Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-1 Business Statistics, 3e by Ken Black Chapter 5 Discrete Distributions

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-2 Learning Objectives Distinguish between discrete random variables and continuous random variables. Know how to determine the mean and variance of a discrete distribution. Identify the type of statistical experiments that can be described by the binomial distribution, and know how to work such problems.

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-3 Learning Objectives -- Continued Decide when to use the Poisson distribution in analyzing statistical experiments, and know how to work such problems. Decide when binomial distribution problems can be approximated by the Poisson distribution, and know how to work such problems. Decide when to use the hypergeometric distribution, and know how to work such problems.

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-4 Discrete vs Continuous Distributions Random Variable -- a variable which contains the outcomes of a chance experiment Discrete Random Variable -- the set of all possible values is at most a finite or a countably infinite number of possible values –Number of new subscribers to a magazine –Number of bad checks received by a restaurant –Number of absent employees on a given day Continuous Random Variable -- takes on values at every point over a given interval –Current Ratio of a motorcycle distributorship –Elapsed time between arrivals of bank customers –Percent of the labor force that is unemployed

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-5 Some Special Distributions Discrete –binomial –Poisson –hypergeometric Continuous –normal –uniform –exponential –t –chi-square –F

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-6 Discrete Distribution -- Example Number of Crises Probability Distribution of Daily Crises ProbabilityProbability Number of Crises

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-7 Requirements for a Discrete Probability Function Probabilities are between 0 and 1, inclusively Total of all probabilities equals 1

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-8 Requirements for a Discrete Probability Function -- Examples XP(X) XP(X) XP(X)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-9 Mean of a Discrete Distribution X P(X) XPX  ()

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-10 Variance and Standard Deviation of a Discrete Distribution X P(X) X 

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-11 Mean of the Crises Data Example XP(X) X  ProbabilityProbability Number of Crises

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-12 Variance and Standard Deviation of Crises Data Example XP(X) (X-  )  ) 2  ) 2  P(X)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-13 Binomial Distribution Experiment involves n identical trials Each trial has exactly two possible outcomes: success and failure Each trial is independent of the previous trials –p is the probability of a success on any one trial –q = (1-p) is the probability of a failure on any one trial –p and q are constant throughout the experiment –X is the number of successes in the n trials Applications –Sampling with replacement –Sampling without replacement -- n < 5% N

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-14 Binomial Distribution Probability function Mean value Variance and standard deviation

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-15 Binomial Distribution: Development Experiment: randomly select, with replacement, two families from the residents of Tiny Town Success is ‘Children in Household:’ p = 0.75 Failure is ‘No Children in Household:’ q = 1- p = 0.25 X is the number of families in the sample with ‘Children in Household’ Family Children in Household Number of Automobiles ABCDABCD Yes No Yes Listing of Sample Space (A,B), (A,C), (A,D), (D,D), (B,A), (B,B), (B,C), (B,D), (C,A), (C,B), (C,C), (C,D), (D,A), (D,B), (D,C), (D,D)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-16 Binomial Distribution: Development Continued Families A, B, and D have children in the household; family C does not Success is ‘Children in Household:’ p = 0.75 Failure is ‘No Children in Household:’ q = 1- p = 0.25 X is the number of families in the sample with ‘Children in Household’ (A,B), (A,C), (A,D), (D,D), (B,A), (B,B), (B,C), (B,D), (C,A), (C,B), (C,C), (C,D), (D,A), (D,B), (D,C), (D,D) Listing of Sample Space X 1/16 P(outcome)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-17 Binomial Distribution: Development Continued (A,B), (A,C), (A,D), (D,D), (B,A), (B,B), (B,C), (B,D), (C,A), (C,B), (C,C), (C,D), (D,A), (D,B), (D,C), (D,D) Listing of Sample Space X 1/16 P(outcome) X /16 6/16 9/16 1 P(X)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-18 Binomial Distribution: Development Continued Families A, B, and D have children in the household; family C does not Success is ‘Children in Household:’ p = 0.75 Failure is ‘No Children in Household:’ q = 1- p = 0.25 X is the number of families in the sample with ‘Children in Household’ X Possible Sequences (F,F) (S,F) (F,S) (S,S) P(sequence) (.) ) (.) 25 2  (.) )2575 (.) )7525 (.) ) (.) 75 2 

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-19 Binomial Distribution: Development Continued X Possible Sequences (F,F) (S,F) (F,S) (S,S) P(sequence) (.) ) (.) 25 2  (.) )2575 (.) )7525 (.) ) (.) 75 2  X P(X) (.) ) =0.375 (.) ) (.) 75 2  = (.) ) (.) 25 2  =0.0625

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-20 Binomial Distribution: Demonstration Problem 5.3

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-21 Binomial Table n = 20PROBABILITY X

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-22 Using the Binomial Table Demonstration Problem 5.4 n = 20PROBABILITY X

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-23 Binomial Distribution using Table: Demonstration Problem 5.3 n = 20PROBABILITY X …………

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-24 Graphs of Selected Binomial Distributions n = 4PROBABILITY X P = X P(X) P = X P(X) P = X P(X)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-25 Poisson Distribution Describes discrete occurrences over a continuum or interval A discrete distribution Describes rare events Each occurrence is independent any other occurrences. The number of occurrences in each interval can vary from zero to infinity. The expected number of occurrences must hold constant throughout the experiment.

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-26 Poisson Distribution: Applications Arrivals at queueing systems –airports -- people, airplanes, automobiles, baggage –banks -- people, automobiles, loan applications –computer file servers -- read and write operations Defects in manufactured goods –number of defects per 1,000 feet of extruded copper wire –number of blemishes per square foot of painted surface –number of errors per typed page

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-27 Poisson Distribution Probability function nMean value nStandard deviation nVariance

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-28 Poisson Distribution: Demonstration Problem 5.7

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-29 Poisson Distribution: Probability Table X

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-30 Poisson Distribution: Using the Poisson Tables X

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-31 Poisson Distribution: Using the Poisson Tables X

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-32 Poisson Distribution: Using the Poisson Tables X

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-33 Poisson Distribution: Graphs

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-34 Poisson Approximation of the Binomial Distribution Binomial probabilities are difficult to calculate when n is large. Under certain conditions binomial probabilities may be approximated by Poisson probabilities. Poisson approximation

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-35 Poisson Approximation of the Binomial Distribution XError XError

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-36 Hypergeometric Distribution Sampling without replacement from a finite population The number of objects in the population is denoted N. Each trial has exactly two possible outcomes, success and failure. Trials are not independent X is the number of successes in the n trials The binomial is an acceptable approximation, if n < 5% N. Otherwise it is not.

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-37 Hypergeometric Distribution Probability function –N is population size –n is sample size –A is number of successes in population –x is number of successes in sample Mean value Variance and standard deviation

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-38 Hypergeometric Distribution: Probability Computations N = 24 X = 8 n = 5 x P(x)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-39 Hypergeometric Distribution: Graph N = 24 X = 8 n = 5 x P(x)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-40 Hypergeometric Distributio: Demonstration Problem 5.11 XP(X) N = 18 n = 3 A = 12

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-41 Hypergeometric Distribution: Binomial Approximation (large n) Hypergeometric N = 24 X = 8 n = 5 Binomial n = 5 p = 8/24 =1/3 xError P(x)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-42 Hypergeometric Distribution: Binomial Approximation (small n) Hypergeometric N = 240 X = 80 n = 5 Binomial n = 5 p = 80/240 =1/3 xP(x)Error P(x)