Presentation on theme: "Economics 105: Statistics Any questions? Go over GH2 Student Information Sheet."— Presentation transcript:
Economics 105: Statistics Any questions? Go over GH2 Student Information Sheet
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
Set Theory Venn diagrams Union A B Intersection A B
Set Theory A′ is the complement of A A and B are mutually exclusive A B =
Set Theory set of events A 1, A 2, A 3 … A N partitions the sample space
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′
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)
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
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′)
Useful Results (cont’d) If A 1, A 2, A 3 … A N partition S, then
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?
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 )
Example Suppose you apply to 3 schools: A, B, and C A) =.20 B) =.40 C) =.60 What is the probability of being rejected at all 3? What is the probability of being accepted somewhere?
Conditional Probability The conditional probability that A occurs given that B is known to have occurred is
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?
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: htmlhttp://gregmankiw.blogspot.com/2006/08/potus html
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?
Medical Diagnosis Problem ideas-on-communicating-risks-to-the-general-public/
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
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)
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)
–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
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)
Example Let C = total cost of building a pool Let X = days to finish the project C = 25, X X P(X = x i ) 10.1Find the mean, std dev, and 11.3 variance of the total cost