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Outline Outline � Definitions � Definitions � Set Operations � Set Operations � Probability Space � Probability Space � Borel � Borel Field Field � Probability Experiment � Probability Experiment ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Subset: b Subset: b is a subset of is a subset of a a if all element of if all element of Set is a collection of elements Set is a collection of elements b are element of are element of a a ) ) b Subset b Subset b of of a a (all element of (all element of b b are element of are element of a a ) ) Space S Space S – – largest set largest set c ⊂ b ⊂ c ⊂ a Null Set O O – – empty set empty set If a & b Null Set If & a = a ⊂ b ⊂ Equality Equality b iff b & a iff & a a c c b b S S ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 1

Elements of union of Elements of union of sets sets a a and and b b are are Elements of intersection of Elements of intersection of sets sets a a and and b b elements of a a or or b b or both or both are elements of both a a and and b. b. elements of are elements of both ∩ = ∩ = ∩ a b b a a a a ∪ = ∪ ∪ = a b b a a a a ∩ 0 = ∩ = a 0 a S a ∪ 0 = ∪ = ( ) ( ) ∪ = ∩ ∩ = ∩ ∩ = ∩ ∩ a a a S S a S S a b c a b c a b c ( ) ( ) ( ) ∩ ∪ = ∩ ∪ ∩ ( ) ( ) a b c a b a c ∪ ∪ = ∪ ∪ = ∪ ∪ a b c a b c a b c b ⊂ ∩ = a b a b If If ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Mutually exclusive sets have no Mutually exclusive sets have no Elements of complement of set Elements of complement of set a a are are common element . . common element elements of S S which are not in which are not in a. a. elements of ∩ b = ∪ = ∩ a = a 0 a a S a 0 = = 0 S S 0 Sets a a 1 , a 2 … are mutually exclusive if Sets 1 , a 2 … are mutually exclusive if b ⊂ b ⊃ a If ∩ = ≠ If a a a 0 for i j i j ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 2

Elements of Elements of a a - - b b are elements of are elements of a a that that ∪ = ∩ are not in b. b. are not in a b a b − = ∩ = − ∩ a b a b a a b ∩ = ∪ ( ) ( ) a b a b = − ∪ ∩ a a b a b ( ) − ∪ = = − a a a a a S a ∪ − = a a a 0 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Random Experiment ℑ ℑ Random Experiment Summary Summary By an experiment ℑ ℑ we mean a set (space) we mean a set (space) S S By an experiment of outcomes ξ ξ . Elements of . Elements of S S are outcomes are outcomes of outcomes or elementary events elementary events . . S S is a probability is a probability or (sample) space. Subsets of S S are called are called (sample) space. Subsets of events . Space . Space S S is the is the sure (certain) sure (certain) event event . . events Empty set O O is the is the impossible event impossible event . . Empty set ∩ b = Mutually Exclusive Events Mutually Exclusive Events a 0 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 3

Axioms of Probability Axioms of Probability Corollaries Corollaries To each event To each event a, a, a measure a measure P(a P(a) ) is assigned is assigned ( ) 0 = P 0 subject to the following axioms subject to the following axioms ( ) 1 ( ) ( ) ≥ = − ≤ i ) P a 0 P a 1 P a ( ) ( ) ( ) ( ) ( ) ∩ b ≠ = ∪ = + − ∩ If a 0 ii ) P S 1 If P a b P a P b P a b ( ) ( ) ( ) ( ) ∩ b = ∪ = + ( ) ( ) ( ) a 0 iii ) P a b P a P b b ⊂ If If = + ∩ ≥ a If If P a P b P a b P b ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Field F is a nonempty class of sets such that Field F is a nonempty class of sets such that If a field has the property that if the If a field has the property that if the a ∈ a ∈ If If F F sets a a 1 , a 2 , …, a n ,… belong to it then belong to it then sets 1 , a 2 , …, a n ,… 1 ∪ ∪ a 2 ∪ ∪ a 3 ∪ ∪ … … ∪ ∪ a ∈ b ∈ ∪ ∈ so does the set a a 1 a 2 a 3 If If F F a b F so does the set & & ∪ …, n ∪ a n a …, then the field is called a then the field is called a Corollaries Corollaries Borel field. Note that the class of all field. Note that the class of all ∩ ∈ Borel a b F a ∈ b ∈ If If F & F & subsets of subsets of S S is a is a Borel Borel field. field. − ∈ a b F ∈ S ∈ Also Also 0 F & & F ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 4

Probability Experiment ℑ ℑ : (S, F, P) : (S, F, P) Probability Experiment Example. Probability Experiment Example. Probability Experiment ℑ : (S, F, P) of Tossing a Coin, ℑ of Tossing a Coin, : (S, F, P) of outcomes ξ ξ ; this set is called space 1. Set S 1. Set S of outcomes ; this set is called space or sure (certain) event or sure (certain) event { } = S h , t 2. Borel Borel field field F F consisting of certain subsets consisting of certain subsets 2. of S S called events called events of { } {} { } F : 0 , h , t , h , t 3. Measure P(a 3. Measure P(a) ) assigned to every event assigned to every event a a ; ; this measure is called probability of this measure is called probability of ( ) {} + q = = = p 1 P h p P t q event a a , satisfies axioms 1 to 3 , satisfies axioms 1 to 3 event ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Concluding Remarks Concluding Remarks � Definitions � Definitions � Set Operations � Set Operations � � Probability Space Probability Space � Borel � Borel Field Field � Probability Experiment � Probability Experiment ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 5