Chapter3_Stats (2).pdf - Chapter 2 Random Variables and Probability Distributions By Joan Llull Probability and Statistics QEM Erasmus Mundus Master

Chapter3_Stats (2).pdf - Chapter 2 Random Variables and...

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Chapter 2: Random Variables and Probability Distributions By Joan Llull * Probability and Statistics. QEM Erasmus Mundus Master. Fall 2015 Main references: — Mood: I.3.2-5; II; III.2.1, III.2.4, III.3.1-2; V.5.1; Appendix A.2.2, A.2.4 — Lindgren: 1.2; 3.1 to 3.3, 3.5; 4.1 to 4.6, 4.9, 4.10; 6.1, 6.5, 6.8; (3.2) I. Preliminaries: An Introduction to Set Theory We start this chapter with the introduction of some tools that we are going to use throughout this course (and you will use in subsequent courses). First, we introduce some definitions, and then describe some operators and properties of these operators. Consider a collection of objects, including all objects under consideration in a given discussion. Each object in our collection is an element or a point. The totality of all these elements is called the space , also known as the universe, or the universal set, and is denoted by Ω. We denote an element of the set Ω by ω . For example, a set can be all the citizens of a country, or all the points in a plane (i.e. Ω = R 2 , and ω = ( x, y ) for any pair of real numbers x and y ). A partition of the space Ω is called a set , which we denote by calligraphic capital Latin letters, with or without subscripts. When we opt for the second, we define the catalog of all possible incides as the index set , which we denote by Λ (for example, if we consider the sets A 1 , A 2 , and A 3 , then Λ = { 1 , 2 , 3 } . To express that an element ω is part of a set A , we write w ∈ A , and to state the opposite, we write w / ∈ A . We can define sets by explicitly specifying all its elements (e.g. A = { 1 , 2 , 3 , 4 , 5 , 6 } ), or implicitly, by specifying properties that describe its elements (e.g. A = { ( x, y ) : x R , y R + } ). The set that includes no elements is called the empty set , and is denoted by . Now we define a list of operators for sets: Subset : when all elements of a set A are also elements of a set B we say that A is a subset of B , denoted by A ⊂ B (“ A is contained in B ”) or B ⊃ A * Departament d’Economia i Hist` oria Econ` omica. Universitat Aut` onoma de Barcelona. Facultat d’Economia, Edifici B, Campus de Bellaterra, 08193, Cerdanyola del Vall` es, Barcelona (Spain). E-mail: joan.llull[at]movebarcelona[dot]eu. URL: . 1
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(“ B contains A ”). Equivalent set: two sets A and B are equivalent or equal, denoted A = B if A ⊂ B and B ⊂ A . Union : the set that consists of all points that are either in A , in B , or in both A and B is defined to be the union between A and B , and is denoted by A ∪ B . More generally, let Λ be an index set, and {A λ } ≡ { A λ : λ Λ } , a collection of subsets of Ω indexed by Λ. The set that consists of all points that belong to A λ for at least one λ Λ is called the union of the sets {A λ } , denoted by λ Λ A λ . If Λ = , we define λ A λ .
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