04 STAT probability 3-SHORT SLIDES

04 STAT probability 3-SHORT SLIDES - PROBABILITY THEORY Dr....

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PROBABILITY THEORY Dr. V.R. Bencivenga Economic Statistics Economics 329 PROBABILITY THEORY PART 3 Outline Probability rules continued (4) Bayes rule Bivariate probabilities Counting rules Objectives Understand “Bayesian updating”— how beliefs are revised as evidence comes in. How to compute probabilities when two characteristics of the basic outcome are of interest. When and how can we compute probabilities using “permutations” and “combinations,” together with the uniform probability model?
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2 PROBABILITY THEORY BAYES RULE
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3 PROBABILITY THEORY Example #10 Job interview A firm hires 4% of job applicants. Only some applicants are interviewed. Of all applicants hired by the firm, 98% are interviewed. Of all applicants not hired by the firm, 1% are interviewed. You have applied to this firm, and you have just received an interview. What is the probability you will be hired? I = interviewed H = hired In this notation, what is 4% (.04)? What is 98% (.98)? What is 1% (.01)? What do we want to calculate?
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4 PROBABILITY THEORY P(H) = .04 => 96 . ) H P( P(I | H) = .98 ) H | P(I = .01 Using the definition of conditional probability: P(I) H) P(I I) | P(H The multiplication rule gives us the numerator: P(H) H) | P(I H) P(I For the denominator, note there are two ways an applicant can be an i nterviewed applicant he can be interviewed and hired , or interviewed and not hired . H H I interviews hired interviewed not hired I not interviewed hired not interviewed not hired Using the multiplication rule again: ) H )P( H | P(I H)P(H) | P(I ) H P(I H) P(I P(I) 803 . ) 96 (. 01 . ) 04 98 . ) 04 98 . ) H ) H | P(I H)P(H) | P(I ) H ( P H) | P(I I) | P(H By getting an interview, the probability of being hired rises from 4% to 80%!
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5 PROBABILITY THEORY (4) Bayes’ rule Bayes rule lets us “update” the probability of an event of interest, given information about conditional probabilities of a related event. In the case where the event of interest (B 1 ) is one of two that might occur (either B 1 or B 2 will occur if the related event A occurs), Bayes rule is given by ) B ( P ) B | A ( P ) B ( P ) B | A ( P ) B ( P ) B | A ( P ) A | B ( P 2 2 1 1 1 1 1 More generally, Bayes rule is given by k 1 i i i j j j ) B ( P ) B | A ( P ) B ( P ) B | A ( P ) A | B ( P .
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6 PROBABILITY THEORY Bayes’ rule follows from the definition of conditional probability ) A ( P ) A B ( P ) A | B ( P j j the multiplication rule … P(B j A) = P(A | B j ) P(B j ) and the law of total probability P(A) = k 1 i i i ) B ( P ) B | A ( P We are familiar with conditional probability and the multiplication rule. The law of total probability is a straightforward application of these two concepts. Let’s see this … Substitute into the numerator. Substitute into the denominator. Bayes rule: k 1 i i i j j j ) B ( P ) B | A ( P ) B ( P ) B | A ( P ) A | B ( P
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7 PROBABILITY THEORY The collection of sets {B 1 , B 2 , …., B k } is a partition of S if (i) B 1 B 2 B k = S (collectively exhaustive) (ii) B i B j = { } for i ≠ j (mutually exclusive) 1 B 2 B 3 B
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8 PROBABILITY THEORY Now derive the law of total probability:
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This note was uploaded on 02/26/2012 for the course ECONOMICS 329 taught by Professor Bencivenga during the Spring '12 term at University of Texas at Austin.

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04 STAT probability 3-SHORT SLIDES - PROBABILITY THEORY Dr....

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