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Lecture 9. Continuous Probability Distributions David R. Merrell 90-786 Intermediate Empirical Methods for Public Policy and Management
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Lecture 9. Continuous Probability Distributions. David R. Merrell 90-786 Intermediate Empirical Methods for Public Policy and Management. Agenda. Normal Distribution Poisson Process Poisson Distribution Exponential Distribution. Continuous Probability Distributions. - PowerPoint PPT Presentation

Transcript of Lecture 9. Continuous Probability Distributions

• Lecture 9. Continuous Probability DistributionsDavid R. Merrell90-786 Intermediate Empirical Methods for Public Policy and Management

• AgendaNormal DistributionPoisson ProcessPoisson DistributionExponential Distribution

• Continuous Probability Distributions

Random variable X can take on any value in a continuous intervalProbability density function: probabilities as areas under curveExample: f(x) = x/8 where 0 x 4 Total area under the curve is 1P(x)1/82/83/84/8 x

• Calculations

Probabilities are areasP(x < 1) is the area to the left of 1 (1/16)P(x > 2) is the area to the right of 2, i.e., between 2 and 4 (1/2)P(1 < x < 3) is the area between 1 and 3 (3/4)In generalP(x > a) is the area to the right of aP(x < 2) = P(x 2)P(x = a) = 0

• Normal DistributionsWhy so important?Many statistical methods are based on the assumption of normalityMany populations are approximately normally distributed

• Characteristics of the Normal DistributionThe graph of the distribution is bell shaped; always symmetricThe mean = median = The spread of the curve depends on , the standard deviationShow this!

• The Shape of the Normal and

• = 5

= 10

= 20

X

Sheet: Sheet1

Sheet: Sheet2

Sheet: Sheet3

Sheet: Sheet4

Education

Gender

Female

Male

Total

0.002109704641350211

0.002070393374741201

0.5

0.5

0.0010449320794148381

0.0010449320794148381

0.0020898641588296763

0.052742616033755275

0.060041407867494824

0.46296296296296297

0.5370370370370371

0.02612330198537095

0.030303030303030304

0.05642633228840126

0.3649789029535865

0.2836438923395445

0.5580645161290323

0.44193548387096776

0.180773249738767

0.14315569487983282

0.3239289446185998

0.10337552742616034

0.06625258799171843

0.6049382716049383

0.3950617283950617

0.051201671891327065

0.03343782654127482

0.08463949843260188

0.16033755274261605

0.18219461697722567

0.4634146341463415

0.5365853658536586

0.0794148380355277

0.09195402298850575

0.17136886102403343

0.31645569620253167

0.4057971014492754

0.43352601156069365

0.5664739884393064

0.15673981191222572

0.20480668756530826

0.361546499477534

Total

0.49529780564263326

0.5047021943573667

1.0

m

50.0

50.0

50.0

0.0

s

5.0

10.0

20.0

1.0

x

z

f

z

f

z

f

z

f

0.0

-10.0

1.538942419179519E-23

-5.0

1.4867414387967507E-7

-2.5

8.764279488228572E-4

-5.0

1.4867414387967506E-6

1.0

-9.8

1.1146164411079066E-22

-4.9

2.438996712279199E-7

-2.45

9.918823462692528E-4

-4.9

2.438996712279199E-6

2.0

-9.6

7.756338241447999E-22

-4.8

3.9613575067371263E-7

-2.4

0.001119743026904524

-4.8

3.961357506737126E-6

3.0

-9.4

5.1858058740934904E-21

-4.7

6.369919112150168E-7

-2.35

0.0012609295899135892

-4.7

6.369919112150168E-6

4.0

-9.2

3.3312251882908004E-20

-4.6

1.0141001608606229E-6

-2.3

0.0014163727734590433

-4.6

1.0141001608606228E-5

5.0

-9.0

2.055985032681737E-19

-4.5

1.598397681279E-6

-2.25

0.0015870059943856164

-4.5

1.598397681279E-5

6.0

-8.8

1.219169604257505E-18

-4.4

2.494283910677543E-6

-2.2

0.0017737557987984318

-4.4

2.4942839106775433E-5

7.0

-8.6

6.946027925938175E-18

-4.3

3.853576500533492E-6

-2.15

0.0019775312409066044

-4.3

3.853576500533493E-5

8.0

-8.4

3.8022191448053397E-17

-4.2

5.894393696655953E-6

-2.1

0.0021992122294527765

-4.2

5.894393696655953E-5

9.0

-8.2

1.999705238123841E-16

-4.1

8.926297348335425E-6

-2.05

0.0024396369047855106

-4.1

8.926297348335424E-5

10.0

-8.0

1.0104691174745598E-15

-4.0

1.3383219930609383E-5

-2.0

0.0026995881348268348

-4.0

1.3383219930609383E-4

11.0

-7.8

4.905782913959083E-15

-3.9

1.9865840088614268E-5

-1.95

0.002979779244414362

-3.9

1.986584008861427E-4

12.0

-7.6

2.28834672520333E-14

-3.8

2.9195123101687136E-5

-1.9

0.003280839119252373

-3.8

2.9195123101687136E-4

13.0

-7.4

1.0255658507487631E-13

-3.7

4.247865346167625E-5

-1.85

0.0036032968524154956

-3.7

4.2478653461676256E-4

14.0

-7.2

4.4160450469770973E-13

-3.6

6.11910953588609E-5

-1.8

0.0039475661273767615

-3.600000000000005

6.119109535885981E-4

15.0

-7.0

1.8269710228919904E-12

-3.5

8.72695564151007E-5

-1.75

0.004313929556193999

-3.5

8.72695564151007E-4

16.0

-6.8

7.262030092279136E-12

-3.4

1.2322373395199035E-4

-1.7

0.004702523214064666

-3.4

0.0012322373395199037

17.0

-6.6

2.773400885969352E-11

-3.3

1.7225943410939618E-4

-1.65

0.005113321631211059

-3.3

0.0017225943410939617

18.0

-6.4

1.0176430628859539E-10

-3.2

2.3841233586670591E-4

-1.6

0.005546123519248438

-3.2

0.002384123358667059

19.0

-6.2

3.587620720366468E-10

-3.1

3.266867230683606E-4

-1.55

0.006000538521103751

-3.1

0.0032668672306836056

20.0

-6.0

1.2151944896911067E-9

-3.0

4.4319137666470844E-4

-1.5

0.006475975280522813

-3.0

0.004431913766647084

21.0

-5.8

3.954697598744414E-9

-2.9

5.95262019940828E-4

-1.45

0.0069716311286284706

-2.9

0.005952620199408279

22.0

-5.6

1.2365423345631717E-8

-2.8

7.915568309002011E-4

-1.4

0.007486483680361923

-2.8

0.007915568309002011

23.0

-5.4

3.714778468666069E-8

-2.7

0.0010421088487811165

-1.35

0.008019284622558768

-2.7

0.010421088487811165

24.0

-5.2

1.0722228803514999E-7

-2.6

0.0013583169536340433

-1.3

0.008568555957620029

-2.6

0.013583169536340433

25.0

-5.0

2.9734828775935014E-7

-2.5

0.0017528558976457145

-1.25

0.009132588942128603

-2.5

0.017528558976457144

26.0

-4.8

7.922715013474253E-7

-2.4

0.002239486053809048

-1.2

0.009709445928393675

-2.4000000000000092

0.022394860538089985

27.0

-4.6

2.0282003217212457E-6

-2.3

0.0028327455469180866

-1.15

0.010296965279018622

-2.3

0.028327455469180866

28.0

-4.4

4.988567821355086E-6

-2.2

0.0035475115975968636

-1.1

0.010892769480605873

-2.2

0.035475115975968634

29.0

-4.2

1.1788787393311906E-5

-2.1

0.004398424458905553

-1.05

0.011494276533244897

-2.1

0.04398424458905553

30.0

-4.0

2.6766439861218765E-5

-2.0

0.0053991762696536695

-1.0

0.012098714638267748

-2.0

0.053991762696536695

31.0

-3.8

5.839024620337427E-5

-1.9

0.006561678238504746

-0.95

0.012703140148861638

-1.9

0.06561678238504745

32.0

-3.6

1.223821907177218E-4

-1.8

0.007895132254753523

-0.9

0.013304458687614934

-1.8

0.07895132254753523

33.0

-3.4

2.464474679039807E-4

-1.7

0.009405046428129333

-0.85

0.013899449273187581

-1.7

0.09405046428129334

34.0

-3.2

4.7682467173341183E-4

-1.6

0.011092247038496875

-0.8

0.014484791236390156

-1.6

0.11092247038496875

35.0

-3.0

8.863827533294169E-4

-1.5

0.012951950561045625

-0.75

0.01505709364544821

-1.5

0.12951950561045625

36.0

-2.8

0.0015831136618004022

-1.4

0.014972967360723847

-0.7

0.015612926902577557

-1.4

0.14972967360723846

37.0

-2.6

0.0027166339072680866

-1.3

0.017137111915240057

-0.65

0.01614885612065408

-1.3

0.17137111915240055

38.0

-2.4

0.004478972107618096

-1.2

0.01941889185678735

-0.6

0.01666147584113638

-1.2

0.19418891856787351

39.0

-2.2

0.007095023195193727

-1.1

0.021785538961211746

-0.55

0.01714744561381322

-1.1

0.21785538961211742

40.0

-2.0

0.010798352539307339

-1.0

0.024197429276535495

-0.5

0.01760352592659513

-0.9999999999999982

0.24197429276535537

41.0

-1.8

0.015790264509507046

-0.9

0.026608917375229867

-0.45

0.01802661395048167

-0.8999999999999981

0.26608917375229907

42.0

-1.6

0.02218449407699375

-0.8

0.028969582472780312

-0.4

0.018413778551843916

-0.799999999999998

0.2896958247278036

43.0

-1.4

0.029945934721447694

-0.7

0.031225853805155114

-0.35

0.01876229402186813

-0.699999999999998

0.31225853805155157

44.0

-1.2

0.0388377837135747

-0.6

0.03332295168227276

-0.3

0.019069671981761346

-0.5999999999999979

0.333229516822728

45.0

-1.0

0.04839485855307099

-0.5

0.03520705185319026

-0.25

0.019333690942202983

-0.4999999999999978

0.35207051853190297

46.0

-0.8

0.057939164945560624

-0.4

0.03682755710368783

-0.2

0.019552423026340957

-0.4

0.3682755710368783

47.0

-0.6

0.06664590336454553

-0.3

0.03813934396352269

-0.15

0.019724257406898635

-0.3

0.3813934396352269

48.0

-0.4

0.07365511420737567

-0.2

0.03910484605268191

-0.1

0.01984792005893192

-0.2

0.39104846052681913

49.0

-0.2

0.07820969210536383

-0.1

0.03969584011786384

-0.05

0.019922489489449104

-0.1

0.3969584011786384

50.0

0.0

0.07978963268897217

0.0

0.039894816344486085

0.0

0.019947408172243043

0.0

0.3989481634448608

51.0

0.2

0.07820969210536383

0.1

0.03969584011786384

0.05

0.019922489489449104

0.1

0.3969584011786384

52.0

0.4

0.07365511420737567

0.2

0.03910484605268191

0.1

0.01984792005893192

0.2

0.39104846052681913

53.0

0.6

0.06664590336454553

0.3

0.03813934396352269

0.15

0.019724257406898635

0.3

0.3813934396352269

54.0

0.8

0.057939164945560624

0.4

0.03682755710368783

0.2

0.019552423026340957

0.4

0.3682755710368783

55.0

1.0

0.04839485855307099

0.5

0.03520705185319026

0.25

0.019333690942202983

0.5000000000000031

0.352070518531902

56.0

1.2

0.0388377837135747

0.6

0.03332295168227276

0.3

0.019069671981761346

0.6000000000000032

0.33322951682272695

57.0

1.4

0.029945934721447694

0.7

0.031225853805155114

0.35

0.01876229402186813

0.7000000000000033

0.3122585380515504

58.0

1.6

0.02218449407699375

0.8

0.028969582472780312

0.4

0.018413778551843916

0.8000000000000034

0.28969582472780236

59.0

1.8

0.015790264509507046

0.9

0.026608917375229867

0.45

0.01802661395048167

0.9000000000000035

0.2660891737522978

60.0

2.0

0.010798352539307339

1.0

0.024197429276535495

0.5

0.01760352592659513

1.0

0.24197429276535493

61.0

2.2

0.007095023195193727

1.1

0.021785538961211746

0.55

0.01714744561381322

1.1

0.21785538961211742

62.0

2.4

0.004478972107618096

1.2

0.01941889185678735

0.6

0.01666147584113638

1.2

0.19418891856787351

63.0

2.6

0.0027166339072680866

1.3

0.017137111915240057

0.65

0.01614885612065408

1.3

0.17137111915240055

64.0

2.8

0.0015831136618004022

1.4

0.014972967360723847

0.7

0.015612926902577557

1.4

0.14972967360723846

65.0

3.0

8.863827533294169E-4

1.5

0.012951950561045625

0.75

0.01505709364544821

1.5

0.12951950561045625

66.0

3.2

4.7682467173341183E-4

1.6

0.011092247038496875

0.8

0.014484791236390156

1.6

0.11092247038496875

67.0

3.4

2.464474679039807E-4

1.7

0.009405046428129333

0.85

0.013899449273187581

1.7

0.09405046428129334

68.0

3.6

1.223821907177218E-4

1.8

0.007895132254753523

0.9

0.013304458687614934

1.8

0.07895132254753523

69.0

3.8

5.839024620337427E-5

1.9

0.006561678238504746

0.95

0.012703140148861638

1.9

0.06561678238504745

70.0

4.0

2.6766439861218765E-5

2.0

0.0053991762696536695

1.0

0.012098714638267748

2.0

0.053991762696536695

71.0

4.2

1.1788787393311906E-5

2.1

0.004398424458905553

1.05

0.011494276533244897

2.100000000000005

0.04398424458905508

72.0

4.4

4.988567821355086E-6

2.2

0.0035475115975968636

1.1

0.010892769480605873

2.2

0.035475115975968634

73.0

4.6

2.0282003217212457E-6

2.3

0.0028327455469180866

1.15

0.010296965279018622

2.3

0.028327455469180866

74.0

4.8

7.922715013474253E-7

2.4

0.002239486053809048

1.2

0.009709445928393675

2.4

0.022394860538090477

75.0

5.0

2.9734828775935014E-7

2.5

0.0017528558976457145

1.25

0.009132588942128603

2.500000000000025

0.017528558976456055

76.0

5.2

1.0722228803514999E-7

2.6

0.0013583169536340433

1.3

0.008568555957620029

2.60000000000003

0.013583169536339385

77.0

5.4

3.714778468666069E-8

2.7

0.0010421088487811165

1.35

0.008019284622558768

2.700000000000035

0.010421088487810187

78.0

5.6

1.2365423345631717E-8

2.8

7.915568309002011E-4

1.4

0.007486483680361923

2.80000000000004

0.007915568309001125

79.0

5.8

3.954697598744414E-9

2.9

5.95262019940828E-4

1.45

0.0069716311286284706

2.9000000000000448

0.0059526201994075075

80.0

6.0

1.2151944896911067E-9

3.0

4.4319137666470844E-4

1.5

0.006475975280522813

2.999999999999994

0.004431913766647168

81.0

6.2

3.587620720366468E-10

3.1

3.266867230683606E-4

1.55

0.006000538521103751

3.0999999999999943

0.0032668672306836667

82.0

6.4

1.0176430628859539E-10

3.2

2.3841233586670591E-4

1.6

0.005546123519248438

3.2

0.002384123358667059

83.0

6.6

2.773400885969352E-11

3.3

1.7225943410939618E-4

1.65

0.005113321631211059

3.3

0.0017225943410939617

84.0

6.8

7.262030092279136E-12

3.4

1.2322373395199035E-4

1.7

0.004702523214064666

3.4

0.0012322373395199037

85.0

7.0

1.8269710228919904E-12

3.5

8.72695564151007E-5

1.75

0.004313929556193999

3.5

8.72695564151007E-4

86.0

7.2

4.4160450469770973E-13

3.6

6.11910953588609E-5

1.8

0.0039475661273767615

3.6

6.119109535886089E-4

87.0

7.4

1.0255658507487631E-13

3.7

4.247865346167625E-5

1.85

0.0036032968524154956

3.7

4.2478653461676256E-4

88.0

7.6

2.28834672520333E-14

3.8

2.9195123101687136E-5

1.9

0.003280839119252373

3.8

2.9195123101687136E-4

89.0

7.8

4.905782913959083E-15

3.9

1.9865840088614268E-5

1.95

0.002979779244414362

3.9

1.986584008861427E-4

90.0

8.0

1.0104691174745598E-15

4.0

1.3383219930609383E-5

2.0

0.0026995881348268348

4.0

1.3383219930609383E-4

91.0

8.2

1.999705238123841E-16

4.1

8.926297348335425E-6

2.05

0.0024396369047855106

4.1

8.926297348335424E-5

92.0

8.4

3.8022191448053397E-17

4.2

5.894393696655953E-6

2.1

0.0021992122294527765

4.2

5.894393696655953E-5

93.0

8.6

6.946027925938175E-18

4.3

3.853576500533492E-6

2.15

0.0019775312409066044

4.3

3.853576500533493E-5

94.0

8.8

1.219169604257505E-18

4.4

2.494283910677543E-6

2.2

0.0017737557987984318

4.4

2.4942839106775433E-5

95.0

9.0

2.055985032681737E-19

4.5

1.598397681279E-6

2.25

0.0015870059943856164

4.5

1.598397681279E-5

96.0

9.2

3.3312251882908004E-20

4.6

1.0141001608606229E-6

2.3

0.0014163727734590433

4.6

1.0141001608606228E-5

97.0

9.4

5.1858058740934904E-21

4.7

6.369919112150168E-7

2.35

0.0012609295899135892

4.7

6.369919112150168E-6

98.0

9.6

7.756338241447999E-22

4.8

3.9613575067371263E-7

2.4

0.001119743026904524

4.8

3.961357506737126E-6

99.0

9.8

1.1146164411079066E-22

4.9

2.438996712279199E-7

2.45

9.918823462692528E-4

4.9

2.438996712279199E-6

• Standard Normal DistributionNormal distribution with = 0 and = 1The standard normal random variable is called ZCan standardize any normal random variable: z score

Z = (X - ) /

• Calculating Probabilities

Table of standard normal distributionPDF template in ExcelExample: X normally distributed with = 20 and = 5 Find:Probability that x is more than 30Probability that x is at least 15Probability that x is between 15 and 25Probability that x is between 10 and 30

• Percentages of the Area Under a Normal CurveShow this!

Z statistics

Range

% of the Area

(1.00

m ( s

68.26%

(1.96

m ( 1.96s

95.00%

(2.00

m ( 2s

95.44%

(2.58

m ( 2.58s

99.00%

(3.00

m ( 3s

99.74%

• Percentages of the Area Under a Normal Curve

• Z score

68.3%

95.5%

99.7%

Sheet: Sheet1

Sheet: Sheet2

Sheet: Sheet3

Sheet: Sheet4

Education

Gender

Female

Male

Total

0.002109704641350211

0.002070393374741201

0.5

0.5

0.0010449320794148381

0.0010449320794148381

0.0020898641588296763

0.052742616033755275

0.060041407867494824

0.46296296296296297

0.5370370370370371

0.02612330198537095

0.030303030303030304

0.05642633228840126

0.3649789029535865

0.2836438923395445

0.5580645161290323

0.44193548387096776

0.180773249738767

0.14315569487983282

0.3239289446185998

0.10337552742616034

0.06625258799171843

0.6049382716049383

0.3950617283950617

0.051201671891327065

0.03343782654127482

0.08463949843260188

0.16033755274261605

0.18219461697722567

0.4634146341463415

0.5365853658536586

0.0794148380355277

0.09195402298850575

0.17136886102403343

0.31645569620253167

0.4057971014492754

0.43352601156069365

0.5664739884393064

0.15673981191222572

0.20480668756530826

0.361546499477534

Total

0.49529780564263326

0.5047021943573667

1.0

m

50.0

50.0

50.0

0.0

s

5.0

10.0

20.0

1.0

x

z

f

z

f

z

f

z

f

0.0

-10.0

1.538942419179519E-23

-5.0

1.4867414387967507E-7

-2.5

8.764279488228572E-4

-5.0

1.4867414387967506E-6

1.0

-9.8

1.1146164411079066E-22

-4.9

2.438996712279199E-7

-2.45

9.918823462692528E-4

-4.9

2.438996712279199E-6

2.0

-9.6

7.756338241447999E-22

-4.8

3.9613575067371263E-7

-2.4

0.001119743026904524

-4.8

3.961357506737126E-6

3.0

-9.4

5.1858058740934904E-21

-4.7

6.369919112150168E-7

-2.35

0.0012609295899135892

-4.7

6.369919112150168E-6

4.0

-9.2

3.3312251882908004E-20

-4.6

1.0141001608606229E-6

-2.3

0.0014163727734590433

-4.6

1.0141001608606228E-5

5.0

-9.0

2.055985032681737E-19

-4.5

1.598397681279E-6

-2.25

0.0015870059943856164

-4.5

1.598397681279E-5

6.0

-8.8

1.219169604257505E-18

-4.4

2.494283910677543E-6

-2.2

0.0017737557987984318

-4.4

2.4942839106775433E-5

7.0

-8.6

6.946027925938175E-18

-4.3

3.853576500533492E-6

-2.15

0.0019775312409066044

-4.3

3.853576500533493E-5

8.0

-8.4

3.8022191448053397E-17

-4.2

5.894393696655953E-6

-2.1

0.0021992122294527765

-4.2

5.894393696655953E-5

9.0

-8.2

1.999705238123841E-16

-4.1

8.926297348335425E-6

-2.05

0.0024396369047855106

-4.1

8.926297348335424E-5

10.0

-8.0

1.0104691174745598E-15

-4.0

1.3383219930609383E-5

-2.0

0.0026995881348268348

-4.0

1.3383219930609383E-4

11.0

-7.8

4.905782913959083E-15

-3.9

1.9865840088614268E-5

-1.95

0.002979779244414362

-3.9

1.986584008861427E-4

12.0

-7.6

2.28834672520333E-14

-3.8

2.9195123101687136E-5

-1.9

0.003280839119252373

-3.8

2.9195123101687136E-4

13.0

-7.4

1.0255658507487631E-13

-3.7

4.247865346167625E-5

-1.85

0.0036032968524154956

-3.7

4.2478653461676256E-4

14.0

-7.2

4.4160450469770973E-13

-3.6

6.11910953588609E-5

-1.8

0.0039475661273767615

-3.600000000000005

6.119109535885981E-4

15.0

-7.0

1.8269710228919904E-12

-3.5

8.72695564151007E-5

-1.75

0.004313929556193999

-3.5

8.72695564151007E-4

16.0

-6.8

7.262030092279136E-12

-3.4

1.2322373395199035E-4

-1.7

0.004702523214064666

-3.4

0.0012322373395199037

17.0

-6.6

2.773400885969352E-11

-3.3

1.7225943410939618E-4

-1.65

0.005113321631211059

-3.3

0.0017225943410939617

18.0

-6.4

1.0176430628859539E-10

-3.2

2.3841233586670591E-4

-1.6

0.005546123519248438

-3.2

0.002384123358667059

19.0

-6.2

3.587620720366468E-10

-3.1

3.266867230683606E-4

-1.55

0.006000538521103751

-3.1

0.0032668672306836056

20.0

-6.0

1.2151944896911067E-9

-3.0

4.4319137666470844E-4

-1.5

0.006475975280522813

-3.0

0.004431913766647084

21.0

-5.8

3.954697598744414E-9

-2.9

5.95262019940828E-4

-1.45

0.0069716311286284706

-2.9

0.005952620199408279

22.0

-5.6

1.2365423345631717E-8

-2.8

7.915568309002011E-4

-1.4

0.007486483680361923

-2.8

0.007915568309002011

23.0

-5.4

3.714778468666069E-8

-2.7

0.0010421088487811165

-1.35

0.008019284622558768

-2.7

0.010421088487811165

24.0

-5.2

1.0722228803514999E-7

-2.6

0.0013583169536340433

-1.3

0.008568555957620029

-2.6

0.013583169536340433

25.0

-5.0

2.9734828775935014E-7

-2.5

0.0017528558976457145

-1.25

0.009132588942128603

-2.5

0.017528558976457144

26.0

-4.8

7.922715013474253E-7

-2.4

0.002239486053809048

-1.2

0.009709445928393675

-2.4000000000000092

0.022394860538089985

27.0

-4.6

2.0282003217212457E-6

-2.3

0.0028327455469180866

-1.15

0.010296965279018622

-2.3

0.028327455469180866

28.0

-4.4

4.988567821355086E-6

-2.2

0.0035475115975968636

-1.1

0.010892769480605873

-2.2

0.035475115975968634

29.0

-4.2

1.1788787393311906E-5

-2.1

0.004398424458905553

-1.05

0.011494276533244897

-2.1

0.04398424458905553

30.0

-4.0

2.6766439861218765E-5

-2.0

0.0053991762696536695

-1.0

0.012098714638267748

-2.0

0.053991762696536695

31.0

-3.8

5.839024620337427E-5

-1.9

0.006561678238504746

-0.95

0.012703140148861638

-1.9

0.06561678238504745

32.0

-3.6

1.223821907177218E-4

-1.8

0.007895132254753523

-0.9

0.013304458687614934

-1.8

0.07895132254753523

33.0

-3.4

2.464474679039807E-4

-1.7

0.009405046428129333

-0.85

0.013899449273187581

-1.7

0.09405046428129334

34.0

-3.2

4.7682467173341183E-4

-1.6

0.011092247038496875

-0.8

0.014484791236390156

-1.6

0.11092247038496875

35.0

-3.0

8.863827533294169E-4

-1.5

0.012951950561045625

-0.75

0.01505709364544821

-1.5

0.12951950561045625

36.0

-2.8

0.0015831136618004022

-1.4

0.014972967360723847

-0.7

0.015612926902577557

-1.4

0.14972967360723846

37.0

-2.6

0.0027166339072680866

-1.3

0.017137111915240057

-0.65

0.01614885612065408

-1.3

0.17137111915240055

38.0

-2.4

0.004478972107618096

-1.2

0.01941889185678735

-0.6

0.01666147584113638

-1.2

0.19418891856787351

39.0

-2.2

0.007095023195193727

-1.1

0.021785538961211746

-0.55

0.01714744561381322

-1.1

0.21785538961211742

40.0

-2.0

0.010798352539307339

-1.0

0.024197429276535495

-0.5

0.01760352592659513

-0.9999999999999982

0.24197429276535537

41.0

-1.8

0.015790264509507046

-0.9

0.026608917375229867

-0.45

0.01802661395048167

-0.8999999999999981

0.26608917375229907

42.0

-1.6

0.02218449407699375

-0.8

0.028969582472780312

-0.4

0.018413778551843916

-0.799999999999998

0.2896958247278036

43.0

-1.4

0.029945934721447694

-0.7

0.031225853805155114

-0.35

0.01876229402186813

-0.699999999999998

0.31225853805155157

44.0

-1.2

0.0388377837135747

-0.6

0.03332295168227276

-0.3

0.019069671981761346

-0.5999999999999979

0.333229516822728

45.0

-1.0

0.04839485855307099

-0.5

0.03520705185319026

-0.25

0.019333690942202983

-0.4999999999999978

0.35207051853190297

46.0

-0.8

0.057939164945560624

-0.4

0.03682755710368783

-0.2

0.019552423026340957

-0.4

0.3682755710368783

47.0

-0.6

0.06664590336454553

-0.3

0.03813934396352269

-0.15

0.019724257406898635

-0.3

0.3813934396352269

48.0

-0.4

0.07365511420737567

-0.2

0.03910484605268191

-0.1

0.01984792005893192

-0.2

0.39104846052681913

49.0

-0.2

0.07820969210536383

-0.1

0.03969584011786384

-0.05

0.019922489489449104

-0.1

0.3969584011786384

50.0

0.0

0.07978963268897217

0.0

0.039894816344486085

0.0

0.019947408172243043

0.0

0.3989481634448608

51.0

0.2

0.07820969210536383

0.1

0.03969584011786384

0.05

0.019922489489449104

0.1

0.3969584011786384

52.0

0.4

0.07365511420737567

0.2

0.03910484605268191

0.1

0.01984792005893192

0.2

0.39104846052681913

53.0

0.6

0.06664590336454553

0.3

0.03813934396352269

0.15

0.019724257406898635

0.3

0.3813934396352269

54.0

0.8

0.057939164945560624

0.4

0.03682755710368783

0.2

0.019552423026340957

0.4

0.3682755710368783

55.0

1.0

0.04839485855307099

0.5

0.03520705185319026

0.25

0.019333690942202983

0.5000000000000031

0.352070518531902

56.0

1.2

0.0388377837135747

0.6

0.03332295168227276

0.3

0.019069671981761346

0.6000000000000032

0.33322951682272695

57.0

1.4

0.029945934721447694

0.7

0.031225853805155114

0.35

0.01876229402186813

0.7000000000000033

0.3122585380515504

58.0

1.6

0.02218449407699375

0.8

0.028969582472780312

0.4

0.018413778551843916

0.8000000000000034

0.28969582472780236

59.0

1.8

0.015790264509507046

0.9

0.026608917375229867

0.45

0.01802661395048167

0.9000000000000035

0.2660891737522978

60.0

2.0

0.010798352539307339

1.0

0.024197429276535495

0.5

0.01760352592659513

1.0

0.24197429276535493

61.0

2.2

0.007095023195193727

1.1

0.021785538961211746

0.55

0.01714744561381322

1.1

0.21785538961211742

62.0

2.4

0.004478972107618096

1.2

0.01941889185678735

0.6

0.01666147584113638

1.2

0.19418891856787351

63.0

2.6

0.0027166339072680866

1.3

0.017137111915240057

0.65

0.01614885612065408

1.3

0.17137111915240055

64.0

2.8

0.0015831136618004022

1.4

0.014972967360723847

0.7

0.015612926902577557

1.4

0.14972967360723846

65.0

3.0

8.863827533294169E-4

1.5

0.012951950561045625

0.75

0.01505709364544821

1.5

0.12951950561045625

66.0

3.2

4.7682467173341183E-4

1.6

0.011092247038496875

0.8

0.014484791236390156

1.6

0.11092247038496875

67.0

3.4

2.464474679039807E-4

1.7

0.009405046428129333

0.85

0.013899449273187581

1.7

0.09405046428129334

68.0

3.6

1.223821907177218E-4

1.8

0.007895132254753523

0.9

0.013304458687614934

1.8

0.07895132254753523

69.0

3.8

5.839024620337427E-5

1.9

0.006561678238504746

0.95

0.012703140148861638

1.9

0.06561678238504745

70.0

4.0

2.6766439861218765E-5

2.0

0.0053991762696536695

1.0

0.012098714638267748

2.0

0.053991762696536695

71.0

4.2

1.1788787393311906E-5

2.1

0.004398424458905553

1.05

0.011494276533244897

2.100000000000005

0.04398424458905508

72.0

4.4

4.988567821355086E-6

2.2

0.0035475115975968636

1.1

0.010892769480605873

2.2

0.035475115975968634

73.0

4.6

2.0282003217212457E-6

2.3

0.0028327455469180866

1.15

0.010296965279018622

2.3

0.028327455469180866

74.0

4.8

7.922715013474253E-7

2.4

0.002239486053809048

1.2

0.009709445928393675

2.4

0.022394860538090477

75.0

5.0

2.9734828775935014E-7

2.5

0.0017528558976457145

1.25

0.009132588942128603

2.500000000000025

0.017528558976456055

76.0

5.2

1.0722228803514999E-7

2.6

0.0013583169536340433

1.3

0.008568555957620029

2.60000000000003

0.013583169536339385

77.0

5.4

3.714778468666069E-8

2.7

0.0010421088487811165

1.35

0.008019284622558768

2.700000000000035

0.010421088487810187

78.0

5.6

1.2365423345631717E-8

2.8

7.915568309002011E-4

1.4

0.007486483680361923

2.80000000000004

0.007915568309001125

79.0

5.8

3.954697598744414E-9

2.9

5.95262019940828E-4

1.45

0.0069716311286284706

2.9000000000000448

0.0059526201994075075

80.0

6.0

1.2151944896911067E-9

3.0

4.4319137666470844E-4

1.5

0.006475975280522813

2.999999999999994

0.004431913766647168

81.0

6.2

3.587620720366468E-10

3.1

3.266867230683606E-4

1.55

0.006000538521103751

3.0999999999999943

0.0032668672306836667

82.0

6.4

1.0176430628859539E-10

3.2

2.3841233586670591E-4

1.6

0.005546123519248438

3.2

0.002384123358667059

83.0

6.6

2.773400885969352E-11

3.3

1.7225943410939618E-4

1.65

0.005113321631211059

3.3

0.0017225943410939617

84.0

6.8

7.262030092279136E-12

3.4

1.2322373395199035E-4

1.7

0.004702523214064666

3.4

0.0012322373395199037

85.0

7.0

1.8269710228919904E-12

3.5

8.72695564151007E-5

1.75

0.004313929556193999

3.5

8.72695564151007E-4

86.0

7.2

4.4160450469770973E-13

3.6

6.11910953588609E-5

1.8

0.0039475661273767615

3.6

6.119109535886089E-4

87.0

7.4

1.0255658507487631E-13

3.7

4.247865346167625E-5

1.85

0.0036032968524154956

3.7

4.2478653461676256E-4

88.0

7.6

2.28834672520333E-14

3.8

2.9195123101687136E-5

1.9

0.003280839119252373

3.8

2.9195123101687136E-4

89.0

7.8

4.905782913959083E-15

3.9

1.9865840088614268E-5

1.95

0.002979779244414362

3.9

1.986584008861427E-4

90.0

8.0

1.0104691174745598E-15

4.0

1.3383219930609383E-5

2.0

0.0026995881348268348

4.0

1.3383219930609383E-4

91.0

8.2

1.999705238123841E-16

4.1

8.926297348335425E-6

2.05

0.0024396369047855106

4.1

8.926297348335424E-5

92.0

8.4

3.8022191448053397E-17

4.2

5.894393696655953E-6

2.1

0.0021992122294527765

4.2

5.894393696655953E-5

93.0

8.6

6.946027925938175E-18

4.3

3.853576500533492E-6

2.15

0.0019775312409066044

4.3

3.853576500533493E-5

94.0

8.8

1.219169604257505E-18

4.4

2.494283910677543E-6

2.2

0.0017737557987984318

4.4

2.4942839106775433E-5

95.0

9.0

2.055985032681737E-19

4.5

1.598397681279E-6

2.25

0.0015870059943856164

4.5

1.598397681279E-5

96.0

9.2

3.3312251882908004E-20

4.6

1.0141001608606229E-6

2.3

0.0014163727734590433

4.6

1.0141001608606228E-5

97.0

9.4

5.1858058740934904E-21

4.7

6.369919112150168E-7

2.35

0.0012609295899135892

4.7

6.369919112150168E-6

98.0

9.6

7.756338241447999E-22

4.8

3.9613575067371263E-7

2.4

0.001119743026904524

4.8

3.961357506737126E-6

99.0

9.8

1.1146164411079066E-22

4.9

2.438996712279199E-7

2.45

9.918823462692528E-4

4.9

2.438996712279199E-6

• Example 1. Normal ProbabilityAn agency is hiring college graduates for analyst positions. Candidate must score in the top 10% of all taking an exam. The mean exam score is 85 and the standard deviation is 6. What is the minimum score needed?Joe scored 90 point on the exam. What percent of the applicants scored above him?The agency changed its criterion to consider all candidates with score of 91 and above. What percent score above 91?

• Example 2. Normal Probability ProblemThe salaries of professional employees in a certain agency are normally distributed with a mean of \$57k and a standard deviation of \$14k.What percentage of employees would have a salary under \$40k?

• Minitab for Probability

Click: Calc > Probability Distributions > Normal Enter: For mean 57, standard deviation 14, input constant 40Output:Cumulative Distribution FunctionNormal with mean = 57.0000 and standard deviation = 14.0000 x P( X

• Plotting a Normal Curve

MTB > set c1DATA > 15:99DATA > endClick: Calc > Probability distributions > Normal > Probability density > Input columnEnter: Input column c1 > Optional storage c2Click: OK > Graph > PlotEnter: Y c2 > X c1Click: Display > Connect > OK

• Normal Curve Output

• Poisson Processtime homogeneityindependenceno clumpingrate xxx0timeAssumptions

• Poisson ProcessEarthquakes strike randomly over time with a rate of = 4 per year.Model time of earthquake strike as a Poisson processCount: How many earthquakes will strike in the next six months?Duration: How long will it take before the next earthquake hits?

• Count: Poisson DistributionWhat is the probability that 3 earthquakes will strike during the next six months?

• Poisson DistributionCount in time period t

• Minitab Probability CalculationClick: Calc > Probability Distributions > PoissonEnter: For mean 2, input constant 3Output:Probability Density FunctionPoisson with mu = 2.00000 x P( X = x) 3.00 0.1804

• Duration: Exponential DistributionTime between occurrences in a Poisson processContinuous probability distributionMean =1/t

• Exponential Probability ProblemWhat is the probability that 9 months will pass with no earthquake?t = 1/12 = 1/31/ t = 3

• Minitab Probability CalculationClick: Calc > Probability Distributions > ExponentialEnter: For mean 3, input constant 9Output:Cumulative Distribution FunctionExponential with mean = 3.00000 x P( X
• Exponential Probability Density Function

MTB > set c1DATA > 0:12000DATA > endLet c1 = c1/1000Click: Calc > Probability distributions > Exponential > Probability density > Input columnEnter: Input column c1 > Optional storage c2Click: OK > Graph > PlotEnter: Y c2 > X c1Click: Display > Connect > OK

• Exponential Probability Density Function

• Next Time: Random Sampling and Sampling DistributionsNormal approximation to binomial distributionPoisson processRandom samplingSampling statistics and sampling distributionsExpected values and standard errors of sample sums and sample means