+ All Categories
Home > Documents > The Multivariate Normal Distribution

The Multivariate Normal Distribution

Date post: 27-May-2017
Category:

Documents
Upload: cleopatra-cristina-doibani
View: 234 times
Download: 1 times
Download Report this document
Share this document with a friend
118
II. The Multivariate Normal Distribution “…it is not enough to know that a sample could have come from a normal population; we must be clear that it is at the same time improbable that it has come from a population differing so much from the normal as to invalidate the use of the ‘normal theory’ tests in further handling of the material.” E. S. Pearson, 1930 (quoted on page 1 in Tests of Normality, Henry C. Thode, Jr., 2002)
Transcript

  • II. The MultivariateNormal Distributionit is not enough to know that a sample could have come from a normal population; we must be clear that it is at the same time improbable that it has come from a population differing so much from the normal as to invalidate the use of the normal theory tests in further handling of the material.

    E. S. Pearson, 1930 (quoted on page 1 in Tests of Normality, Henry C. Thode, Jr., 2002)

  • A.Review of the Univariate Normal Distribution

    Normal Probability Distribution - expresses the probabilities of outcomes for a continuous random variable x with a particular symmetric and unimodal distribution. This density function is given bywhere = mean = standard deviation = 3.14159 e = 2.71828

  • but the probability is given byThis looks like a difficult integration problem! Will I have to integrate this function every time I want to calculate probabilities for some normal random variable?

  • Characteristics of the normal probability distribution are: -there are an infinite number of normal distributions, each defined by their unique combination of the mean and standard deviation - determines the central location and determines the spread or width -the distribution is symmetric about -it is unimodal - = Md = Mo -it is asymptotic with respect to the horizontal axis -the area under the curve is 1.0 -it is neither platykurtic nor leptokurtic -it follows the empirical rule:

  • Normal distributions with the same mean but different standard deviations:

    Chart1

    0.00548531950.00000389560.0000000001

    0.01275263420.00002599960.0000000034

    0.02785205770.00015076320.0000000765

    0.0571444530.00075955310.0000013551

    0.11014132150.00332471650.0000187022

    0.19942789050.01264400760.0002010176

    0.33921958390.04177808650.0016827291

    0.54204500650.11993502070.0109706389

    0.81367165630.29914183580.0557041154

    1.14742005720.6482491640.2202826431

    1.52004027681.22050748450.6784391678

    1.89167778351.99651510061.6273433127

    2.21155956192.83751667533.0400805535

    2.42889939383.50378876974.4231191238

    2.50599281723.75898922585.0119856345

    2.42889939383.50378876974.4231191238

    2.21155956192.83751667533.0400805535

    1.89167778351.99651510061.6273433127

    1.52004027681.22050748450.6784391678

    1.14742005720.6482491640.2202826431

    0.81367165630.29914183580.0557041154

    0.54204500650.11993502070.0109706389

    0.33921958390.04177808650.0016827291

    0.19942789050.01264400760.0002010176

    0.11014132150.00332471650.0000187022

    0.0571444530.00075955310.0000013551

    0.02785205770.00015076320.0000000765

    0.01275263420.00002599960.0000000034

    0.00548531950.00000389560.0000000001

    m x

    f(x)

    Sheet1

    000

    11.52

    -3.5182.304787610.0054853195256701.9601512720.00000389568691640725.506530.0000000001

    -3.2578.41517185580.012752634238462.16619577360.0000259996297515683.4305660.0000000034

    -335.90398996990.02785205776632.9179428810.000150763213076162.99982720.0000000765

    -2.7517.49951127870.0571444531316.56358371720.0007595531737926.5734041250.0000013551

    -2.59.07924460990.1101413215300.77752492690.003324716553469.77263904550.0000187022

    -2.255.01434376770.199427890579.08884867910.01264400764974.68845716070.0002010176

    -22.94794300650.339219583923.93599331320.0417780865594.27272977890.0016827291

    -1.751.84486525670.54204500658.33784822880.119935020791.1523938050.0109706389

    -1.51.22899696970.81367165633.34289584510.299141835817.95199498490.0557041154

    -1.250.87152041121.14742005721.54261672130.6482491644.5396223050.2202826431

    -10.65787730451.52004027680.81933131321.22050748451.47397150330.6784391678

    -0.750.52863125461.89167778350.50087274561.99651510060.61449848491.6273433127

    -0.50.45216959892.21155956190.35242083642.83751667530.32893865223.0400805535

    -0.250.41170910682.42889939380.28540533283.50378876970.22608479944.4231191238

    00.39904344222.50599281720.26602896153.75898922580.19952172115.0119856345

    0.250.41170910682.42889939380.28540533283.50378876970.22608479944.4231191238

    0.50.45216959892.21155956190.35242083642.83751667530.32893865223.0400805535

    0.750.52863125461.89167778350.50087274561.99651510060.61449848491.6273433127

    10.65787730451.52004027680.81933131321.22050748451.47397150330.6784391678

    1.250.87152041121.14742005721.54261672130.6482491644.5396223050.2202826431

    1.51.22899696970.81367165633.34289584510.299141835817.95199498490.0557041154

    1.751.84486525670.54204500658.33784822880.119935020791.1523938050.0109706389

    22.94794300650.339219583923.93599331320.0417780865594.27272977890.0016827291

    2.255.01434376770.199427890579.08884867910.01264400764974.68845716070.0002010176

    2.59.07924460990.1101413215300.77752492690.003324716553469.77263904550.0000187022

    2.7517.49951127870.0571444531316.56358371720.0007595531737926.5734041250.0000013551

    335.90398996990.02785205776632.9179428810.000150763213076162.99982720.0000000765

    3.2578.41517185580.012752634238462.16619577360.0000259996297515683.4305660.0000000034

    3.5182.304787610.0054853195256701.9601512720.00000389568691640725.506530.0000000001

    00.51

    111

    -3.5182.304787610.00548531951188.54545955790.00084136459949.37691432150.0001005088

    -3.2578.41517185580.0127526342451.16628739040.00221647773333.00177608720.0003000298

    -335.90398996990.0278520577182.304787610.00548531951188.54545955790.0008413645

    -2.7517.49951127870.05714445378.41517185580.0127526342451.16628739040.0022164777

    -2.59.07924460990.110141321535.90398996990.0278520577182.304787610.0054853195

    -2.255.01434376770.199427890517.49951127870.05714445378.41517185580.0127526342

    -22.94794300650.33921958399.07924460990.110141321535.90398996990.0278520577

    -1.751.84486525670.54204500655.01434376770.199427890517.49951127870.057144453

    -1.51.22899696970.81367165632.94794300650.33921958399.07924460990.1101413215

    -1.250.87152041121.14742005721.84486525670.54204500655.01434376770.1994278905

    -10.65787730451.52004027681.22899696970.81367165632.94794300650.3392195839

    -0.750.52863125461.89167778350.87152041121.14742005721.84486525670.5420450065

    -0.50.45216959892.21155956190.65787730451.52004027681.22899696970.8136716563

    -0.250.41170910682.42889939380.52863125461.89167778350.87152041121.1474200572

    00.39904344222.50599281720.45216959892.21155956190.65787730451.5200402768

    0.250.41170910682.42889939380.41170910682.42889939380.52863125461.8916777835

    0.50.45216959892.21155956190.39904344222.50599281720.45216959892.2115595619

    0.750.52863125461.89167778350.41170910682.42889939380.41170910682.4288993938

    10.65787730451.52004027680.45216959892.21155956190.39904344222.5059928172

    1.250.87152041121.14742005720.52863125461.89167778350.41170910682.4288993938

    1.51.22899696970.81367165630.65787730451.52004027680.45216959892.2115595619

    1.751.84486525670.54204500650.87152041121.14742005720.52863125461.8916777835

    22.94794300650.33921958391.22899696970.81367165630.65787730451.5200402768

    2.255.01434376770.19942789051.84486525670.54204500650.87152041121.1474200572

    2.59.07924460990.11014132152.94794300650.33921958391.22899696970.8136716563

    2.7517.49951127870.0571444535.01434376770.19942789051.84486525670.5420450065

    335.90398996990.02785205779.07924460990.11014132152.94794300650.3392195839

    3.2578.41517185580.012752634217.49951127870.0571444535.01434376770.1994278905

    3.5182.304787610.005485319535.90398996990.02785205779.07924460990.1101413215

    &A

    Page &P

    Sheet1

    &A

    Page &P

    m x

    f(x)

    m m m x

    f(x)

  • Normal distributions with the same standard deviation but different means:

    Chart3

    0.00548531950.00084136450.0001005088

    0.01275263420.00221647770.0003000298

    0.02785205770.00548531950.0008413645

    0.0571444530.01275263420.0022164777

    0.11014132150.02785205770.0054853195

    0.19942789050.0571444530.0127526342

    0.33921958390.11014132150.0278520577

    0.54204500650.19942789050.057144453

    0.81367165630.33921958390.1101413215

    1.14742005720.54204500650.1994278905

    1.52004027680.81367165630.3392195839

    1.89167778351.14742005720.5420450065

    2.21155956191.52004027680.8136716563

    2.42889939381.89167778351.1474200572

    2.50599281722.21155956191.5200402768

    2.42889939382.42889939381.8916777835

    2.21155956192.50599281722.2115595619

    1.89167778352.42889939382.4288993938

    1.52004027682.21155956192.5059928172

    1.14742005721.89167778352.4288993938

    0.81367165631.52004027682.2115595619

    0.54204500651.14742005721.8916777835

    0.33921958390.81367165631.5200402768

    0.19942789050.54204500651.1474200572

    0.11014132150.33921958390.8136716563

    0.0571444530.19942789050.5420450065

    0.02785205770.11014132150.3392195839

    0.01275263420.0571444530.1994278905

    0.00548531950.02785205770.1101413215

    m m m x

    f(x)

    Sheet1

    000

    11.52

    -3.5182.304787610.0054853195256701.9601512720.00000389568691640725.506530.0000000001

    -3.2578.41517185580.012752634238462.16619577360.0000259996297515683.4305660.0000000034

    -335.90398996990.02785205776632.9179428810.000150763213076162.99982720.0000000765

    -2.7517.49951127870.0571444531316.56358371720.0007595531737926.5734041250.0000013551

    -2.59.07924460990.1101413215300.77752492690.003324716553469.77263904550.0000187022

    -2.255.01434376770.199427890579.08884867910.01264400764974.68845716070.0002010176

    -22.94794300650.339219583923.93599331320.0417780865594.27272977890.0016827291

    -1.751.84486525670.54204500658.33784822880.119935020791.1523938050.0109706389

    -1.51.22899696970.81367165633.34289584510.299141835817.95199498490.0557041154

    -1.250.87152041121.14742005721.54261672130.6482491644.5396223050.2202826431

    -10.65787730451.52004027680.81933131321.22050748451.47397150330.6784391678

    -0.750.52863125461.89167778350.50087274561.99651510060.61449848491.6273433127

    -0.50.45216959892.21155956190.35242083642.83751667530.32893865223.0400805535

    -0.250.41170910682.42889939380.28540533283.50378876970.22608479944.4231191238

    00.39904344222.50599281720.26602896153.75898922580.19952172115.0119856345

    0.250.41170910682.42889939380.28540533283.50378876970.22608479944.4231191238

    0.50.45216959892.21155956190.35242083642.83751667530.32893865223.0400805535

    0.750.52863125461.89167778350.50087274561.99651510060.61449848491.6273433127

    10.65787730451.52004027680.81933131321.22050748451.47397150330.6784391678

    1.250.87152041121.14742005721.54261672130.6482491644.5396223050.2202826431

    1.51.22899696970.81367165633.34289584510.299141835817.95199498490.0557041154

    1.751.84486525670.54204500658.33784822880.119935020791.1523938050.0109706389

    22.94794300650.339219583923.93599331320.0417780865594.27272977890.0016827291

    2.255.01434376770.199427890579.08884867910.01264400764974.68845716070.0002010176

    2.59.07924460990.1101413215300.77752492690.003324716553469.77263904550.0000187022

    2.7517.49951127870.0571444531316.56358371720.0007595531737926.5734041250.0000013551

    335.90398996990.02785205776632.9179428810.000150763213076162.99982720.0000000765

    3.2578.41517185580.012752634238462.16619577360.0000259996297515683.4305660.0000000034

    3.5182.304787610.0054853195256701.9601512720.00000389568691640725.506530.0000000001

    00.51

    111

    -3.5182.304787610.00548531951188.54545955790.00084136459949.37691432150.0001005088

    -3.2578.41517185580.0127526342451.16628739040.00221647773333.00177608720.0003000298

    -335.90398996990.0278520577182.304787610.00548531951188.54545955790.0008413645

    -2.7517.49951127870.05714445378.41517185580.0127526342451.16628739040.0022164777

    -2.59.07924460990.110141321535.90398996990.0278520577182.304787610.0054853195

    -2.255.01434376770.199427890517.49951127870.05714445378.41517185580.0127526342

    -22.94794300650.33921958399.07924460990.110141321535.90398996990.0278520577

    -1.751.84486525670.54204500655.01434376770.199427890517.49951127870.057144453

    -1.51.22899696970.81367165632.94794300650.33921958399.07924460990.1101413215

    -1.250.87152041121.14742005721.84486525670.54204500655.01434376770.1994278905

    -10.65787730451.52004027681.22899696970.81367165632.94794300650.3392195839

    -0.750.52863125461.89167778350.87152041121.14742005721.84486525670.5420450065

    -0.50.45216959892.21155956190.65787730451.52004027681.22899696970.8136716563

    -0.250.41170910682.42889939380.52863125461.89167778350.87152041121.1474200572

    00.39904344222.50599281720.45216959892.21155956190.65787730451.5200402768

    0.250.41170910682.42889939380.41170910682.42889939380.52863125461.8916777835

    0.50.45216959892.21155956190.39904344222.50599281720.45216959892.2115595619

    0.750.52863125461.89167778350.41170910682.42889939380.41170910682.4288993938

    10.65787730451.52004027680.45216959892.21155956190.39904344222.5059928172

    1.250.87152041121.14742005720.52863125461.89167778350.41170910682.4288993938

    1.51.22899696970.81367165630.65787730451.52004027680.45216959892.2115595619

    1.751.84486525670.54204500650.87152041121.14742005720.52863125461.8916777835

    22.94794300650.33921958391.22899696970.81367165630.65787730451.5200402768

    2.255.01434376770.19942789051.84486525670.54204500650.87152041121.1474200572

    2.59.07924460990.11014132152.94794300650.33921958391.22899696970.8136716563

    2.7517.49951127870.0571444535.01434376770.19942789051.84486525670.5420450065

    335.90398996990.02785205779.07924460990.11014132152.94794300650.3392195839

    3.2578.41517185580.012752634217.49951127870.0571444535.01434376770.1994278905

    3.5182.304787610.005485319535.90398996990.02785205779.07924460990.1101413215

    &A

    Page &P

    Sheet1

    &A

    Page &P

    m x

    f(x)

    m m m x

    f(x)

  • The Standard Normal Probability Distribution - the probability distribution associated with any normal random variable (usually denoted z) that has = 0 and = 1.

    There are tables that can be used to obtain the results of the integrationfor the standard normal random variable.

  • Some of the tables work from the cumulative standard normal probability distribution (the probability that a random value selected from the standard normal random variable falls between and some given value b > 0, i.e., P(- z b))There are tables that give the results of the integration (Table 1 of the Appendices in J&W).

  • Cumulative Standard Normal Distribution (J&W Table 1)

    Sheet17

    &A

    Page &P

    Normal Distribution Graphs

    0.50.251.5

    -30.00134996720.00000000100.01

    -2.750.00297981860.00000001900.02

    -2.50.00620967990.000000287100.03

    -2.250.01222443340.000003400800.04

    -20.0227500620.0000316866.66133814775094E-160.06

    -1.750.04005911360.000232673400.09

    -1.50.06680722880.00134996720.0000000010.12

    -1.250.10564983910.00620967990.00000028710.15

    -10.15865525980.0227500620.0000316860.19

    -0.750.22662727970.06680722880.00134996720.240.52.5

    -0.50.330.15865525980.0227500620.29-30.00134996720.0010

    -0.250.420.30853753260.15865525980.33-2.750.10.080

    00.450.49999999980.54999999980.36-2.50.75210.60

    -0.250.420.30853753260.15865525980.33-2.250.71050.550

    -0.50.330.15865525980.0227500620.29-20.22510.30

    -0.750.22662727970.06680722880.00134996720.24-1.750.10.20

    -10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025

    -1.250.10564983910.00620967990.00000028710.15-1.250.290.260.0000007178

    -1.50.06680722880.00134996720.0000000010.12-10.430.330.0000792151

    -1.750.04005911360.000232673400.09-0.750.290.250.0033749181

    -20.0227500620.0000316866.66133814775094E-160.06-0.50.050.230.0568751551

    -2.250.01222443340.000003400800.04-0.250.080.330.3966381494

    -2.50.00620967990.000000287100.0300.190.851.3749999995

    -2.750.00297981860.00000001900.02-0.250.230.430.3966381494

    -30.00134996720.00000000100.01-0.50.270.350.0568751551

    -0.750.380.290.0033749181

    -10.440.310.0000792151

    -1.250.50.330.0000007178

    -1.50.420.220.0000000025

    -1.750.210.150

    -20.090.050

    -2.250.070.030

    -2.50.030.020

    -2.750.020.010

    -30.010.0010

    -40.000031686

    -3.750.0000884446

    -3.50.0002326734

    -3.250.0005770865

    -30.00134996720.00057708650.000031686

    -2.750.00297981860.00134996720.0000884446

    -2.50.00620967990.00297981860.0002326734

    -2.250.01222443340.00620967990.0005770865

    -20.0227500620.01222443340.0013499672

    -1.750.04005911360.0227500620.0029798186

    -1.50.06680722880.04005911360.0062096799

    -1.250.10564983910.06680722880.0122244334

    -10.15865525980.10564983910.022750062

    -0.750.22662727970.15865525980.0400591136

    -0.50.330.22662727970.0668072288

    -0.250.420.330.1056498391

    00.450.420.1586552598

    -0.250.420.450.2266272797

    -0.50.330.420.33

    -0.750.22662727970.330.42

    -10.15865525980.22662727970.45

    -1.250.10564983910.15865525980.42

    -1.50.06680722880.10564983910.33

    -1.750.04005911360.06680722880.2266272797

    -20.0227500620.04005911360.1586552598

    -2.250.01222443340.0227500620.1056498391

    -2.50.00620967990.01222443340.0668072288

    -2.750.00297981860.00620967990.0400591136

    -30.00134996720.00297981860.022750062

    &A

    Page &P

    Normal Distribution Graphs

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    &A

    Page &P

    Normal Distribution Table

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    &A

    Page &P

    m

    x

    f(x)

    t-Distribution Table

    0.0013499672

    0.1

    0.7521

    0.7105

    0.2251

    0.1

    0.15

    0.29

    0.43

    0.29

    0.05

    0.08

    0.19

    0.23

    0.27

    0.38

    0.44

    0.5

    0.42

    0.21

    0.09

    0.07

    0.03

    0.02

    0.01

    &A

    Page &P

    m

    x

    f(x)

    Sheet3

    0.00134996720.0010

    0.10.080

    0.75210.60

    0.71050.550

    0.22510.30

    0.10.20

    0.150.220.0000000025

    0.290.260.0000007178

    0.430.330.0000792151

    0.290.250.0033749181

    0.050.230.0568751551

    0.080.330.3966381494

    0.190.851.3749999995

    0.230.430.3966381494

    0.270.350.0568751551

    0.380.290.0033749181

    0.440.310.0000792151

    0.50.330.0000007178

    0.420.220.0000000025

    0.210.150

    0.090.050

    0.070.030

    0.030.020

    0.020.010

    0.010.0010

    &A

    Page &P

    Sheet4

    0.00134996720.000000001

    0.00297981860.000000019

    0.00620967990.0000002871

    0.01222443340.0000034008

    0.0227500620.000031686

    0.04005911360.0002326734

    0.06680722880.0013499672

    0.10564983910.0062096799

    0.15865525980.022750062

    0.22662727970.0668072288

    0.330.1586552598

    0.420.3085375326

    0.450.4999999998

    0.420.3085375326

    0.330.1586552598

    0.22662727970.0668072288

    0.15865525980.022750062

    0.10564983910.0062096799

    0.06680722880.0013499672

    0.04005911360.0002326734

    0.0227500620.000031686

    0.01222443340.0000034008

    0.00620967990.0000002871

    0.00297981860.000000019

    0.00134996720.000000001

    &A

    Page &P

    Sheet5

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet6

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet7

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet8

    0

    0

    0

    0

    6.66133814775094E-16

    0

    0.000000001

    0.0000002871

    0.000031686

    0.0013499672

    0.022750062

    0.1586552598

    0.5499999998

    0.1586552598

    0.022750062

    0.0013499672

    0.000031686

    0.0000002871

    0.000000001

    0

    6.66133814775094E-16

    0

    0

    0

    0

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet9

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    m

    x

    f(x)

    Sheet10

    0.0013499672

    0.0029798186

    0.0062096799

    0.0122244334

    0.022750062

    0.0400591136

    0.0668072288

    0.1056498391

    0.1586552598

    0.2266272797

    0.33

    0.42

    0.45

    0.42

    0.33

    0.2266272797

    0.1586552598

    0.1056498391

    0.0668072288

    0.0400591136

    0.022750062

    0.0122244334

    0.0062096799

    0.0029798186

    0.0013499672

    0

    z

    f(z)

    Sheet11

    0.0013499672

    0.0029798186

    0.0062096799

    0.0122244334

    0.022750062

    0.0400591136

    0.0668072288

    0.1056498391

    0.1586552598

    0.2266272797

    0.33

    0.42

    0.45

    0.42

    0.33

    0.2266272797

    0.1586552598

    0.1056498391

    0.0668072288

    0.0400591136

    0.022750062

    0.0122244334

    0.0062096799

    0.0029798186

    0.0013499672

    m=5

    x

    f(x)

    Sheet12

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    &A

    Page &P

    m

    m

    m

    x

    f(x)

    Sheet13

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    Sheet14

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    Sheet15

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    Sheet16

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    f(x)

    m

    x

    z0.000.010.020.030.040.050.060.070.080.09z0.000.010.020.030.040.050.060.070.080.09

    0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359:::::::::::

    0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753:::::::::::

    0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61411.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177

    0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65171.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319

    0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68791.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441

    0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72241.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545

    0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75491.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633

    0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78521.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706

    0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81331.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767

    0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.83892.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817

    1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.86212.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857

    1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88302.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890

    1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90152.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916

    1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91772.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936

    1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93192.50.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015

    1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94412.60.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177

    1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95452.70.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319

    1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96332.80.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441

    1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97062.90.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545

    1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.97673.00.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633

    2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.98173.10.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706

    2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98573.20.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767

    2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890

    2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916

    2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936

    2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952

    2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964

    2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974

    2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981

    2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986

    3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

    &A

    Page &P

    degreesArea in Upper tail

    of freedom0.100.050.0250.010.005

    13.0786.31412.70631.82163.656

    21.8862.9204.3036.9659.925

    31.6382.3533.1824.5415.841

    41.5332.1322.7763.7474.604

    51.4762.0152.5713.3654.032

    61.4401.9432.4473.1433.707

    71.4151.8952.3652.9983.499

    81.3971.8602.3062.8963.355

    91.3831.8332.2622.8213.250

    101.3721.8122.2282.7643.169

    111.3631.7962.2012.7183.106

    121.3561.7822.1792.6813.055

    131.3501.7712.1602.6503.012

    141.3451.7612.1452.6242.977

    151.3411.7532.1312.6022.947

    161.3371.7462.1202.5832.921

    171.3331.7402.1102.5672.898

    181.3301.7342.1012.5522.878

    191.3281.7292.0932.5392.861

    201.3251.7252.0862.5282.845

    211.3231.7212.0802.5182.831

    221.3211.7172.0742.5082.819

    231.3191.7142.0692.5002.807

    241.3181.7112.0642.4922.797

    251.3161.7082.0602.4852.787

    261.3151.7062.0562.4792.779

    271.3141.7032.0522.4732.771

    281.3131.7012.0482.4672.763

    291.3111.6992.0452.4622.756

    301.3101.6972.0422.4572.750

    401.3031.6842.0212.4232.704

    601.2961.6712.0002.3902.660

    1201.2891.6581.9802.3582.617

    1.2821.6451.9602.3262.576

    degreesArea in Upper tail

    of freedom0.100.050.0250.010.005

    13.0786.31412.70631.82163.656

    ......

    ......

    ......

    151.3411.7532.1312.6022.947

    161.3371.7462.1202.5832.921

    171.3331.7402.1102.5672.898

    181.3301.7342.1012.5522.878

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

  • Lets focus on a small part of the Cumulative Standard Normal Probability Distribution Table

    Example: for a standard normal random variable z, what is the probability that z is between - and 0.43?

    Sheet17

    &A

    Page &P

    Normal Distribution Graphs

    0.50.251.5

    -30.00134996720.00000000100.01

    -2.750.00297981860.00000001900.02

    -2.50.00620967990.000000287100.03

    -2.250.01222443340.000003400800.04

    -20.0227500620.0000316866.66133814775094E-160.06

    -1.750.04005911360.000232673400.09

    -1.50.06680722880.00134996720.0000000010.12

    -1.250.10564983910.00620967990.00000028710.15

    -10.15865525980.0227500620.0000316860.19

    -0.750.22662727970.06680722880.00134996720.240.52.5

    -0.50.330.15865525980.0227500620.29-30.00134996720.0010

    -0.250.420.30853753260.15865525980.33-2.750.10.080

    00.450.49999999980.54999999980.36-2.50.75210.60

    -0.250.420.30853753260.15865525980.33-2.250.71050.550

    -0.50.330.15865525980.0227500620.29-20.22510.30

    -0.750.22662727970.06680722880.00134996720.24-1.750.10.20

    -10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025

    -1.250.10564983910.00620967990.00000028710.15-1.250.290.260.0000007178

    -1.50.06680722880.00134996720.0000000010.12-10.430.330.0000792151

    -1.750.04005911360.000232673400.09-0.750.290.250.0033749181

    -20.0227500620.0000316866.66133814775094E-160.06-0.50.050.230.0568751551

    -2.250.01222443340.000003400800.04-0.250.080.330.3966381494

    -2.50.00620967990.000000287100.0300.190.851.3749999995

    -2.750.00297981860.00000001900.02-0.250.230.430.3966381494

    -30.00134996720.00000000100.01-0.50.270.350.0568751551

    -0.750.380.290.0033749181

    -10.440.310.0000792151

    -1.250.50.330.0000007178

    -1.50.420.220.0000000025

    -1.750.210.150

    -20.090.050

    -2.250.070.030

    -2.50.030.020

    -2.750.020.010

    -30.010.0010

    -40.000031686

    -3.750.0000884446

    -3.50.0002326734

    -3.250.0005770865

    -30.00134996720.00057708650.000031686

    -2.750.00297981860.00134996720.0000884446

    -2.50.00620967990.00297981860.0002326734

    -2.250.01222443340.00620967990.0005770865

    -20.0227500620.01222443340.0013499672

    -1.750.04005911360.0227500620.0029798186

    -1.50.06680722880.04005911360.0062096799

    -1.250.10564983910.06680722880.0122244334

    -10.15865525980.10564983910.022750062

    -0.750.22662727970.15865525980.0400591136

    -0.50.330.22662727970.0668072288

    -0.250.420.330.1056498391

    00.450.420.1586552598

    -0.250.420.450.2266272797

    -0.50.330.420.33

    -0.750.22662727970.330.42

    -10.15865525980.22662727970.45

    -1.250.10564983910.15865525980.42

    -1.50.06680722880.10564983910.33

    -1.750.04005911360.06680722880.2266272797

    -20.0227500620.04005911360.1586552598

    -2.250.01222443340.0227500620.1056498391

    -2.50.00620967990.01222443340.0668072288

    -2.750.00297981860.00620967990.0400591136

    -30.00134996720.00297981860.022750062

    &A

    Page &P

    Normal Distribution Graphs

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    &A

    Page &P

    Normal Distribution Table

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    &A

    Page &P

    m

    x

    f(x)

    t-Distribution Table

    0.0013499672

    0.1

    0.7521

    0.7105

    0.2251

    0.1

    0.15

    0.29

    0.43

    0.29

    0.05

    0.08

    0.19

    0.23

    0.27

    0.38

    0.44

    0.5

    0.42

    0.21

    0.09

    0.07

    0.03

    0.02

    0.01

    &A

    Page &P

    m

    x

    f(x)

    Sheet3

    0.00134996720.0010

    0.10.080

    0.75210.60

    0.71050.550

    0.22510.30

    0.10.20

    0.150.220.0000000025

    0.290.260.0000007178

    0.430.330.0000792151

    0.290.250.0033749181

    0.050.230.0568751551

    0.080.330.3966381494

    0.190.851.3749999995

    0.230.430.3966381494

    0.270.350.0568751551

    0.380.290.0033749181

    0.440.310.0000792151

    0.50.330.0000007178

    0.420.220.0000000025

    0.210.150

    0.090.050

    0.070.030

    0.030.020

    0.020.010

    0.010.0010

    &A

    Page &P

    Sheet4

    0.00134996720.000000001

    0.00297981860.000000019

    0.00620967990.0000002871

    0.01222443340.0000034008

    0.0227500620.000031686

    0.04005911360.0002326734

    0.06680722880.0013499672

    0.10564983910.0062096799

    0.15865525980.022750062

    0.22662727970.0668072288

    0.330.1586552598

    0.420.3085375326

    0.450.4999999998

    0.420.3085375326

    0.330.1586552598

    0.22662727970.0668072288

    0.15865525980.022750062

    0.10564983910.0062096799

    0.06680722880.0013499672

    0.04005911360.0002326734

    0.0227500620.000031686

    0.01222443340.0000034008

    0.00620967990.0000002871

    0.00297981860.000000019

    0.00134996720.000000001

    &A

    Page &P

    Sheet5

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet6

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet7

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet8

    0

    0

    0

    0

    6.66133814775094E-16

    0

    0.000000001

    0.0000002871

    0.000031686

    0.0013499672

    0.022750062

    0.1586552598

    0.5499999998

    0.1586552598

    0.022750062

    0.0013499672

    0.000031686

    0.0000002871

    0.000000001

    0

    6.66133814775094E-16

    0

    0

    0

    0

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet9

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    m

    x

    f(x)

    Sheet10

    0.0013499672

    0.0029798186

    0.0062096799

    0.0122244334

    0.022750062

    0.0400591136

    0.0668072288

    0.1056498391

    0.1586552598

    0.2266272797

    0.33

    0.42

    0.45

    0.42

    0.33

    0.2266272797

    0.1586552598

    0.1056498391

    0.0668072288

    0.0400591136

    0.022750062

    0.0122244334

    0.0062096799

    0.0029798186

    0.0013499672

    0

    z

    f(z)

    Sheet11

    0.0013499672

    0.0029798186

    0.0062096799

    0.0122244334

    0.022750062

    0.0400591136

    0.0668072288

    0.1056498391

    0.1586552598

    0.2266272797

    0.33

    0.42

    0.45

    0.42

    0.33

    0.2266272797

    0.1586552598

    0.1056498391

    0.0668072288

    0.0400591136

    0.022750062

    0.0122244334

    0.0062096799

    0.0029798186

    0.0013499672

    m=5

    x

    f(x)

    Sheet12

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    &A

    Page &P

    m

    m

    m

    x

    f(x)

    Sheet13

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    Sheet14

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    Sheet15

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    Sheet16

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    f(x)

    m

    x

    z0.000.010.020.030.040.050.060.070.080.09z0.000.010.020.030.040.050.060.070.080.09

    0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359:::::::::::

    0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753:::::::::::

    0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61411.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177

    0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65171.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319

    0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68791.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441

    0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72241.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545

    0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75491.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633

    0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78521.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706

    0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81331.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767

    0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.83892.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817

    1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.86212.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857

    1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88302.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890

    1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90152.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916

    1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91772.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936

    1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93192.50.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015

    1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94412.60.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177

    1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95452.70.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319

    1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96332.80.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441

    1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97062.90.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545

    1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.97673.00.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633

    2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.98173.10.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706

    2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98573.20.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767

    2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890

    2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916

    2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936

    2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952

    2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964

    2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974

    2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981

    2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986

    3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

    &A

    Page &P

    degreesArea in Upper tail

    of freedom0.100.050.0250.010.005

    13.0786.31412.70631.82163.656

    21.8862.9204.3036.9659.925

    31.6382.3533.1824.5415.841

    41.5332.1322.7763.7474.604

    51.4762.0152.5713.3654.032

    61.4401.9432.4473.1433.707

    71.4151.8952.3652.9983.499

    81.3971.8602.3062.8963.355

    91.3831.8332.2622.8213.250

    101.3721.8122.2282.7643.169

    111.3631.7962.2012.7183.106

    121.3561.7822.1792.6813.055

    131.3501.7712.1602.6503.012

    141.3451.7612.1452.6242.977

    151.3411.7532.1312.6022.947

    161.3371.7462.1202.5832.921

    171.3331.7402.1102.5672.898

    181.3301.7342.1012.5522.878

    191.3281.7292.0932.5392.861

    201.3251.7252.0862.5282.845

    211.3231.7212.0802.5182.831

    221.3211.7172.0742.5082.819

    231.3191.7142.0692.5002.807

    241.3181.7112.0642.4922.797

    251.3161.7082.0602.4852.787

    261.3151.7062.0562.4792.779

    271.3141.7032.0522.4732.771

    281.3131.7012.0482.4672.763

    291.3111.6992.0452.4622.756

    301.3101.6972.0422.4572.750

    401.3031.6842.0212.4232.704

    601.2961.6712.0002.3902.660

    1201.2891.6581.9802.3582.617

    1.2821.6451.9602.3262.576

    degreesArea in Upper tail

    of freedom0.100.050.0250.010.005

    13.0786.31412.70631.82163.656

    ......

    ......

    ......

    151.3411.7532.1312.6022.947

    161.3371.7462.1202.5832.921

    171.3331.7402.1102.5672.898

    181.3301.7342.1012.5522.878

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

  • Example: for a standard normal random variable z, what is the probability that z is between 0 and 2.0?2.0

    Chart2

    0.0013499672

    0.0029798186

    0.0062096799

    0.0122244334

    0.022750062

    0.0400591136

    0.0668072288

    0.1056498391

    0.1586552598

    0.2266272797

    0.33

    0.42

    0.45

    0.42

    0.33

    0.2266272797

    0.1586552598

    0.1056498391

    0.0668072288

    0.0400591136

    0.022750062

    0.0122244334

    0.0062096799

    0.0029798186

    0.0013499672

    0

    z

    f(z)

    Sheet17

    &A

    Page &P

    Normal Distribution Graphs

    0.50.251.5

    -30.00134996720.00000000100.01

    -2.750.00297981860.00000001900.02

    -2.50.00620967990.000000287100.03

    -2.250.01222443340.000003400800.04

    -20.0227500620.0000316866.66133814775094E-160.06

    -1.750.04005911360.000232673400.09

    -1.50.06680722880.00134996720.0000000010.12

    -1.250.10564983910.00620967990.00000028710.15

    -10.15865525980.0227500620.0000316860.19

    -0.750.22662727970.06680722880.00134996720.240.52.5

    -0.50.330.15865525980.0227500620.29-30.00134996720.0010

    -0.250.420.30853753260.15865525980.33-2.750.10.080

    00.450.49999999980.54999999980.36-2.50.75210.60

    -0.250.420.30853753260.15865525980.33-2.250.71050.550

    -0.50.330.15865525980.0227500620.29-20.22510.30

    -0.750.22662727970.06680722880.00134996720.24-1.750.10.20

    -10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025

    -1.250.10564983910.00620967990.00000028710.15-1.250.290.260.0000007178

    -1.50.06680722880.00134996720.0000000010.12-10.430.330.0000792151

    -1.750.04005911360.000232673400.09-0.750.290.250.0033749181

    -20.0227500620.0000316866.66133814775094E-160.06-0.50.050.230.0568751551

    -2.250.01222443340.000003400800.04-0.250.080.330.3966381494

    -2.50.00620967990.000000287100.0300.190.851.3749999995

    -2.750.00297981860.00000001900.02-0.250.230.430.3966381494

    -30.00134996720.00000000100.01-0.50.270.350.0568751551

    -0.750.380.290.0033749181

    -10.440.310.0000792151

    -1.250.50.330.0000007178

    -1.50.420.220.0000000025

    -1.750.210.150

    -20.090.050

    -2.250.070.030

    -2.50.030.020

    -2.750.020.010

    -30.010.0010

    -40.000031686

    -3.750.0000884446

    -3.50.0002326734

    -3.250.0005770865

    -30.00134996720.00057708650.000031686

    -2.750.00297981860.00134996720.0000884446

    -2.50.00620967990.00297981860.0002326734

    -2.250.01222443340.00620967990.0005770865

    -20.0227500620.01222443340.0013499672

    -1.750.04005911360.0227500620.0029798186

    -1.50.06680722880.04005911360.0062096799

    -1.250.10564983910.06680722880.0122244334

    -10.15865525980.10564983910.022750062

    -0.750.22662727970.15865525980.0400591136

    -0.50.330.22662727970.0668072288

    -0.250.420.330.1056498391

    00.450.420.1586552598

    -0.250.420.450.2266272797

    -0.50.330.420.33

    -0.750.22662727970.330.42

    -10.15865525980.22662727970.45

    -1.250.10564983910.15865525980.42

    -1.50.06680722880.10564983910.33

    -1.750.04005911360.06680722880.2266272797

    -20.0227500620.04005911360.1586552598

    -2.250.01222443340.0227500620.1056498391

    -2.50.00620967990.01222443340.0668072288

    -2.750.00297981860.00620967990.0400591136

    -30.00134996720.00297981860.022750062

    &A

    Page &P

    Normal Distribution Graphs

    &A

    Page &P

    Normal Distribution Table

    &A

    Page &P

    m

    x

    f(x)

    t-Distribution Table

    &A

    Page &P

    m

    f(x)

    Sheet3

    &A

    Page &P

    Sheet4

    &A

    Page &P

    Sheet5

    &A

    Page &P

    Sheet6

    &A

    Page &P

    Sheet7

    0.00134996720.00000000100.01

    0.00297981860.00000001900.02

    0.00620967990.000000287100.03

    0.01222443340.000003400800.04

    0.0227500620.0000316866.66133814775094E-160.06

    0.04005911360.000232673400.09

    0.06680722880.00134996720.0000000010.12

    0.10564983910.00620967990.00000028710.15

    0.15865525980.0227500620.0000316860.19

    0.22662727970.06680722880.00134996720.24

    0.330.15865525980.0227500620.29

    0.420.30853753260.15865525980.33

    0.450.49999999980.54999999980.36

    0.420.30853753260.15865525980.33

    0.330.15865525980.0227500620.29

    0.22662727970.06680722880.00134996720.24

    0.15865525980.0227500620.0000316860.19

    0.10564983910.00620967990.00000028710.15

    0.06680722880.00134996720.0000000010.12

    0.04005911360.000232673400.09

    0.0227500620.0000316866.66133814775094E-160.06

    0.01222443340.000003400800.04

    0.00620967990.000000287100.03

    0.00297981860.00000001900.02

    0.00134996720.00000000100.01

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet8

    0.001349967200.01

    0.002979818600.02

    0.006209679900.03

    0.012224433400.04

    0.0227500626.66133814775094E-160.06

    0.040059113600.09

    0.06680722880.0000000010.12

    0.10564983910.00000028710.15

    0.15865525980.0000316860.19

    0.22662727970.00134996720.24

    0.330.0227500620.29

    0.420.15865525980.33

    0.450.54999999980.36

    0.420.15865525980.33

    0.330.0227500620.29

    0.22662727970.00134996720.24

    0.15865525980.0000316860.19

    0.10564983910.00000028710.15

    0.06680722880.0000000010.12

    0.040059113600.09

    0.0227500626.66133814775094E-160.06

    0.012224433400.04

    0.006209679900.03

    0.002979818600.02

    0.001349967200.01

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet9

    00.01

    00.02

    00.03

    00.04

    6.66133814775094E-160.06

    00.09

    0.0000000010.12

    0.00000028710.15

    0.0000316860.19

    0.00134996720.24

    0.0227500620.29

    0.15865525980.33

    0.54999999980.36

    0.15865525980.33

    0.0227500620.29

    0.00134996720.24

    0.0000316860.19

    0.00000028710.15

    0.0000000010.12

    00.09

    6.66133814775094E-160.06

    00.04

    00.03

    00.02

    00.01

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet10

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet11

    m

    x

    f(x)

    Sheet12

    0

    z

    f(z)

    Sheet13

    m=5

    x

    f(x)

    Sheet14

    &A

    Page &P

    m

    m

    m

    x

    f(x)

    Sheet15

    z0.000.010.020.030.040.050.060.070.080.09

    0.0-0.00000.00400.00800.01200.01600.01990.02390.02790.03190.0359

    0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753

    0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141

    0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517

    0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879

    0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224

    0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549

    0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852

    0.80.28810.29100.29390.29670.29950.30230.30510.30780.31060.3133

    0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389

    1.00.34130.34380.34610.34850.35080.35310.35540.35770.35990.3621

    1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830

    1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015

    1.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177

    1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319

    1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441

    1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545

    1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633

    1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706

    1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767

    2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817

    2.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857

    2.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890

    2.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916

    2.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936

    2.50.49380.49400.49410.49430.49450.49460.49480.49490.49510.4952

    2.60.49530.49550.49560.49570.49590.49600.49610.49620.49630.4964

    2.70.49650.49660.49670.49680.49690.49700.49710.49720.49730.4974

    2.80.49740.49750.49760.49770.49770.49780.49790.49790.49800.4981

    2.90.49810.49820.49820.49830.49840.49840.49850.49850.49860.4986

    3.00.49870.49870.49870.49880.49880.49890.49890.49890.49900.4990

    &A

    Page &P

    Sheet16

    degreesArea in Upper tail

    of freedom0.100.050.0250.010.005

    13.0786.31412.70631.82163.656

    21.8862.9204.3036.9659.925

    31.6382.3533.1824.5415.841

    41.5332.1322.7763.7474.604

    51.4762.0152.5713.3654.032

    61.4401.9432.4473.1433.707

    71.4151.8952.3652.9983.499

    81.3971.8602.3062.8963.355

    91.3831.8332.2622.8213.250

    101.3721.8122.2282.7643.169

    111.3631.7962.2012.7183.106

    121.3561.7822.1792.6813.055

    131.3501.7712.1602.6503.012

    141.3451.7612.1452.6242.977

    151.3411.7532.1312.6022.947

    161.3371.7462.1202.5832.921

    171.3331.7402.1102.5672.898

    181.3301.7342.1012.5522.878

    191.3281.7292.0932.5392.861

    201.3251.7252.0862.5282.845

    211.3231.7212.0802.5182.831

    221.3211.7172.0742.5082.819

    231.3191.7142.0692.5002.807

    241.3181.7112.0642.4922.797

    251.3161.7082.0602.4852.787

    261.3151.7062.0562.4792.779

    271.3141.7032.0522.4732.771

    281.3131.7012.0482.4672.763

    291.3111.6992.0452.4622.756

    301.3101.6972.0422.4572.750

    401.3031.6842.0212.4232.704

    601.2961.6712.0002.3902.660

    1201.2891.6581.9802.3582.617

    1.2821.6451.9602.3262.576

    degreesArea in Upper tail

    of freedom0.100.050.0250.010.005

    13.0786.31412.70631.82163.656

    ......

    ......

    ......

    151.3411.7532.1312.6022.947

    161.3371.7462.1202.5832.921

    171.3331.7402.1102.5672.898

    181.3301.7342.1012.5522.878

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

  • Again, looking at a small part of the Cumulative Standard Normal Probability Distribution Table, we find the probability that a standard normal random variable z is between - and 2.00?

    Sheet17

    &A

    Page &P

    Normal Distribution Graphs

    0.50.251.5

    -30.00134996720.00000000100.01

    -2.750.00297981860.00000001900.02

    -2.50.00620967990.000000287100.03

    -2.250.01222443340.000003400800.04

    -20.0227500620.0000316866.66133814775094E-160.06

    -1.750.04005911360.000232673400.09

    -1.50.06680722880.00134996720.0000000010.12

    -1.250.10564983910.00620967990.00000028710.15

    -10.15865525980.0227500620.0000316860.19

    -0.750.22662727970.06680722880.00134996720.240.52.5

    -0.50.330.15865525980.0227500620.29-30.00134996720.0010

    -0.250.420.30853753260.15865525980.33-2.750.10.080

    00.450.49999999980.54999999980.36-2.50.75210.60

    -0.250.420.30853753260.15865525980.33-2.250.71050.550

    -0.50.330.15865525980.0227500620.29-20.22510.30

    -0.750.22662727970.06680722880.00134996720.24-1.750.10.20

    -10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025

    -1.250.10564983910.00620967990.00000028710.15-1.250.290.260.0000007178

    -1.50.06680722880.00134996720.0000000010.12-10.430.330.0000792151

    -1.750.04005911360.000232673400.09-0.750.290.250.0033749181

    -20.0227500620.0000316866.66133814775094E-160.06-0.50.050.230.0568751551

    -2.250.01222443340.000003400800.04-0.250.080.330.3966381494

    -2.50.00620967990.000000287100.0300.190.851.3749999995

    -2.750.00297981860.00000001900.02-0.250.230.430.3966381494

    -30.00134996720.00000000100.01-0.50.270.350.0568751551

    -0.750.380.290.0033749181

    -10.440.310.0000792151

    -1.250.50.330.0000007178

    -1.50.420.220.0000000025

    -1.750.210.150

    -20.090.050

    -2.250.070.030

    -2.50.030.020

    -2.750.020.010

    -30.010.0010

    -40.000031686

    -3.750.0000884446

    -3.50.0002326734

    -3.250.0005770865

    -30.00134996720.00057708650.000031686

    -2.750.00297981860.00134996720.0000884446

    -2.50.00620967990.00297981860.0002326734

    -2.250.01222443340.00620967990.0005770865

    -20.0227500620.01222443340.0013499672

    -1.750.04005911360.0227500620.0029798186

    -1.50.06680722880.04005911360.0062096799

    -1.250.10564983910.06680722880.0122244334

    -10.15865525980.10564983910.022750062

    -0.750.22662727970.15865525980.0400591136

    -0.50.330.22662727970.0668072288

    -0.250.420.330.1056498391

    00.450.420.1586552598

    -0.250.420.450.2266272797

    -0.50.330.420.33

    -0.750.22662727970.330.42

    -10.15865525980.22662727970.45

    -1.250.10564983910.15865525980.42

    -1.50.06680722880.10564983910.33

    -1.750.04005911360.06680722880.2266272797

    -20.0227500620.04005911360.1586552598

    -2.250.01222443340.0227500620.1056498391

    -2.50.00620967990.01222443340.0668072288

    -2.750.00297981860.00620967990.0400591136

    -30.00134996720.00297981860.022750062

    &A

    Page &P

    Normal Distribution Graphs

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    &A

    Page &P

    Normal Distribution Table

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    &A

    Page &P

    m

    x

    f(x)

    t-Distribution Table

    0.0013499672

    0.1

    0.7521

    0.7105

    0.2251

    0.1

    0.15

    0.29

    0.43

    0.29

    0.05

    0.08

    0.19

    0.23

    0.27

    0.38

    0.44

    0.5

    0.42

    0.21

    0.09

    0.07

    0.03

    0.02

    0.01

    &A

    Page &P

    m

    x

    f(x)

    Sheet3

    0.00134996720.0010

    0.10.080

    0.75210.60

    0.71050.550

    0.22510.30

    0.10.20

    0.150.220.0000000025

    0.290.260.0000007178

    0.430.330.0000792151

    0.290.250.0033749181

    0.050.230.0568751551

    0.080.330.3966381494

    0.190.851.3749999995

    0.230.430.3966381494

    0.270.350.0568751551

    0.380.290.0033749181

    0.440.310.0000792151

    0.50.330.0000007178

    0.420.220.0000000025

    0.210.150

    0.090.050

    0.070.030

    0.030.020

    0.020.010

    0.010.0010

    &A

    Page &P

    Sheet4

    0.00134996720.000000001

    0.00297981860.000000019

    0.00620967990.0000002871

    0.01222443340.0000034008

    0.0227500620.000031686

    0.04005911360.0002326734

    0.06680722880.0013499672

    0.10564983910.0062096799

    0.15865525980.022750062

    0.22662727970.0668072288

    0.330.1586552598

    0.420.3085375326

    0.450.4999999998

    0.420.3085375326

    0.330.1586552598

    0.22662727970.0668072288

    0.15865525980.022750062

    0.10564983910.0062096799

    0.06680722880.0013499672

    0.04005911360.0002326734

    0.0227500620.000031686

    0.01222443340.0000034008

    0.00620967990.0000002871

    0.00297981860.000000019

    0.00134996720.000000001

    &A

    Page &P

    Sheet5

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    0000

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet6

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet7

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    00

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet8

    0

    0

    0

    0

    6.66133814775094E-16

    0

    0.000000001

    0.0000002871

    0.000031686

    0.0013499672

    0.022750062

    0.1586552598

    0.5499999998

    0.1586552598

    0.022750062

    0.0013499672

    0.000031686

    0.0000002871

    0.000000001

    0

    6.66133814775094E-16

    0

    0

    0

    0

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet9

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    m

    x

    f(x)

    Sheet10

    0.0013499672

    0.0029798186

    0.0062096799

    0.0122244334

    0.022750062

    0.0400591136

    0.0668072288

    0.1056498391

    0.1586552598

    0.2266272797

    0.33

    0.42

    0.45

    0.42

    0.33

    0.2266272797

    0.1586552598

    0.1056498391

    0.0668072288

    0.0400591136

    0.022750062

    0.0122244334

    0.0062096799

    0.0029798186

    0.0013499672

    0

    z

    f(z)

    Sheet11

    0.0013499672

    0.0029798186

    0.0062096799

    0.0122244334

    0.022750062

    0.0400591136

    0.0668072288

    0.1056498391

    0.1586552598

    0.2266272797

    0.33

    0.42

    0.45

    0.42

    0.33

    0.2266272797

    0.1586552598

    0.1056498391

    0.0668072288

    0.0400591136

    0.022750062

    0.0122244334

    0.0062096799

    0.0029798186

    0.0013499672

    m=5

    x

    f(x)

    Sheet12

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    &A

    Page &P

    m

    m

    m

    x

    f(x)

    Sheet13

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    Sheet14

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    Sheet15

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    Sheet16

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    000

    f(x)

    m

    x

    z0.000.010.020.030.040.050.060.070.080.09z0.000.010.020.030.040.050.060.070.080.09

    0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359:::::::::::

    0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753:::::::::::

    0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61411.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177

    0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65171.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319

    0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68791.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441

    0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72241.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545

    0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75491.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633

    0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78521.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706

    0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81331.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767

    0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.83892.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817

    1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.86212.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857

    1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88302.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890

    1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90152.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916

    1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91772.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936

    1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93192.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952

    1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94412.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964

    1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95452.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974

    1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96332.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981

    1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97062.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986

    1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.97673.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

    2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.98173.10.99900.99910.99910.99910.99920.99920.99920.99920.99930.9993

    2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98573.20.99930.99930.99940.99940.99940.99940.99940.99950.99950.9995

    2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890

    2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916

    2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936

    2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952

    2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964

    2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974

    2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981

    2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986

    3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

    &A

    Page &P

    degreesArea in Upper tail

    of freedom0.100.050.0250.010.005

    13.0786.31412.70631.82163.656

    21.8862.9204.3036.9659.925

    31.6382.3533.1824.5415.841

    41.5332.1322.7763.7474.604

    51.4762.0152.5713.3654.032

    61.4401.9432.4473.1433.707

    71.4151.8952.3652.9983.499

    81.3971.8602.3062.8963.355

    91.3831.8332.2622.8213.250

    101.3721.8122.2282.7643.169

    111.3631.7962.2012.7183.106

    121.3561.7822.1792.6813.055

    131.3501.7712.1602.6503.012

    141.3451.7612.1452.6242.977

    151.3411.7532.1312.6022.947

    161.3371.7462.1202.5832.921

    171.3331.7402.1102.5672.898

    181.3301.7342.1012.5522.878

    191.3281.7292.0932.5392.861

    201.3251.7252.0862.5282.845

    211.3231.7212.0802.5182.831

    221.3211.7172.0742.5082.819

    231.3191.7142.0692.5002.807

    241.3181.7112.0642.4922.797

    251.3161.7082.0602.4852.787

    261.3151.7062.0562.4792.779

    271.3141.7032.0522.4732.771

    281.3131.7012.0482.4672.763

    291.3111.6992.0452.4622.756

    301.3101.6972.0422.4572.750

    401.3031.6842.0212.4232.704

    601.2961.6712.0002.3902.660

    1201.2891.6581.9802.3582.617

    1.2821.6451.9602.3262.576

    degreesArea in Upper tail

    of freedom0.100.050.0250.010.005

    13.0786.31412.70631.82163.656

    ......

    ......

    ......

    151.3411.7532.1312.6022.947

    161.3371.7462.1202.5832.921

    171.3331.7402.1102.5672.898

    181.3301.7342.1012.5522.878

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

  • Example: for a standard normal random variable z, what is the probability that z is between 0 and 2.0?2.0}Area of Probability =0.9772 0.5000 = 0.4772Area ofProbability = 0.5000{

    Chart2

    0.0013499672

    0.0029798186

    0.0062096799

    0.0122244334

    0.022750062

    0.0400591136

    0.0668072288

    0.1056498391

    0.1586552598

    0.2266272797

    0.33

    0.42

    0.45

    0.42

    0.33

    0.2266272797

    0.1586552598

    0.1056498391

    0.0668072288

    0.0400591136

    0.022750062

    0.0122244334

    0.0062096799

    0.0029798186

    0.0013499672

    0

    z

    f(z)

    Sheet17

    &A

    Page &P

    Normal Distribution Graphs

    0.50.251.5

    -30.00134996720.00000000100.01

    -2.750.00297981860.00000001900.02

    -2.50.00620967990.000000287100.03

    -2.250.01222443340.000003400800.04

    -20.0227500620.0000316866.66133814775094E-160.06

    -1.750.04005911360.000232673400.09

    -1.50.06680722880.00134996720.0000000010.12

    -1.250.10564983910.00620967990.00000028710.15

    -10.15865525980.0227500620.0000316860.19

    -0.750.22662727970.06680722880.00134996720.240.52.5

    -0.50.330.15865525980.0227500620.29-30.00134996720.0010

    -0.250.420.30853753260.15865525980.33-2.750.10.080

    00.450.49999999980.54999999980.36-2.50.75210.60

    -0.250.420.30853753260.15865525980.33-2.250.71050.550

    -0.50.330.15865525980.0227500620.29-20.22510.30

    -0.750.22662727970.06680722880.00134996720.24-1.750.10.20

    -10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025

    -1.250.10564983910.00620967990.00000028710.15-1.250.290.260.0000007178

    -1.50.06680722880.00134996720.0000000010.12-10.430.330.0000792151

    -1.750.04005911360.000232673400.09-0.750.290.250.0033749181

    -20.0227500620.0000316866.66133814775094E-160.06-0.50.050.230.0568751551

    -2.250.01222443340.000003400800.04-0.250.080.330.3966381494

    -2.50.00620967990.000000287100.0300.190.851.3749999995

    -2.750.00297981860.00000001900.02-0.250.230.430.3966381494

    -30.00134996720.00000000100.01-0.50.270.350.0568751551

    -0.750.380.290.0033749181

    -10.440.310.0000792151

    -1.250.50.330.0000007178

    -1.50.420.220.0000000025

    -1.750.210.150

    -20.090.050

    -2.250.070.030

    -2.50.030.020

    -2.750.020.010

    -30.010.0010

    -40.000031686

    -3.750.0000884446

    -3.50.0002326734

    -3.250.0005770865

    -30.00134996720.00057708650.000031686

    -2.750.00297981860.00134996720.0000884446

    -2.50.00620967990.00297981860.0002326734

    -2.250.01222443340.00620967990.0005770865

    -20.0227500620.01222443340.0013499672

    -1.750.04005911360.0227500620.0029798186

    -1.50.06680722880.04005911360.0062096799

    -1.250.10564983910.06680722880.0122244334

    -10.15865525980.10564983910.022750062

    -0.750.22662727970.15865525980.0400591136

    -0.50.330.22662727970.0668072288

    -0.250.420.330.1056498391

    00.450.420.1586552598

    -0.250.420.450.2266272797

    -0.50.330.420.33

    -0.750.22662727970.330.42

    -10.15865525980.22662727970.45

    -1.250.10564983910.15865525980.42

    -1.50.06680722880.10564983910.33

    -1.750.04005911360.06680722880.2266272797

    -20.0227500620.04005911360.1586552598

    -2.250.01222443340.0227500620.1056498391

    -2.50.00620967990.01222443340.0668072288

    -2.750.00297981860.00620967990.0400591136

    -30.00134996720.00297981860.022750062

    &A

    Page &P

    Normal Distribution Graphs

    &A

    Page &P

    Normal Distribution Table

    &A

    Page &P

    m

    x

    f(x)

    t-Distribution Table

    &A

    Page &P

    m

    f(x)

    Sheet3

    &A

    Page &P

    Sheet4

    &A

    Page &P

    Sheet5

    &A

    Page &P

    Sheet6

    &A

    Page &P

    Sheet7

    0.00134996720.00000000100.01

    0.00297981860.00000001900.02

    0.00620967990.000000287100.03

    0.01222443340.000003400800.04

    0.0227500620.0000316866.66133814775094E-160.06

    0.04005911360.000232673400.09

    0.06680722880.00134996720.0000000010.12

    0.10564983910.00620967990.00000028710.15

    0.15865525980.0227500620.0000316860.19

    0.22662727970.06680722880.00134996720.24

    0.330.15865525980.0227500620.29

    0.420.30853753260.15865525980.33

    0.450.49999999980.54999999980.36

    0.420.30853753260.15865525980.33

    0.330.15865525980.0227500620.29

    0.22662727970.06680722880.00134996720.24

    0.15865525980.0227500620.0000316860.19

    0.10564983910.00620967990.00000028710.15

    0.06680722880.00134996720.0000000010.12

    0.04005911360.000232673400.09

    0.0227500620.0000316866.66133814775094E-160.06

    0.01222443340.000003400800.04

    0.00620967990.000000287100.03

    0.00297981860.00000001900.02

    0.00134996720.00000000100.01

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet8

    0.001349967200.01

    0.002979818600.02

    0.006209679900.03

    0.012224433400.04

    0.0227500626.66133814775094E-160.06

    0.040059113600.09

    0.06680722880.0000000010.12

    0.10564983910.00000028710.15

    0.15865525980.0000316860.19

    0.22662727970.00134996720.24

    0.330.0227500620.29

    0.420.15865525980.33

    0.450.54999999980.36

    0.420.15865525980.33

    0.330.0227500620.29

    0.22662727970.00134996720.24

    0.15865525980.0000316860.19

    0.10564983910.00000028710.15

    0.06680722880.0000000010.12

    0.040059113600.09

    0.0227500626.66133814775094E-160.06

    0.012224433400.04

    0.006209679900.03

    0.002979818600.02

    0.001349967200.01

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet9

    00.01

    00.02

    00.03

    00.04

    6.66133814775094E-160.06

    00.09

    0.0000000010.12

    0.00000028710.15

    0.0000316860.19

    0.00134996720.24

    0.0227500620.29

    0.15865525980.33

    0.54999999980.36

    0.15865525980.33

    0.0227500620.29

    0.00134996720.24

    0.0000316860.19

    0.00000028710.15

    0.0000000010.12

    00.09

    6.66133814775094E-160.06

    00.04

    00.03

    00.02

    00.01

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet10

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet11

    m

    x

    f(x)

    Sheet12

    0

    z

    f(z)

    Sheet13

    m=5

    x

    f(x)

    Sheet14

    &A

    Page &P

    m

    m

    m

    x

    f(x)

    Sheet15

    z0.000.010.020.030.040.050.060.070.080.09

    0.0-0.00000.00400.00800.01200.01600.01990.02390.02790.03190.0359

    0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753

    0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141

    0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517

    0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879

    0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224

    0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549

    0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852

    0.80.28810.29100.29390.29670.29950.30230.30510.30780.31060.3133

    0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389

    1.00.34130.34380.34610.34850.35080.35310.35540.35770.35990.3621

    1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830

    1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015

    1.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177

    1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319

    1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441

    1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545

    1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633

    1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706

    1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767

    2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817

    2.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857

    2.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890

    2.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916

    2.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936

    2.50.49380.49400.49410.49430.49450.49460.49480.49490.49510.4952

    2.60.49530.49550.49560.49570.49590.49600.49610.49620.49630.4964

    2.70.49650.49660.49670.49680.49690.49700.49710.49720.49730.4974

    2.80.49740.49750.49760.49770.49770.49780.49790.49790.49800.4981

    2.90.49810.49820.49820.49830.49840.49840.49850.49850.49860.4986

    3.00.49870.49870.49870.49880.49880.49890.49890.49890.49900.4990

    &A

    Page &P

    Sheet16

    degreesArea in Upper tail

    of freedom0.100.050.0250.010.005

    13.0786.31412.70631.82163.656

    21.8862.9204.3036.9659.925

    31.6382.3533.1824.5415.841

    41.5332.1322.7763.7474.604

    51.4762.0152.5713.3654.032

    61.4401.9432.4473.1433.707

    71.4151.8952.3652.9983.499

    81.3971.8602.3062.8963.355

    91.3831.8332.2622.8213.250

    101.3721.8122.2282.7643.169

    111.3631.7962.2012.7183.106

    121.3561.7822.1792.6813.055

    131.3501.7712.1602.6503.012

    141.3451.7612.1452.6242.977

    151.3411.7532.1312.6022.947

    161.3371.7462.1202.5832.921

    171.3331.7402.1102.5672.898

    181.3301.7342.1012.5522.878

    191.3281.7292.0932.5392.861

    201.3251.7252.0862.5282.845

    211.3231.7212.0802.5182.831

    221.3211.7172.0742.5082.819

    231.3191.7142.0692.5002.807

    241.3181.7112.0642.4922.797

    251.3161.7082.0602.4852.787

    261.3151.7062.0562.4792.779

    271.3141.7032.0522.4732.771

    281.3131.7012.0482.4672.763

    291.3111.6992.0452.4622.756

    301.3101.6972.0422.4572.750

    401.3031.6842.0212.4232.704

    601.2961.6712.0002.3902.660

    1201.2891.6581.9802.3582.617

    1.2821.6451.9602.3262.576

    degreesArea in Upper tail

    of freedom0.100.050.0250.010.005

    13.0786.31412.70631.82163.656

    ......

    ......

    ......

    151.3411.7532.1312.6022.947

    161.3371.7462.1202.5832.921

    171.3331.7402.1102.5672.898

    181.3301.7342.1012.5522.878

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

    &A

    Page &P

  • 2.0Area of Probability =1.0000 - 0.9772 = 0.0228What is the probability that z is at least 2.0?{

    Chart2

    0.0013499672

    0.0029798186

    0.0062096799

    0.0122244334

    0.022750062

    0.0400591136

    0.0668072288

    0.1056498391

    0.1586552598

    0.2266272797

    0.33

    0.42

    0.45

    0.42

    0.33

    0.2266272797

    0.1586552598

    0.1056498391

    0.0668072288

    0.0400591136

    0.022750062

    0.0122244334

    0.0062096799

    0.0029798186

    0.0013499672

    0

    z

    f(z)

    Sheet17

    &A

    Page &P

    Normal Distribution Graphs

    0.50.251.5

    -30.00134996720.00000000100.01

    -2.750.00297981860.00000001900.02

    -2.50.00620967990.000000287100.03

    -2.250.01222443340.000003400800.04

    -20.0227500620.0000316866.66133814775094E-160.06

    -1.750.04005911360.000232673400.09

    -1.50.06680722880.00134996720.0000000010.12

    -1.250.10564983910.00620967990.00000028710.15

    -10.15865525980.0227500620.0000316860.19

    -0.750.22662727970.06680722880.00134996720.240.52.5

    -0.50.330.15865525980.0227500620.29-30.00134996720.0010

    -0.250.420.30853753260.15865525980.33-2.750.10.080

    00.450.49999999980.54999999980.36-2.50.75210.60

    -0.250.420.30853753260.15865525980.33-2.250.71050.550

    -0.50.330.15865525980.0227500620.29-20.22510.30

    -0.750.22662727970.06680722880.00134996720.24-1.750.10.20

    -10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025

    -1.250.10564983910.00620967990.00000028710.15-1.250.290.260.0000007178

    -1.50.06680722880.00134996720.0000000010.12-10.430.330.0000792151

    -1.750.04005911360.000232673400.09-0.750.290.250.0033749181

    -20.0227500620.0000316866.66133814775094E-160.06-0.50.050.230.0568751551

    -2.250.01222443340.000003400800.04-0.250.080.330.3966381494

    -2.50.00620967990.000000287100.0300.190.851.3749999995

    -2.750.00297981860.00000001900.02-0.250.230.430.3966381494

    -30.00134996720.00000000100.01-0.50.270.350.0568751551

    -0.750.380.290.0033749181

    -10.440.310.0000792151

    -1.250.50.330.0000007178

    -1.50.420.220.0000000025

    -1.750.210.150

    -20.090.050

    -2.250.070.030

    -2.50.030.020

    -2.750.020.010

    -30.010.0010

    -40.000031686

    -3.750.0000884446

    -3.50.0002326734

    -3.250.0005770865

    -30.00134996720.00057708650.000031686

    -2.750.00297981860.00134996720.0000884446

    -2.50.00620967990.00297981860.0002326734

    -2.250.01222443340.00620967990.0005770865

    -20.0227500620.01222443340.0013499672

    -1.750.04005911360.0227500620.0029798186

    -1.50.06680722880.04005911360.0062096799

    -1.250.10564983910.06680722880.0122244334

    -10.15865525980.10564983910.022750062

    -0.750.22662727970.15865525980.0400591136

    -0.50.330.22662727970.0668072288

    -0.250.420.330.1056498391

    00.450.420.1586552598

    -0.250.420.450.2266272797

    -0.50.330.420.33

    -0.750.22662727970.330.42

    -10.15865525980.22662727970.45

    -1.250.10564983910.15865525980.42

    -1.50.06680722880.10564983910.33

    -1.750.04005911360.06680722880.2266272797

    -20.0227500620.04005911360.1586552598

    -2.250.01222443340.0227500620.1056498391

    -2.50.00620967990.01222443340.0668072288

    -2.750.00297981860.00620967990.0400591136

    -30.00134996720.00297981860.022750062

    &A

    Page &P

    Normal Distribution Graphs

    &A

    Page &P

    Normal Distribution Table

    &A

    Page &P

    m

    x

    f(x)

    t-Distribution Table

    &A

    Page &P

    m

    f(x)

    Sheet3

    &A

    Page &P

    Sheet4

    &A

    Page &P

    Sheet5

    &A

    Page &P

    Sheet6

    &A

    Page &P

    Sheet7

    0.00134996720.00000000100.01

    0.00297981860.00000001900.02

    0.00620967990.000000287100.03

    0.01222443340.000003400800.04

    0.0227500620.0000316866.66133814775094E-160.06

    0.04005911360.000232673400.09

    0.06680722880.00134996720.0000000010.12

    0.10564983910.00620967990.00000028710.15

    0.15865525980.0227500620.0000316860.19

    0.22662727970.06680722880.00134996720.24

    0.330.15865525980.0227500620.29

    0.420.30853753260.15865525980.33

    0.450.49999999980.54999999980.36

    0.420.30853753260.15865525980.33

    0.330.15865525980.0227500620.29

    0.22662727970.06680722880.00134996720.24

    0.15865525980.0227500620.0000316860.19

    0.10564983910.00620967990.00000028710.15

    0.06680722880.00134996720.0000000010.12

    0.04005911360.000232673400.09

    0.0227500620.0000316866.66133814775094E-160.06

    0.01222443340.000003400800.04

    0.00620967990.000000287100.03

    0.00297981860.00000001900.02

    0.00134996720.00000000100.01

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet8

    0.001349967200.01

    0.002979818600.02

    0.006209679900.03

    0.012224433400.04

    0.0227500626.66133814775094E-160.06

    0.040059113600.09

    0.06680722880.0000000010.12

    0.10564983910.00000028710.15

    0.15865525980.0000316860.19

    0.22662727970.00134996720.24

    0.330.0227500620.29

    0.420.15865525980.33

    0.450.54999999980.36

    0.420.15865525980.33

    0.330.0227500620.29

    0.22662727970.00134996720.24

    0.15865525980.0000316860.19

    0.10564983910.00000028710.15

    0.06680722880.0000000010.12

    0.040059113600.09

    0.0227500626.66133814775094E-160.06

    0.012224433400.04

    0.006209679900.03

    0.002979818600.02

    0.001349967200.01

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet9

    00.01

    00.02

    00.03

    00.04

    6.66133814775094E-160.06

    00.09

    0.0000000010.12

    0.00000028710.15

    0.0000316860.19

    0.00134996720.24

    0.0227500620.29

    0.15865525980.33

    0.54999999980.36

    0.15865525980.33

    0.0227500620.29

    0.00134996720.24

    0.0000316860.19

    0.00000028710.15

    0.0000000010.12

    00.09

    6.66133814775094E-160.06

    00.04

    00.03

    00.02

    00.01

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet10

    &A

    Page &P

    0

    f(z,t)

    z,t

    Sheet11

    m

    x

    f(x)

    Sheet12

    0

    z

    f(z)

    Sheet13

    m=5

    x

    f(x)

    Sheet14

    &A

    Page &P

    m

    m

    m

    x

    f(x)

    Sheet15

    z0.000.010.020.030.040.050.060.070.080.09

    0.0-0.00000.00400.00800.01200.01600.01990.02390.02790.03190.0359

    0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753

    0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141

    0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517

    0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879

    0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224

    0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549

    0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852

    0.80.28810.29100.29390.29670.29950.30230.30510.30780.31060.3133

    0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389

    1.00.34130.34380.34610.34850.35080.35310.35540.35770.35990.3621

    1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830

    1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015

    1.30.40320.40490.40660.40820.40990.41150.4


Recommended
Multivariate Bernoulli Distribution Models

Multivariate Bernoulli Distribution Models

Documents

Bayesian Inference for the Multivariate Extended-Skew ... · CREST Florian Pelgriny EDHEC Business School, France CIRANO The multivariate extended skew-normal distribution allows

Bayesian Inference for the Multivariate Extended-Skew ... · CREST Florian Pelgriny EDHEC Business School, France CIRANO The multivariate extended skew-normal distribution allows

Documents

MULTIVARIATE NORMAL DISTRIBUTION - York … · J. Elder CSE 4404/5327 Introduction to Machine Learning and Pattern Recognition MULTIVARIATE NORMAL DISTRIBUTION Last updated: Sept

MULTIVARIATE NORMAL DISTRIBUTION - York … · J. Elder CSE 4404/5327 Introduction to Machine Learning and Pattern Recognition MULTIVARIATE NORMAL DISTRIBUTION Last updated: Sept

Documents

MULTIVARIATE DISTRIBUTION CORRECTION OF CLIMATE …

MULTIVARIATE DISTRIBUTION CORRECTION OF CLIMATE …

Documents

Marginal and Conditional distributions. Theorem: (Marginal distributions for the Multivariate Normal distribution) have p-variate Normal distribution.

Marginal and Conditional distributions. Theorem: (Marginal distributions for the Multivariate Normal distribution) have p-variate Normal distribution.

Documents

Maximum-likelihood estimation of multivariate normal ...€¦ · Maximum-likelihood estimation of multivariate normal graphical models: large-scale numerical implementation and topology

Maximum-likelihood estimation of multivariate normal ...€¦ · Maximum-likelihood estimation of multivariate normal graphical models: large-scale numerical implementation and topology

Documents

The multivariate normal distribution - Freie Universität · In this section we summarize some basic properties of the multivariate normal distribution, ... (42, −31, 58, 54, −22,

The multivariate normal distribution - Freie Universität · In this section we summarize some basic properties of the multivariate normal distribution, ... (42, −31, 58, 54, −22,

Documents

The Multivariate Normal Distribution Edps/Soc 584 and ...courses.education.illinois.edu/EdPsy584/lectures/Multivariate... · Multivariate Normal Distribution Slide 1 of 54 The Multivariate

The Multivariate Normal Distribution Edps/Soc 584 and ...courses.education.illinois.edu/EdPsy584/lectures/Multivariate... · Multivariate Normal Distribution Slide 1 of 54 The Multivariate

Documents

3.4 Multivariate normal distributions 3.4.1 Density ...

3.4 Multivariate normal distributions 3.4.1 Density ...

Documents

3. The Multivariate Normal Distributionhpeng/Math3806/Lecture_note3.pdf · 3. The Multivariate Normal Distribution 3.1 Introduction • A generalization of the familiar bell shaped

3. The Multivariate Normal Distributionhpeng/Math3806/Lecture_note3.pdf · 3. The Multivariate Normal Distribution 3.1 Introduction • A generalization of the familiar bell shaped

Documents

Individual Loss Reserving with the Multivariate Skew ... · Keywords: Stochastic loss reserving, general insurance, Multivariate Skew Normal distribution, chain–ladder, individual

Individual Loss Reserving with the Multivariate Skew ... · Keywords: Stochastic loss reserving, general insurance, Multivariate Skew Normal distribution, chain–ladder, individual

Documents

SF2940: Probability theory Lecture 8: Multivariate Normal ...

SF2940: Probability theory Lecture 8: Multivariate Normal ...

Documents

The Multivariate Normal Distribution, Part 1 BMTRY 726 1/10/2014.

The Multivariate Normal Distribution, Part 1 BMTRY 726 1/10/2014.

Documents

Generalized Variance Multivariate Normal Distribution

Generalized Variance Multivariate Normal Distribution

Documents

Data Analysis: TD multivariate normal correction

Data Analysis: TD multivariate normal correction

Documents

Multivariate Normal Distribution - Home Page | Jonathan ... · PDF fileThe multivariate normal distribution function is: f(x ... The mean vector is µ The covariance matrix is ...

Multivariate Normal Distribution - Home Page | Jonathan ... · PDF fileThe multivariate normal distribution function is: f(x ... The mean vector is µ The covariance matrix is ...

Documents

Multivariate Normal Distributioncc.ee.ntu.edu.tw/.../MultivariateNormalDistribution6s.pdf · 2012. 9. 19. · Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical

Multivariate Normal Distributioncc.ee.ntu.edu.tw/.../MultivariateNormalDistribution6s.pdf · 2012. 9. 19. · Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical

Documents

The Multivariate Normal Distribution - Paul Johnson Homepage

The Multivariate Normal Distribution - Paul Johnson Homepage

Documents