Lec4MoreProbHowCount

# Lec4MoreProbHowCount - ORIE 270 Engineering Probability and...

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Review Properties of P Equal probabilities How to count. Title Page Page 1 of 34 Go Back Full Screen Close Quit ORIE 270 Engineering Probability and Statistics Lecture 4: More Probability; How to Count Sidney Resnick School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA sid [email protected] September 3, 2007

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Review Properties of P Equal probabilities How to count. Title Page Page 2 of 34 Go Back Full Screen Close Quit 1. Review 1.1. Definition A probability model has 3 components: 1. A sample space S –abstract set containing a subset which can be put in 1-1 correspondence with the outcomes of experiment we wish to model. 2. A distinguished collection of subsets A of S which we designate as events . When S is discrete, A is all subsets of S . 3. A probability measure P . This is a rule assigning numbers between 0 and 1 to events. Formally, P is a function with domain A and range [0 , 1] such that (a) P ( S ) = 1. (b) 0 P ( A ) 1 , for all events A ∈ A . (c) Additivity: If A 1 , A 2 , . . . are disjoint events then P ( A 1 A 2 A 3 . . . ) = P ( A 1 ) + P ( A 2 ) + P ( A 3 ) + . . . . (1)
Review Properties of P Equal probabilities How to count. Title Page Page 3 of 34 Go Back Full Screen Close Quit Remember: A probabilist builds a probability model. A statistician confronts a range of models with the intent to pick the one(s) which best fit the data. How many hats will you wear? 1.2. Methods of constructing probability models. Two common ways: S is discrete: If S = { s 1 , s 2 , . . . } , we suppose we have numbers p k satisfying p k 0; i =1 p k = 1 . Schematically S s 1 s 2 . . . P p 1 p 2 . . . We define P by P ( A ) = i : s i A P ( { s i } ) = i : s i A p i .

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Review Properties of P Equal probabilities How to count. Title Page Page 4 of 34 Go Back Full Screen Close Quit When S is continuous, say a subset of R : We suppose there is a density function f ( x ) satisfying f ( x ) 0 , -∞ f ( u ) du = 1 . We define P by P (( a, b ]) = b a f ( u ) du. Where does { p k } or f ( x ) come from? No magic: Most common reasons for specifying probabilities: Symmetry. Feeling that we want to model a situation where all outcomes are equally likely. 1. Example. Toss a red die and then a blue die. Then S = { ( i, j ) : 1 i 6; 1 j 6 } has 36 outcomes. Because we feel each outcome should be equally likely, we specify the p k ’s as 1/36 for each: S (1 , 1) (1 , 2) . . . (6 , 6) P 1/36 1/36 . . . 1/36
Review Properties of P Equal probabilities How to count. Title Page Page 5 of 34 Go Back Full Screen Close Quit 2. Example. Pick a number at random from [0 , 1]. Then the set of outcomes is S = [0 , 1] . Symmetry implies P ([ a, b ]) = b - a. Then the density f ( x ) is f ( x ) = 1 , if 0 x 1 , 0 , otherwise . Measurement and estimation. Example: Suppose S = { R, D, I } corresponding to the out- comes of the election that have a Republican, Democrat or Independent win. Based on a sampling experiment we esti- mate the following model (numbers fictitious): S R D I P 0.47 0.49 0.04

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