1. © 2002 Thomson / South-Western Slide 5-1 Chapter 5 Discrete Probability Distributions

2. © 2002 Thomson / South-Western Slide 5-2 Learning Objectives Distinguish between discrete random variables and continuous random variables. Identify the type of statistical experiments that can be described by the binomial distribution, and know how to work such problems.

3. © 2002 Thomson / South-Western Slide 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 to use the hypergeometric distribution, and know how to work such problems.

4. © 2002 Thomson / South-Western Slide 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 countable infinite number of possible values Continuous Random Variable -- takes on values at every point over a given interval

5. © 2002 Thomson / South-Western Slide 5-5 Some Special Distributions Discrete distributions are constructed from discrete ransom variables. The binomial, Poisson, and hypergeometric distributions are discrete distributions Continuous distributions are based on continuous random variables. The normal, uniform, exponential, t, chi- square, and F distributions are continuous distributions

6. © 2002 Thomson / South-Western Slide 5-6 Binomial Distribution A widely known discrete distribution constructed by determining the probabilities of X successes in n trials.

7. © 2002 Thomson / South-Western Slide 5-7 Assumptions of the Binomial Distribution The experiment involves n identical trials Each trial has only two possible outcomes: success and failure Each trial is independent of the previous trials The terms p and q remain constant throughout the experiment –p is the probability of a success on any one trial –q = (1-p) is the probability of a failure on any one trial

8. © 2002 Thomson / South-Western Slide 5-8 Assumptions of the Binomial Distribution, continued In the n trials X is the number of successes possible where X is a whole number between 0 and n. Applications –Sampling with replacement –Sampling without replacement causes p to change but if the sample size n < 5% N, the independence assumption is not a great concern.

9. © 2002 Thomson / South-Western Slide 5-9 Binomial Distribution: Development Experiment: randomly select, with replacement, two families from the residents of the four-family 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’

10. © 2002 Thomson / South-Western Slide 5-10 Binomial Distribution: Development continued (2) Family Children in Household Number of Automobiles ABCDABCD Yes No Yes 32123212 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)

11. © 2002 Thomson / South-Western Slide 5-11 Binomial Distribution: Development continued (3) 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 21222212110122122122221211012212 X 1/16 P(outcome)

12. © 2002 Thomson / South-Western Slide 5-12 Binomial Distribution: Development continued (4) (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 21222212110122122122221211012212 X 1/16 P(outcome) X 012012 1/16 6/16 9/16 1 P(X)

13. © 2002 Thomson / South-Western Slide 5-13 Binomial Distribution: Development continued (5) 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 01120112 (F,F) (S,F) (F,S) (S,S) P(sequence) (.) ) (.) 25 2  (.) )2575 (.) )7525 (.) ) (.) 75 2 

14. © 2002 Thomson / South-Western Slide 5-14 Binomial Distribution: Development continued (6) X Possible Sequences 01120112 (F,F) (S,F) (F,S) (S,S) P(sequence) (.) ) (.) 25 2  (.) )2575 (.) )7525 (.) ) (.) 75 2  X 012012 P(X) (.) )2575 2 =0.375 (.) ) (.) 75 2  =0.5625 (.) ) (.) 25 2  =0.0625

15. © 2002 Thomson / South-Western Slide 5-15 Binomial Distribution: Demonstration Problem 5.2

16. © 2002 Thomson / South-Western Slide 5-16 Binomial Table n = 20PROBABILITY X0.10.20.30.40.50.60.70.80.9 00.1220.0120.0010.000 10.2700.0580.0070.000 20.2850.1370.0280.0030.000 30.1900.2050.0720.0120.0010.000 40.0900.2180.1300.0350.0050.000 50.0320.1750.1790.0750.0150.0010.000 60.0090.1090.1920.1240.0370.0050.000 70.0020.0550.1640.1660.0740.0150.0010.000 8 0.0220.1140.1800.1200.0350.0040.000 9 0.0070.0650.160 0.0710.0120.000 100.0000.0020.0310.1170.1760.1170.0310.0020.000 110.000 0.0120.0710.160 0.0650.0070.000 120.000 0.0040.0350.1200.1800.1140.0220.000 130.000 0.0010.0150.0740.1660.1640.0550.002 140.000 0.0050.0370.1240.1920.1090.009 150.000 0.0010.0150.0750.1790.1750.032 160.000 0.0050.0350.1300.2180.090 170.000 0.0010.0120.0720.2050.190 180.000 0.0030.0280.1370.285 190.000 0.0070.0580.270 200.000 0.0010.0120.122

17. © 2002 Thomson / South-Western Slide 5-17 Using the Binomial Table: Demonstration Problem 5.3 n = 20PROBABILITY X0.10.20.30.4 00.1220.0120.0010.000 10.2700.0580.0070.000 20.2850.1370.0280.003 30.1900.2050.0720.012 40.0900.2180.1300.035 50.0320.1750.1790.075 60.0090.1090.1920.124 70.0020.0550.1640.166 80.0000.0220.1140.180 90.0000.0070.0650.160 100.0000.0020.0310.117 110.000 0.0120.071 120.000 0.0040.035 130.000 0.0010.015 140.000 0.005 150.000 0.001 160.000 170.000 180.000 190.000 200.000

18. © 2002 Thomson / South-Western Slide 5-18 Binomial Distribution Probability function Mean value Variance and standard deviation

19. © 2002 Thomson / South-Western Slide 5-19 Graphs of Selected Binomial Distributions n = 4PROBABILITY X0.10.50.9 00.6560.0630.000 10.2920.2500.004 20.0490.3750.049 30.0040.2500.292 40.0000.0630.656 P = 0.1 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 01234 X P(X) P = 0.5 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 01234 X P(X) P = 0.9 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 01234 X P(X)

20. © 2002 Thomson / South-Western Slide 5-20 Assumptions of the 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.

21. © 2002 Thomson / South-Western Slide 5-21 Poisson Distribution Probability function nMean value nStandard deviation nVariance

22. © 2002 Thomson / South-Western Slide 5-22 Poisson Distribution: Demonstration Problem 5.7

23. © 2002 Thomson / South-Western Slide 5-23 Poisson Distribution: Probability Table X0.51.51.63.03.26.46.57.08.0 00.60650.22310.20190.04980.04080.00170.00150.00090.0003 10.30330.33470.32300.14940.13040.01060.00980.00640.0027 20.07580.25100.25840.22400.20870.03400.03180.02230.0107 30.01260.12550.13780.22400.22260.07260.06880.05210.0286 40.00160.04710.05510.16800.17810.11620.11180.09120.0573 50.00020.01410.01760.10080.11400.14870.14540.12770.0916 60.00000.00350.00470.05040.06080.15860.15750.14900.1221 70.00000.00080.00110.02160.02780.14500.14620.14900.1396 80.00000.00010.00020.00810.01110.11600.11880.13040.1396 90.0000 0.00270.00400.08250.08580.10140.1241 100.0000 0.00080.00130.05280.05580.07100.0993 110.0000 0.00020.00040.03070.03300.04520.0722 120.0000 0.0001 0.01640.01790.02630.0481 130.0000 0.00810.00890.01420.0296 140.0000 0.00370.00410.00710.0169 150.0000 0.00160.00180.00330.0090 160.0000 0.00060.00070.00140.0045 170.0000 0.00020.00030.00060.0021 180.0000 0.0001 0.00020.0009

24. © 2002 Thomson / South-Western Slide 5-24 Using the Poisson Tables: Demonstration Problem 5.7 X0.51.51.63.0 00.60650.22310.20190.0498 10.30330.33470.32300.1494 20.07580.25100.25840.2240 30.01260.12550.13780.2240 40.00160.04710.05510.1680 50.00020.01410.01760.1008 60.00000.00350.00470.0504 70.00000.00080.00110.0216 80.00000.00010.00020.0081 90.0000 0.0027 100.0000 0.0008 110.0000 0.0002 120.0000 0.0001

25. © 2002 Thomson / South-Western Slide 5-25 Poisson Distribution: Graphs 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 012345678 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0246810121416

26. © 2002 Thomson / South-Western Slide 5-26 Assumptions of the Hypergeometric Distribution It is a discrete distribution. Sampling is done without replacement. The number of objects in the population, N, is finite and known. Each trial has exactly two possible outcomes: success and failure. Trials are not independent X is the number of successes in the n trials

27. © 2002 Thomson / South-Western Slide 5-27 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

28. © 2002 Thomson / South-Western Slide 5-28 Hypergeometric Distribution: Probability Computations N = 24 X = 8 n = 5 x 00.1028 10.3426 20.3689 30.1581 40.0264 50.0013 P(x)

29. © 2002 Thomson / South-Western Slide 5-29 Hypergeometric Distribution: Graph N = 24 X = 8 n = 5 x 00.1028 10.3426 20.3689 30.1581 40.0264 50.0013 P(x) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 012345

30. © 2002 Thomson / South-Western Slide 5-30 Hypergeometric Distributio: Demonstration Problem 5.8 XP(X) 00.0245 10.2206 20.4853 3 0.2696 N = 18 n = 3 A = 12

31. © 2002 Thomson / South-Western Slide 5-31 The Hypergeometric Distribution and the Binomial Distribution Because the hypergeometric distribution is described by three parameters N, A and n, it is practically impossible to create tables for easy use. The binomial (which has tables) is an acceptable approximation, if n < 5% N. Otherwise it is not. Excel has eliminated all the tedious calculations and allows the user to compute the exact probabilities for the hypergeometric.