D내문서학부확률및통계1확통1AncillaryLectureNoteonEssentialProbabilityTheory.pdf

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Essential Probability Theory 1
Probability theory : Terminology Sample space 𝛀 : 𝛀 is the set of all possible outcomes 𝑒 of an experiment. Example 1. In tossing of a die we have 𝛀 = {1, 2, 3, 4, 5, 6}. Example 2. The life-time of a bulb 𝛀 = {? ∈ 𝑅 | ? > 0}. Event: An event is a subset of the sample space 𝛀 . An event is usually denoted by a capital letter ?, ?, ⋯ . If the outcome of an experiment is a member of event ? , we say that ? has occurred. An impossible event is an empty subset of 𝛀 . Example 1. The outcome of tossing a die is an even number: ? = {2, 4, 6} ⊂ 𝛀. Example 2. The life-time of a bulb is at least 3000 hours: ? = {? ∈ 𝑅 | ? > 3000} ⊂ 𝛀. Probability theory is a subset of measure theory. Probability makes extensive use of set operations, so let us introduce the more relevant notations. 2
Algebra of events (Boolean Algebra) Union: ? or ? ”. ? ∪ ? = {𝑒 ∈ 𝛀 | 𝑒 ∈ ? or 𝑒 ∈ ?} Intersection: (joint event) “A and B”. ? ∩ ? = {𝑒 ∈ 𝛀 | 𝑒 ∈ ? and 𝑒 ∈ ?} Events ? and ? are mutually exclusive , if ? ∩ ? = ∅ . Complement: “not ? ”. ? = ? 𝑐 = {𝑒 ∈ 𝛀 | 𝑒 ∉ ?} Partition of the sample space A set of events ? 1 , ? 2 , ⋯ is a partition of the sample space 𝛀 if 1. The events are mutually exclusive, ? ? ∩ ? ? = ∅ , when ? ≠ ? . 2. Together they cover the whole sample space, ڂ ? ? ? = 𝛀 . 3
Algebra of events (continued) In many ways the algebra of events differs from the algebra of numbers, as some of the identities below indicate. ? ∪ ? = ?, ? ∪ ? 𝑐 = 𝛀, ? ∩ ? = ?, ? ∩ ? 𝑐 = ∅ ? ∩ 𝛀 = A, A ∪ ∅ = ?, ? ∪ 𝛀 = 𝛀, ? ∩ ∅ = ∅ A method of verifying relations among events involves algebraic manipulation, using the identities. Four examples are given below; in working out the identities, it may be helpful to write ? ∪ ? as ? + ? and ? ∩ ? as ??. 1. ? ∪ ? ∩ ? = ? Proof . ? + ?? = ?𝛀 + ?? = ? 𝛀 + ? = ?𝛀 = ? 2. ? ∪ ? ? ∪ ? = ? ∪ ? ∩ ? Proof . ? + ? ? + ? = ? + ? ? + ? + ? ? = ?? + ?? + ?? + ?? note ?? = ?? = ? + ?? + ?? + ?? = ? 𝛀 + ? + ? + ?? = ?𝛀 + ?? = ? + ?? 3. ? ∪ ? ∩ ? 𝑐 = 𝛀 Proof . ? + (??) 𝑐 = ? + ? 𝑐 + ? 𝑐 = ? + ? 𝑐 + ? 𝑐 = 𝛀 + ? 𝑐 = 𝛀 4. ? ∩ ? 𝑐 ? ∩ ? ? 𝑐 ∩ ? = ? ∪ ? Proof . ?? 𝑐 + ?? + ? 𝑐 ? = ?? 𝑐 + ?? + ?? + ? 𝑐 ? = ? ? 𝑐 + ? + ? + ? 𝑐 ? = ?𝛀 + 𝛀? = ? + ? 4
Algebra of events (continued) Example: Let 𝛀 be the set of nonnegative real numbers. Let ? 𝑛 = 0,1 − 1 𝑛 = ? ∈ 𝛀 ∶ 0 ≤ ? ≤ 1 − 1 𝑛 𝑛 = 1,2, … Then, 𝑛=1 ? 𝑛 = 0,1 = ?: 0 ≤ ? < 1 , 𝑛=1 ? 𝑛 = {0} As an illustration of the DeMorgan laws, 𝑛=1 ? 𝑛 𝑐 = [0,1) 𝑐 = 1, ∞ = ?: ? ≥ 1 , 𝑛=1 ? 𝑛 𝑐 = ሩ 𝑛=1 1 − 1 𝑛 , ∞ = 1, ∞ (Notice that ? > 1 − 1/𝑛 for all 𝑛 = 1,2, … ⇔ ? ≥ 1). Also 𝑛=1 ? 𝑛 𝑐 = {0} 𝑐 = 0, ∞ = ?: ? > 0 , 𝑛=1 ? 𝑛 𝑐 = ራ 𝑛=1 1 − 1 𝑛 , ∞ = (0, ∞) 5
Probability model Probability model : A probabilistic model is a mathematical description of an uncertain situation. A probability model has three components: A sample space 𝛀 , which is the set of all possible outcomes of the experiment that is modelled by the probability model; A family of sets 𝓕 representing the class of allowable events , where each set is a subset of the sample space 𝛀 ; and A probability function 𝑃: 𝓕 ⟼ 𝓡[0,1] which, also called as a probability measure , satisfies some probability axioms given in the next slide. Definition : A probability space is a triplet ( 𝛀, 𝓕, 𝑃 ). We can also consider more general measures 𝜇 , not just probability measures, in which case ( 𝛀, 𝓕, 𝜇 ) is called a measure space . You probably have some familiarity with 𝛀 and 𝑃 , but not necessarily with 𝓕 because you may not have been aware of the need to restrict the class (set) of events you could consider. In a given problem, there will be a particular class of subsets of 𝛀 called as class of events .