Ai=S,then these n events are said to be collectively exhaustive. Foundation of Probability Theory Introduction to Statistics and Econometrics May22,2019 27/248
Foundation of Probability Theory Introduction to Statistics and Econometrics May 22, 2019 27/248 Review of Set Theory Foundation of Probability Theory Review of Set Theory Definition 7. [ Collective Exhaustiveness ] Suppose 𝐴1, 𝐴2, … , 𝐴𝑛 are 𝑛 events in the sample space S, where 𝑛 is any positive integer. If U𝑖=1 𝑛 𝐴𝑖 = 𝑆, then these 𝑛 events are said to be collectively exhaustive
Foundation of Probability Theory Review of Set Theory Review of Set Theory Definition 8.Complement The complement of A is the set of basic outcomes of a random experiment belonging to s but not to A,denoted as Ac. A A Complement:Ac Foundation of Probability Theory Introduction to Statistics and Econometrics May22,2019 28/248
Foundation of Probability Theory Introduction to Statistics and Econometrics May 22, 2019 28/248 Review of Set Theory Foundation of Probability Theory Review of Set Theory Definition 8. [ Complement ] The complement of A is the set of basic outcomes of a random experiment belonging to S but not to A, denoted as 𝐴 𝑐
Foundation of Probability Theory Review of Set Theory Review of Set Theory Remarks: The complement of event A is also called the negation of A. Any event A and its complement Ac are mutually exclusive and collectively exhaustive.That is, A∩Ac=0 and AUAc=S. Foundation of Probability Theory Introduction to Statistics and Econometrics May22,2019 29/248
Foundation of Probability Theory Introduction to Statistics and Econometrics May 22, 2019 29/248 Review of Set Theory Foundation of Probability Theory Review of Set Theory Remarks: ● The complement of event A is also called the negation of A. ● Any event A and its complement 𝐴 𝑐 are mutually exclusive and collectively exhaustive. That is, 𝐴 ∩ 𝐴 𝑐 = ∅ and 𝐴 ∪ 𝐴 𝑐 = 𝑆
Foundation of Probability Theory Review of Set Theory Review of Set Theory Definition 9.Difference The difference of A and B,denoted as A-B=An BC,is the set of basic outcomes in S that belong to a but not to B. A Foundation of Probability Theory Introduction to Statistics and Econometrics May22,2019 30/248
Foundation of Probability Theory Introduction to Statistics and Econometrics May 22, 2019 30/248 Review of Set Theory Foundation of Probability Theory Review of Set Theory Definition 9. [ Difference ] The difference of A and B, denoted as 𝐴 − 𝐵 = 𝐴 ∩ 𝐵 𝑐 , is the set of basic outcomes in S that belong to A but not to B