Uncertainty, Data, and Judgment

Transcription

1 Uncertainty, Data, and Judgment Session 04 Structure of the Course Topic Session Probability 2-5 Estimation 6-8 Hypothesis Testing 9-10 Regression

2 Key Questions in UDJ 1. How to look at Data? (1-3) Judgment issues Measures of location & dispersion Measures of extremes (Chebyshev s T., Empirical Rule) How to model Uncertainty? (4-5) Distributions: Binomial, Poisson, Normal Functions of Random Variables 2. How to make Estimations? (6-8) Make educated guesses about population parameters Say how confident you are in those guesses 3. How to make Decisions based on data? (9-10) Hypothesis testing / Correct Observations/Theories 4. How to make Predictions based on data? (11-16) Regression / forecasting Today: Probability Distributions 1. Binomial 2. Poisson 3. Normal 2

3 Interested in = # of Heads in n tosses of a fair coin = # of consumers in a sample of n people who prefer to use Brand A of a product versus all other brands = # of defective items in a lot of n items = # of times a missile will hit its target, out of n launches Binomial Distribution, if n Bernoulli trials: n trials such that each trial has only two possible outcomes: Success or Failure ; 2. P(Success) = p is the same for all trials P(Failure) = 1 - p = q; 3. All trials are independent. Interested in the probability of observing x = # of Successes in n trials 3

4 BINOMIAL DISTRIBUTION In a given trial: P(Success) = p P(Failure)= q = 1-p Interested in: x = # of Successes in n trials n! x!(n-x)! p x (1-p) n-x Summary Measures for : µ x = Mean of x = Expected Value of x = E(x) = np σ x = Standard Deviation of x = npq Example: BINOMIAL DISTRIBUTION It is known that 10% of the items produced are defective. A random lot of 5 items is picked. Success = an item is defective x = # of Successes in n trials = # of defective items in a lot of 5 In this example: p = 0.1 q = 0.9 n = 5 P(x=0) = P(x=2) = From the tables: x Probability

5 Question 3.2 A particular analyst claims that he is good at picking winners in the US stock market -- stocks with annual returns in the top 10 percent of the population of all stocks in the US. You have heard similar stories many times before and you are, of course, very suspicious. However, you are willing to give the analyst a chance. The analyst picks fourteen stocks for you, and a year later four of them are in the top 10%. Assuming that the analyst is really no better than chance (i.e., has a 10% chance of picking a winner), what is the probability of observing exactly four winners in a sample of 14? What is the probability of observing at least four winners? Question 3.3 U.S. News & World Report (July 8, 1996) reported that of the 900 diversified equity funds in existence at the end of 1990, only 14 managed to beat the S&P 500 year in, year out i.e., only 14 of the 900 equity fund managers were able to outperform the S&P 500 in each of the six years from 1991 to Assume that each fund manager uses a strategy that is no more sophisticated than the toss of a coin that is, a fund manager has exactly a 0.5 probability of beating the S&P 500 in any given year, and the fund manager s performance in any given year is independent of his/her performance in any of the other years. What is then the probability that a fund manager will beat the S&P 500 in six out of six years? Based on this probability, how many of the 900 fund managers would you expect to beat the S&P 500 in six out of six years? Would you invest your money in an equity fund? 5

6 Interested in = # of Arrivals in a given period of time (a) At a bank (b) At hospital emergency room (c)... = # of Accidents / Breakdowns in a given period of time (a) in machinery (b) on a given road (traffic accidents) (c)... = # of defects in a continuous manufacturing process (a) # of blemishes in a bolt of fabric (b) # of bubbles in fiber optic cables (c)... Poisson Distribution, if 1. Observe # of successes over some continuum (such as time or space) 2. Average # of successes is constant in the unit of measure 3. Successes are independent Interested in the probability of observing = # of Successes in a given period of time or space. 6

7 POISSON DISTRIBUTION In a given interval (such as time): λ = average # of successes in the interval. Interested in: = # of Successes in the interval e λ P() = λ! Summary Measures for : µ x = Mean of = Expected Value of = E() = λ σ x = standard deviation of = λ POISSON DISTRIBUTION Example: In the Fontainebleau Hospital, on average, 2 cases arrive at the emergency room during an 8-hour shift on a Saturday. Success = an emergency case arrives = # of Successes in an 8-hour shift on a Saturday = # of emergency cases in an 8-hour shift on a Saturday In this example: λ =? P(=0) =?? P(=5) =?? 7

8 POISSON DISTRIBUTION Example: To get extra cash you have taken a week-end job as a staff physician at the Fontainebleau hospital. On a Saturday morning, you want to sleep in. What is the probability of no emergencies in the first 4 hours? Student Project:Does hot-hand exist in Italian football? Goal per match per team If hot-hand exists λ = 1.32, aggregating across teams and games occurrence frequency Number of goals per match: Poisson Hot-hand # of goals per match per team 8

9 Student Project:Does hot-hand exist in Italian football? Goal per match per team Theoretical expectation vs actual occurrence frequency Number of goals per match: Poisson Actual # of goals per match per team Probability of War An international arms dealer wants to estimate the probability of a new war in the next five years. The following information is available: Between 1500 and 1931, war broke out somewhere in the world a total of 299 times. A military action was defined as a war if It was legally declared, or It involved over 50,000 troops, or It resulted in significant boundary realignments Large confrontations were split into smaller subwars. For example, WW I was treated as 5 separate wars. Lewis F. Richardson (1944). The Distribution of Wars in Time. Journal of the Royal Statistical Society, 107,

10 Outbreaks of War Is it Poisson? Number of Wars Observed Expected Beginning in a Given Year Frequency Frequency Lewis F. Richardson (1944). The Distribution of Wars in Time. Journal of the Royal Statistical Society, 107, Problem 4.2 On the basis of past data regarding sales, the owner of a car dealership finds that on average 3.75 cars are sold per day on Saturdays and Sundays during the months of January and February. The dealership is open for ten hours on Saturdays and Sundays. The sales rate is relatively stable for different hours of the day and purchases appear to be independent of one another. The owner has no reason to believe that this year s car-selling process will be different from that in the past years covered by the data. On Saturday, February 4, the dealership will open at 9am. What is the probability that the first sale of the day will occur before 11am? 10

11 Binomial or Poisson? The number of stars in a sky zone The number of meals served by the INSEAD restaurant in an hour The number of defective parts in a new car The number of deaths in a year among holders of life insurance policies The total number of goals scored in a football game Discrete or Continuous? Your grade in this class The number of arrivals at BNP Bank in 1 hour The return on an investment Q: How many values can each take? 11

12 Normal Distribution Two Kinds of Problems Find the probability that...will exceed a given cutoff value...will be less than a given cutoff value...will be between two given cutoff values Example: What is the probability that IBM stock will have a return greater than 8%? Find a cutoff value such that...will exceed it with a prespecified level of probability...will be less than it with a prespecified level of probability Find two cutoff values such that will be between them with a prespecified level of probability Example: There is a 0.99 probability that IBM Stock will have a return greater than? (fill in the blank) Empirical Rule µ 3σ µ 2σ µ 1σ µ µ+1σ µ+2σ µ+3σ 68.26% 95.44% 99.74% 12

13 P(Investment return > 8%) =? Investment µ = 5 σ = Z P(Investment return > 12%) =? Investment µ = 11 σ =

14 Subtract mean, divide by standard deviation µ 3σ µ 2σ µ 1σ µ µ+1σ µ+2σ µ+3σ Z Z-score tells you how many standard deviations you are from the mean Transformations ~ N(µ, σ 2 ) Z = µ σ = σ Z + µ Z ~ N(0, 1) 14

15 P (2< < 8) =? µ = 5 σ = Z Find cut-off c such that P( < c) = 0.05 µ = 5 σ = Z 15

16 Tips on Using Standard Normal Tables Probability corresponds to area Symmetry Probabilities sum to 1 For intervals, use subtraction Problem 4.5 The time it takes a student to go from her house to school is normally distributed with a mean of 20 minutes and a standard deviation of 5. Estimate the percentage of time she will be late for class if she leaves her house 30 minutes before the start of class. 16

17 Probability Distributions Distribution Parameters Summary Measures Binomial n, p µ= np σ = np( 1 p) Poisson λ µ = λ σ = λ Normal µ, σ µ σ 17