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# Ch2 - Statistics 427 Introduction to Probability and...

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Statistics 427 Introduction to Probability and Statistics I Instructor: Prof. Mark Berliner Email: Office: CH 205B (Cockins Hall) Phone: 614.292.0291 http://www.stat.osu.edu/~mb Course information available at:

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Chapter 2 Intro to basic prob. theory Point: Mathematical models and analyses for treating uncertainty Key topics: 1. Formulation 2. Set theory and Venn diagrams 3. Probability measures (PM) 4. Basic properties and examples of PM 5. Conditional prob. and Bayes‟ Theorem
3 1. Formulation Outcome: a possible result of the random phenomenon studied. Experiment: any process leading to an uncertain (= “random”) outcome. Sample space S: set of all possible outcomes. Event: a subset of S. (Notation: A, B, etc.) Ex) Expt: observe temperature at 5pm at Columbus airport. S: all real numbers (OK to make S too big) Event: temp is below freezing A={x | x < 32F}

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4 Ex) Expt: flip a coin twice. (remark: silly examples just for rapid illustration!) S = {(h,h), (h,t), (t,h), (t,t)} or S 2 = {0 h‟s, 1 h, 2 h‟s} (remark: sample spaces aren‟t unique. “Art” in choosing S to be useful and efficient. ) Events (using S): A: “one h” …. {(h,t), (t,h)} B: “match” …. {(h,h), (t,t)}
2. Events, Sets, and Venn Diagrams Box represents S A B

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Main operations Intersection: A I B Union: A U B
Complement: A A, B mutually exclusive or disjoint A A A B

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8 3. Probability measures A probability measure (PM) is a function that assigns numbers between 0 and 1 to events. P: events [0,1] Note: “regular” functions, say f. y = f(x) where x and y are reals PM: p = P(A) where p is a real number in [0, 1] and A is a subset of S. Graphs of f ‟s are useful, but we can‟t generally graph P . Events are not typically “well - ordered”
9 Rules (“axioms”) for all PM (p. 51) 1. For any event A, P(A) > 0 2. P(S) = 1 3. Additivity. For any collection of disjoint sets: A 1 , A 2 , A 3 , …. P(A 1 U A 2 U A 3 U …. ) = P(A 1 ) + P(A 2 ) + …. i.e., for disjoint events, Prob(union) = sum of individual prob‟s

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10 4. Basic properties of PM P(A ) = 1 P(A) Since A and A are disjoint and their union is S. Remark: sometimes easier to find P(A ) than to find P(A) directly. P( f ) = 0. (“empty set”, f ) If A is a subset of B, P(A) < P(B) General addition rules P(A U B) = P(A) + P(B) - P(A I B)
P(A U B) = P(A) + P(B) - P(A I B) A B

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12 Types of sample spaces 1. Discrete (or countable) S = {E 1 , E 2 , E 3 , …. } 2. Continuous S is some interval of real numbers Cover discrete case first: Fundamental fact for discrete S (p. 56): For any event A, P(A) = S E i in A P({E i }) (Continuous case: integrate to find prob‟s )
13 Ex) Geiger counter. Record number of particles hitting plate in some fixed time period.

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