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Probablity Models have 3 components: Sample Space (Ω) Events on Sample Space (F) Functions defined on the sample space (P) ⇒Probability Model: (Ω,F,P) Why consider sets? Probability models involve families of sets - combining and operating on a family of sets, using unions, intersections, complements. The results of such operations must then also be in the family Set Algebra Results: Cont’d Recall: Sample Space Subsets, Union, Intersection Cummutative , Associative, Distributive Laws Recall: A partition of Ω is any collection of disjoint sets A α that fill out the whole space: ie U A α = Ω, and A α A β = ∅, α ≠ β { } Recall: An event is a subset of Ω. All Events are subsets - but all subsets are not events. (Here, we want to assign probabilities to events in a coherent fashion - cannot always do this if all subsets are events. A partition of Ω serves to partition an event in Ω, ie if {A α }is a partition of Ω (from before) and let B be an event, the sets BA α constitute a partition of B: ( )( ) U BA α = B, and BA α BA β = ∅, α ≠ β Related Result on ascending/descending sequences: { } The sequence A n + 1 − A n constitutes a partition of their union, A= U A n { } Ex: Consider the disks Bn defined by { } x 2 + y 2 ≤ 1 − 1 / n, n = 1, 2,...The sequence Bn is (?).Bn − Bn −1 are rings between concentric circles which constitute a partion of their union (the closed disk) Definition: A Field is a family of sets closed under complementation, finite unions, and finite intersections. Functions on a sample space: X(ω ) maps from Ω to a new set. The “image point” is the value of X(ω) corresponding to ω. For a given value of X (ie x), the set of ω ‘s such that X(ω)=x is the pre-image of x. A set of values of X - say B, defines a pre-image set in Ω . Pre-image sets may or may not be events. A function defined on a sample space induces a partition of the sample space. Recall: Probability Axioms: Given a sample space Ω, a collection of subsets called events, the Probability of an event E is P(E). The set function P must satisfy the Axioms of Probability: 1. 2. 3. For every event E, 0 ≤ P(E) ≤ 1 P(Ω) = 1 If EF= E∩F= 0, then P(E∪F) = P(E) +P(F) Related Probability Results on ascending/descending sequences: { } Suppose En an ascending sequence: E1, E 2 ,....En , Ei ⊂ E j,, i<j, the P(U En ) = lim P(En ) n→∞ { } { } Likewise, En is a descending sequence, En , E1, E 2 ,....En , Ei ⊂ E j,,the i>j, then P(I En ) = lim P(En ) n→∞ { } Axiom 3a: For any sequence Ei such that Ei E j =0 for n ∞ i ≠ j, P U E i = ∑ P E i i=1 i=1 ( ) This is called the axiom of countable additivity. Law of Total Probability If {En} is a countable partition of Ω, then n P(F) = ∑ P(FEn ) i=1 Outcomes of Symmetric Experiments: Given a total of N experiments, if outcomes are equally likely (random), then the odds "probability"of one outcome is 1/N. (lottery or coin tossing examples). The Probability of an event E, P(E) = 1/N Law of Large Numbers: Suppose repeat a long sequence of trials and the results of any set do not depend (independent) on results of other trials, the proportion of occurrences of an event stabilizes and tends to a limiting value. Consider probability to be the long run limit of proportions of "successes." Ex: Consider a baseball player’s succie times “at bat” as successive trials of an experiment. Batting average (no hits/no times at bat) fluctuates, but tends to a limiting value. Ex: Tosses of a coin - consider the proportion of “heads”