Lecture 9. Continuous Probability Distributions
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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
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Lecture 9. Continuous Probability DistributionsDavid R. Merrell90-786 Intermediate Empirical Methods for Public Policy and Management
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AgendaNormal DistributionPoisson ProcessPoisson DistributionExponential Distribution
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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
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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
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Normal DistributionsWhy so important?Many statistical methods are based on the assumption of normalityMany populations are approximately normally distributed
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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!
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The Shape of the Normal and
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= 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?
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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?
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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
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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
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Normal Curve Output
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Poisson Processtime homogeneityindependenceno clumpingrate xxx0timeAssumptions
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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?
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Count: Poisson DistributionWhat is the probability that 3 earthquakes will strike during the next six months?
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Poisson DistributionCount in time period t
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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
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Duration: Exponential DistributionTime between occurrences in a Poisson processContinuous probability distributionMean =1/t
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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
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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
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Exponential Probability Density Function
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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
