Foundation of Probability Theory Basic Concepts of Probability Basic Concepts of Probability Example 5 Suppose to is the lowest temperature in an area,and t1 is the highest temperature of the area. Let T denotes the possible temperature of the area. Then the sample space of T is S={t∈R:to≤t≤t1} where R denotes the real line. 61381401d42 4 73g Foundation of Probability Theory Introduction to Statistics and Econometrics May22,2019 16/248
Foundation of Probability Theory Introduction to Statistics and Econometrics May 22, 2019 16/248 Example 5 Suppose 𝑡0 is the lowest temperature in an area, and 𝑡1 is the highest temperature of the area. Let T denotes the possible temperature of the area. Then the sample space of T is 𝑆 = {𝑡 ∈ ℝ ∶ 𝑡0 ≤ 𝑡 ≤ 𝑡1} where ℝ denotes the real line. Basic Concepts of Probability Foundation of Probability Theory Basic Concepts of Probability
Foundation of Probability Theory Basic Concepts of Probability Basic Concepts of Probability Remarks: A sample space s can be countable or uncountable. The distinction between a countable sample space and an uncountable sample space dictates the ways in which probabilities will be assigned. Foundation of Probability Theory Introduction to Statistics and Econometrics May22,2019 171248
Foundation of Probability Theory Introduction to Statistics and Econometrics May 22, 2019 17/248 Remarks: ● A sample space S can be countable or uncountable. ● The distinction between a countable sample space and an uncountable sample space dictates the ways in which probabilities will be assigned. Basic Concepts of Probability Foundation of Probability Theory Basic Concepts of Probability
Foundation of Probability Theory Basic Concepts of Probability Basic Concepts of Probability Definition 3.Event An event A is a collection of basic outcomes from the sample space s that share certain common features or equivalently obey certain restrictions. The event A is said to occur if the random experiment gives rise to one (and only one)of the constituent basic outcomes in A. Sample Point Event A (has two Sample Points) Sample Space A Foundation of Probability Theory Introduction to Statistics and Econometrics May22,2019 18/248
Foundation of Probability Theory Introduction to Statistics and Econometrics May 22, 2019 18/248 Basic Concepts of Probability Foundation of Probability Theory Basic Concepts of Probability Definition 3. [ Event ] An event A is a collection of basic outcomes from the sample space S that share certain common features or equivalently obey certain restrictions. The event A is said to occur if the random experiment gives rise to one (and only one) of the constituent basic outcomes in A
Foundation of Probability Theory Basic Concepts of Probability Basic Concepts of Probability Remark: Conceptually speaking an event is equivalent to a set. Universal set Sample space,S Event B Event A Foundation of Probability Theory Introduction to Statistics and Econometrics May22,2019 19/248
Foundation of Probability Theory Introduction to Statistics and Econometrics May 22, 2019 19/248 Remark: ● Conceptually speaking an event is equivalent to a set. Basic Concepts of Probability Foundation of Probability Theory Basic Concepts of Probability
Foundation of Probability Theory Basic Concepts of Probability Basic Concepts of Probability Example 6:Rolling a Die Event A is defined as "the number resulting is even". Event B is "the number resulting is at least 4". A={2,4,6 B={4,5,6} Remark:Basic outcome Event Sample space. Foundation of Probability Theory Introduction to Statistics and Econometrics May22,2019 20/248
Foundation of Probability Theory Introduction to Statistics and Econometrics May 22, 2019 20/248 Example 6: [ Rolling a Die ] Event A is defined as "the number resulting is even". Event B is "the number resulting is at least 4". Basic Concepts of Probability Foundation of Probability Theory Basic Concepts of Probability 𝐴 = {2,4,6} 𝐵 = {4,5,6} Remark: Basic outcome ⊆ Event ⊆ Sample space