Business Statistics Chapter 5 Discrete Distributions

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

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

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

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

Discrete Distribution -- Example 1 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

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

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

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

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

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

Binomial Distribution Probability function Mean value Variance and standard deviation 14

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

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

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

Excel’s Binomial Function 20 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)

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

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

Poisson Distribution Probability function Mean value Standard deviation Variance 34

Poisson Distribution: Demonstration Problem 5.7 35

Poisson Distribution: Probability Table X 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 

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 

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

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

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

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)

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

Poisson Approximation of the Binomial Distribution Error 0.2231 0.2181 -0.0051 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 -0.0011 5 0.0141 0.0131 -0.0010 6 0.0035 0.0030 -0.0005 7 0.0008 0.0006 -0.0002 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