1. Business Statistics
Chapter 5
Discrete
Distributions

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. 2

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. 3

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 4

5. Some Special Distributions
Discrete
binomial
Poisson
hypergeometric
Continuous
normal
uniform
exponential t
chi-square F 5

6. Discrete Distribution -- Example1 2 3 4 5
0.37
0.31
0.18
0.09
0.04
0.01
Number of Crises
Probability
Distribution of Daily Crises
0.1
0.2
0.3
0.4
0.5 1 2 3 4 5 P r o b a i l t y
Number of Crises 6

7. Requirements for a Discrete Probability Function
Probabilities are between 0 and 1, inclusively
Total of all probabilities equals 1 7

8. Requirements for a Discrete Probability Function -- Examples
P(X) -1 1 2 3 .1 .2 .4
1.0 X
P(X) -1 1 2 3
-.1 .3 .4 .1
1.0 X
P(X) -1 1 2 3 .1 .3 .4
1.2 8

9. Mean of a Discrete DistributionX -1 1 2 3
P(X) .1 .2 .4
-.1 .0 .3
1.0 P  ( ) 9

10. Variance and Standard Deviation of a Discrete DistributionX -1 1 2 3
P(X) .1 .2 .4 -2   4 .0
1.2 10

11. 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 13

12. Binomial Distribution
Probability function
Mean value
Variance and standard deviation 14

13. Binomial Table n = 20 PROBABILITY X 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.9
0.122
0.012
0.001
0.000 1
0.270
0.058
0.007 2
0.285
0.137
0.028
0.003 3
0.190
0.205
0.072 4
0.090
0.218
0.130
0.035
0.005 5
0.032
0.175
0.179
0.075
0.015 6
0.009
0.109
0.192
0.124
0.037 7
0.002
0.055
0.164
0.166
0.074 8
0.022
0.114
0.180
0.120
0.004 9
0.065
0.160
0.071 10
0.031
0.117
0.176 11 12 13 14 15 16 17 18 19 20
Binomial Table

14. Using the Binomial Table Demonstration Problem 5.4
PROBABILITY X
0.1
0.2
0.3
0.4
0.122
0.012
0.001
0.000 1
0.270
0.058
0.007 2
0.285
0.137
0.028
0.003 3
0.190
0.205
0.072 4
0.090
0.218
0.130
0.035 5
0.032
0.175
0.179
0.075 6
0.009
0.109
0.192
0.124 7
0.002
0.055
0.164
0.166 8
0.022
0.114
0.180 9
0.065
0.160 10
0.031
0.117 11
0.071 12
0.004 13
0.015 14
0.005 15 16 17 18 19 20

15. Binomial Distribution using Table: Demonstration Problem 5.3
PROBABILITY X
0.05
0.06
0.07
0.3585
0.2901
0.2342 1
0.3774
0.3703
0.3526 2
0.1887
0.2246
0.2521 3
0.0596
0.0860
0.1139 4
0.0133
0.0233
0.0364 5
0.0022
0.0048
0.0088 6
0.0003
0.0008
0.0017 7
0.0000
0.0001
0.0002 8 … 20 23

16. Excel’s Binomial Function20
p =
0.06 X
P(X)
=BINOMDIST(A5,B$1,B$2,FALSE) 1
=BINOMDIST(A6,B$1,B$2,FALSE) 2
=BINOMDIST(A7,B$1,B$2,FALSE) 3
=BINOMDIST(A8,B$1,B$2,FALSE) 4
=BINOMDIST(A9,B$1,B$2,FALSE) 5
=BINOMDIST(A10,B$1,B$2,FALSE) 6
=BINOMDIST(A11,B$1,B$2,FALSE) 7
=BINOMDIST(A12,B$1,B$2,FALSE) 8
=BINOMDIST(A13,B$1,B$2,FALSE) 9
=BINOMDIST(A14,B$1,B$2,FALSE)

17. 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. 32

18. Poisson Distribution: Applications
Arrivals at queuing 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 33

19. Poisson Distribution Probability function Mean value
Standard deviation
Variance 34

20. Poisson Distribution: Demonstration Problem 5.735

21. Poisson Distribution: Probability TableX
0.5
1.5
1.6
3.0
3.2
6.4
6.5
7.0
8.0
0.6065
0.2231
0.2019
0.0498
0.0408
0.0017
0.0015
0.0009
0.0003 1
0.3033
0.3347
0.3230
0.1494
0.1304
0.0106
0.0098
0.0064
0.0027 2
0.0758
0.2510
0.2584
0.2240
0.2087
0.0340
0.0318
0.0223
0.0107 3
0.0126
0.1255
0.1378
0.2226
0.0726
0.0688
0.0521
0.0286 4
0.0016
0.0471
0.0551
0.1680
0.1781
0.1162
0.1118
0.0912
0.0573 5
0.0002
0.0141
0.0176
0.1008
0.1140
0.1487
0.1454
0.1277
0.0916 6
0.0000
0.0035
0.0047
0.0504
0.0608
0.1586
0.1575
0.1490
0.1221 7
0.0008
0.0011
0.0216
0.0278
0.1450
0.1462
0.1396 8
0.0001
0.0081
0.0111
0.1160
0.1188 9
0.0040
0.0825
0.0858
0.1014
0.1241 10
0.0013
0.0528
0.0558
0.0710
0.0993 11
0.0004
0.0307
0.0330
0.0452
0.0722 12
0.0164
0.0179
0.0263
0.0481 13
0.0089
0.0142
0.0296 14
0.0037
0.0041
0.0071
0.0169 15
0.0018
0.0033
0.0090 16
0.0006
0.0007
0.0014
0.0045 17
0.0021 18 

22. Poisson Distribution: Using the Poisson TablesX
0.5
1.5
1.6
3.0
0.6065
0.2231
0.2019
0.0498 1
0.3033
0.3347
0.3230
0.1494 2
0.0758
0.2510
0.2584
0.2240 3
0.0126
0.1255
0.1378 4
0.0016
0.0471
0.0551
0.1680 5
0.0002
0.0141
0.0176
0.1008 6
0.0000
0.0035
0.0047
0.0504 7
0.0008
0.0011
0.0216 8
0.0001
0.0081 9
0.0027 10 11 12 

23. Poisson Distribution: Using the Poisson Tables X
0.5
1.5
1.6
3.0
0.6065
0.2231
0.2019
0.0498 1
0.3033
0.3347
0.3230
0.1494 2
0.0758
0.2510
0.2584
0.2240 3
0.0126
0.1255
0.1378 4
0.0016
0.0471
0.0551
0.1680 5
0.0002
0.0141
0.0176
0.1008 6
0.0000
0.0035
0.0047
0.0504 7
0.0008
0.0011
0.0216 8
0.0001
0.0081 9
0.0027 10 11 12

24. Poisson Distribution: Using the Poisson Tables X
0.5
1.5
1.6
3.0
0.6065
0.2231
0.2019
0.0498 1
0.3033
0.3347
0.3230
0.1494 2
0.0758
0.2510
0.2584
0.2240 3
0.0126
0.1255
0.1378 4
0.0016
0.0471
0.0551
0.1680 5
0.0002
0.0141
0.0176
0.1008 6
0.0000
0.0035
0.0047
0.0504 7
0.0008
0.0011
0.0216 8
0.0001
0.0081 9
0.0027 10 11 12

25. Poisson Distribution: Graphs
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35 1 2 3 4 5 6 7 8
0.02
0.04
0.06
0.08
0.12
0.14
0.16 10 12 14 16 40

26. Excel’s Poisson Function
1.6 X
P(X)
=POISSON(D5,E$1,FALSE) 1
=POISSON(D6,E$1,FALSE) 2
=POISSON(D7,E$1,FALSE) 3
=POISSON(D8,E$1,FALSE) 4
=POISSON(D9,E$1,FALSE) 5
=POISSON(D10,E$1,FALSE) 6
=POISSON(D11,E$1,FALSE) 7
=POISSON(D12,E$1,FALSE) 8
=POISSON(D13,E$1,FALSE) 9
=POISSON(D14,E$1,FALSE)

27. 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 41

28. Poisson Approximation of the Binomial Distribution
Error
0.2231
0.2181 1
0.3347
0.3372
0.0025 2
0.2510
0.2555
0.0045 3
0.1255
0.1264
0.0009 4
0.0471
0.0459 5
0.0141
0.0131 6
0.0035
0.0030 7
0.0008
0.0006 8
0.0001
0.0000 9
0.0498
0.1494
0.1493
0.2240
0.2241
0.1680
0.1681
0.1008
0.0504
0.0216
0.0081
0.0027 10 11
0.0002 12 13 42