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Economics 105: Statistics Any questions? Go over GH2 Student Information Sheet

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Intro to Probability: Basic Definitions Random trials – multiple outcomes & uncertainty Basic outcome Sample space Event Examples: coin toss, die roll, dice roll, deck of cards, etc. Deck of cards will be defined as 52 cards, 13 of each suit ( ♠♣♥♦ ), 2, 3,..., 10, J, K, Q, A

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Set Theory Venn diagrams Union A B Intersection A B

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Set Theory A′ is the complement of A A and B are mutually exclusive A B =

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Set Theory set of events A 1, A 2, A 3 … A N partitions the sample space

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Rules for Set Operations A B = B A Commutative A B = B A A A = AIdempotency A A = A A A′ = S, A A′ = Complementation (A B)′ = A′ B′ (A B)′ = A′ B′

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Rules for Set Operations Associative A (B C) = (A B) C A (B C) = (A B) C Distributive A (B C) = (A B) (A C) A (B C) = (A B) (A C)

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Fundamental Postulates 1: P(A) ≥ 0 [Impossible events cannot occur] 2: P(S) = 1 [Some outcome must occur] 3: If A 1, A 2, A 3 … A N are N mutually exclusive events then or P(A) should satisfy certain postulates

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Useful Results P(A′) = 1 – P(A) P( ) = 0 If A B, then P(A) ≤ P(B) 0 ≤ P(A) ≤ 1 P(A B) = P(A) + P(B) – P(A B) – avoid double counting P(A B) = 1 - P(A B)′ = 1 - P(A′ B′)

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Useful Results (cont’d) If A 1, A 2, A 3 … A N partition S, then

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Example Problem Example (Problem 4.8, p. 134, BLK 10e) – 824 Homeowners, of 1000 asked, drive to work – 681 Renters, of 1000 asked, drive to work 1.Make a contingency (cross-classification) table 2.If a respondent is selected at random, what is the probability that they drive to work? 3.… that they drive and are a homeowner? 4.… that they drive or are a homeowner?

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Statistical Independence Events A and B are statistically independent when P(A|B) = P(A) (Multiplication Rule): Events A and B are statistically independent when P(A B) = P(A)*P(B) If A 1, A 2, A 3 … A N are independent events then P(A 1 A 2 A 3 … A N ) = P(A 1 )P(A 2 )P(A 3 )… P(A N )

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Example Suppose you apply to 3 schools: A, B, and C P(accepted @ A) =.20 P(accepted @ B) =.40 P(accepted @ C) =.60 What is the probability of being rejected at all 3? What is the probability of being accepted somewhere?

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Conditional Probability The conditional probability that A occurs given that B is known to have occurred is

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Conditional Probability Probability a beginning golfer makes a good shot if she selects the correct club is 1/3. The probability of a good shot with the wrong club is 1/5. There are 4 clubs in her golf bag, one of which is the correct club for the next shot. Club selection is random. What is the probability of a good shot? Given that she hit a good shot, what is the probability that she chose the wrong club?

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Bayes’ Theorem If A and B are two events with P(A) > 0 and P(B) > 0 then, P(A|B) = P(B|A)*P(A) P(B) Example: Auditor found that historically 15% of a firm’s account balances have an error. Of those balances with an error, 60% were unusual values based on historical figures. Of all balances, 20% were unusual values. If the number for a particular balance appears unusual, what is the probability it is in error? Example: http://gregmankiw.blogspot.com/2006/08/potus- 2008.htmlhttp://gregmankiw.blogspot.com/2006/08/potus- 2008.html

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Medical Diagnosis Problem The following question was asked of 60 students and staff at Harvard Medical School Assume that a test to detect a disease, which has prevalence in the population of 1/1000, has a false positive rate of 5%, and a true positive rate of 100%. what is the probability that a person found to have a positive test actually has the disease, assuming you know nothing about the person’s symptoms?

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Medical Diagnosis Problem http://www.decisionsciencenews.com/2010/12/03/some- ideas-on-communicating-risks-to-the-general-public/

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Discrete Random Variables Take on a limited number of distinct values Each outcome has an associated probability We can represent the probability distribution function in 3 ways – function ƒ(x i ) = P(X = x i ) – graph – table Bernoulli distribution – graph & table ? Cumulative distribution function

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Discrete Random Variable Summary Measures Expected Value (or mean) of a discrete distribution (Weighted Average) –Example: Toss 2 coins, X = # of heads, compute expected value of X: E(X) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0 X P(X) 0 0.25 1 0.50 2 0.25

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Variance of a discrete random variable Standard Deviation of a discrete random variable where: E(X) = Expected value of the discrete random variable X X i = the i th outcome of X P(X i ) = Probability of the i th occurrence of X Discrete Random Variable Summary Measures (continued)

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–Example: Toss 2 coins, X = # heads, compute standard deviation (recall E(X) = 1) Discrete Random Variable Summary Measures (continued) Possible number of heads = 0, 1, or 2

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Properties of Expected Values E(a + bX) = a + bE(X), where a and b are constants If Y = a + bX, then var(Y) = var(a + bX) = b 2 var(X)

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Example Let C = total cost of building a pool Let X = days to finish the project C = 25,000 + 900X X P(X = x i ) 10.1Find the mean, std dev, and 11.3 variance of the total cost. 12.3 13.2 14.1

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Review of Final Part I Sections 2.2 -- 4.5 Jiaping Wang Department of Mathematics 02/29/2013, Monday.

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