Prob_Set_Fun - I NTRODUCTION TO P ROBABILITY T HEORY-eld...

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I NTRODUCTION TO P ROBABILITY T HEORY σ -field Interested in assigning probabilities to events, complements of events, and union and intersections of events. We want our collection of events to include these combinations of events. σ -field: Let B be a collection of subsets of C , (1). ∅ ∈ B , (2). If C ∈ B then C c ∈ B , (3). If the sequence of sets { C 1 , C 2 , . . . } is in B then i =1 C i ∈ B .
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I NTRODUCTION TO P ROBABILITY T HEORY By (1) and (2), a σ -field always contains and C . By (2) and (3), it follows from DeMorgan’s laws that σ -field is closed under countable intersections, besides countable unions. Ex. B = {∅ , C} Ex. B = {∅ , C , C, C c } Ex. B = { power set of C}
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I NTRODUCTION TO P ROBABILITY T HEORY Probability Set Function Probability Set Function: Let C be a sample space and let B be a σ -field on C . P is a probability set function if P ( C ) 0 for all C ∈ B . P ( C ) = 1 .
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