slide2 - ECON321 Econometrics Lecture 2 Probability Theory...

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Unformatted text preview: ECON321 : Econometrics Lecture 2 : Probability Theory Sasan Bakhtiari University of Maryland, College Park Summer 2007 Session II, Set and Set Operations A set is a collection of objects : ◮ Set of books taught in ECON321 = { Devore,McClave } , ◮ Set of natural numbers less than 6 = { 1,2,3,4,5 } , ◮ Set of non-positive integers = { 0,-1,-2,-3,-4, . . . } . Some notation ◮ Ω : Universal set - set of all sample space. ◮ ∅ : Empty set - the set with no members. ◮ a ∈ A : membership - a is a member of A . ◮ A ⊂ B : subset - A is a subset of B , i.e., every member of A belongs to B too (but not necessarily vice versa) Subsets ◮ A = {− 1 , − 5 , − 101 } and B = { , − 1 , − 2 , . . . } then A ⊂ B . ◮ A = { 1 , 2 , 6 } and B = { 1 , 2 , 3 , 4 , 5 } , then A not a subset of B . ◮ If A ⊂ B and B ⊂ A then A = B Venn Diagrams Visualizing Sets Ω Ω A Set Operations Complement of a Set A c = { ω ∈ Ω : ω / ∈ A } . ◮ Example: Ω = { 1 , 2 , 3 , 4 , 5 , 6 } , A = { 1 , 5 , 6 } then A c = { 2 , 3 , 4 } . ◮ Venn diagram A A c Ω Set Operations Union of two sets A uniontext B = { ω ∈ Ω : ω ∈ A OR ω ∈ B } . ◮ Example: A = { 1 , 5 , 6 } , B = {− 1 , 1 , 2 , 6 } then A uniontext B = {− 1 , 1 , 2 , 5 , 6 } . ◮ Venn diagram Ω B A Set Operations Intersect of two sets A intersectiontext B = { ω ∈ Ω : ω ∈ A AND ω ∈ B } . ◮ Example: A = { 1 , 5 , 6 } , B = {− 1 , 1 , 2 , 6 } then A intersectiontext B = { 1 , 6 } . ◮ Venn diagram B A Ω Definition Disjoint Sets If A intersectiontext B = ∅ then sets A and B are called disjoint . Ω B A Properties ◮ ( A c ) c = A , ◮ Commutative Property A uniondisplay B = B uniondisplay A , A intersectiondisplay B = B intersectiondisplay A , ◮ Associative Property ( A uniondisplay B ) uniondisplay C = A uniondisplay ( B uniondisplay C ) , ( A intersectiondisplay B ) intersectiondisplay C = A intersectiondisplay ( B intersectiondisplay C ) , ◮ Distributive Property ( A uniondisplay B ) intersectiondisplay C = ( A intersectiondisplay C ) uniondisplay ( B intersectiondisplay C ) , ( A intersectiondisplay B ) uniondisplay C = ( A uniondisplay C ) intersectiondisplay ( B uniondisplay C ) , ◮ Remember : A uniontext A c = Ω , A intersectiontext A c = ∅ . More Properties De Morgan’s Law ( A uniondisplay B ) c = A c intersectiondisplay B c ( A intersectiondisplay B ) c = A c uniondisplay B c ....
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slide2 - ECON321 Econometrics Lecture 2 Probability Theory...

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