INSTRUCTOR’S SOLUTIONS
MANUAL
AN I NTRODUCTION TO
MATHEMATICAL S TATISTICS
AND ITS APPLICATIONS
F IFTH E DITION
Richard J. Larsen
Vanderbilt University
Morris L. Marx
University of West Florida
,This should be only distributed free of cost. If you have paid for this from an online solution manual vendor, you
have been cheated.
Copyright © 2012, 2006, 2001 Pearson Education, Inc.
Publishing as Prentice Hall, 75 Arlington Street, Boston, MA 02116.
All rights reserved. This manual may be reproduced for classroom use only.
ISBN-13: 978-0-321-69401-0
ISBN-10: 0-321-69401-5
, Contents
Chapter 2: Probability..................................................................................................................................................1
2.2 Samples Spaces and the Algebra of Sets ..........................................................................................................1
2.3 The Probability Function ..................................................................................................................................5
2.4 Conditional Probability.....................................................................................................................................7
2.5 Independence..................................................................................................................................................13
2.6 Combinatorics.................................................................................................................................................17
2.7 Combinatorial Probability ..............................................................................................................................23
Chapter 3: Random Variables....................................................................................................................................27
3.2 Binomial and Hypergeometric Probabilities ..................................................................................................27
3.3 Discrete Random Variables ............................................................................................................................34
3.4 Continuous Random Variables.......................................................................................................................37
3.5 Expected Values .............................................................................................................................................39
3.6 The Variance...................................................................................................................................................45
3.7 Joint Densities.................................................................................................................................................49
3.8 Transforming and Combining Random Variables..........................................................................................58
3.9 Further Properties of the Mean and Variance.................................................................................................60
3.10 Order Statistics ............................................................................................................................................64
3.11 Conditional Densities ..................................................................................................................................67
3.12 Moment-Generating Functions....................................................................................................................71
Chapter 4: Special Distributions ................................................................................................................................75
4.2 The Poisson Distribution ................................................................................................................................75
4.3 The Normal Distribution ................................................................................................................................80
4.4 The Geometric Distribution............................................................................................................................87
4.5 The Negative Binomial Distribution ..............................................................................................................89
4.6 The Gamma Distribution ................................................................................................................................91
Chapter 5: Estimation ................................................................................................................................................93
5.2 Estimating Parameters: The Method of Maximum Likelihood and Method of Moments .............................93
5.3 Interval Estimation .........................................................................................................................................98
5.4 Properties of Estimators................................................................................................................................102
5.5 Minimum-Variance Estimators: The Cramér-Rao Lower Bound ................................................................105
5.6 Sufficient Estimators ....................................................................................................................................107
5.7 Consistency...................................................................................................................................................109
5.8 Bayesian Estimation .....................................................................................................................................111
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
, ii Contents
Chapter 6: Hypothesis Testing.................................................................................................................................113
6.2 The Decision Rule ........................................................................................................................................113
6.3 Testing Binomial Data - H0: p = po.......................................................................................................................................................................... 114
6.4 Type I and Type II Errors .............................................................................................................................115
6.5 A Notion of Optimality: The Generalized Likelihood Ratio........................................................................119
Chapter 7: Inferences Based on the Normal Distribution ........................................................................................121
7.3 Deriving the Distribution of Y −µ ...............................................................................................................121
S/ n
7.4 Drawing Inferences about µ .........................................................................................................................123
2
7.5 Drawing Inferences about σ ........................................................................................................................127
Chapter 8: Types of Data: A Brief Overview ..........................................................................................................131
8.2 Classifying Data ...........................................................................................................................................131
Chapter 9: Two-Sample Inference ...........................................................................................................................133
9.2 Testing H 0 : µ X = µY .......................................................................................................................................133
9.3 Testing H 0 : σ X2 = σ Y2 —The F Test .................................................................................................................136
9.4 Binomial Data: Testing H 0 : p X = pY ...........................................................................................................138
9.5 Confidence Intervals for the Two-Sample Problem .....................................................................................140
Chapter 10: Goodness-of-Fit Tests ..........................................................................................................................143
10.2 The Multinomial Distribution ...................................................................................................................143
10.3 Goodness-of-Fit Tests: All Parameters Known.........................................................................................145
10.4 Goodness-of-Fit Tests: Parameters Unknown...........................................................................................148
10.5 Contingency Tables...................................................................................................................................154
Chapter 11: Regression............................................................................................................................................159
11.2 The Method of Least Squares....................................................................................................................159
11.3 The Linear Model......................................................................................................................................169
11.4 Covariance and Correlation.......................................................................................................................174
11.5 The Bivariate Normal Distribution............................................................................................................178
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
MANUAL
AN I NTRODUCTION TO
MATHEMATICAL S TATISTICS
AND ITS APPLICATIONS
F IFTH E DITION
Richard J. Larsen
Vanderbilt University
Morris L. Marx
University of West Florida
,This should be only distributed free of cost. If you have paid for this from an online solution manual vendor, you
have been cheated.
Copyright © 2012, 2006, 2001 Pearson Education, Inc.
Publishing as Prentice Hall, 75 Arlington Street, Boston, MA 02116.
All rights reserved. This manual may be reproduced for classroom use only.
ISBN-13: 978-0-321-69401-0
ISBN-10: 0-321-69401-5
, Contents
Chapter 2: Probability..................................................................................................................................................1
2.2 Samples Spaces and the Algebra of Sets ..........................................................................................................1
2.3 The Probability Function ..................................................................................................................................5
2.4 Conditional Probability.....................................................................................................................................7
2.5 Independence..................................................................................................................................................13
2.6 Combinatorics.................................................................................................................................................17
2.7 Combinatorial Probability ..............................................................................................................................23
Chapter 3: Random Variables....................................................................................................................................27
3.2 Binomial and Hypergeometric Probabilities ..................................................................................................27
3.3 Discrete Random Variables ............................................................................................................................34
3.4 Continuous Random Variables.......................................................................................................................37
3.5 Expected Values .............................................................................................................................................39
3.6 The Variance...................................................................................................................................................45
3.7 Joint Densities.................................................................................................................................................49
3.8 Transforming and Combining Random Variables..........................................................................................58
3.9 Further Properties of the Mean and Variance.................................................................................................60
3.10 Order Statistics ............................................................................................................................................64
3.11 Conditional Densities ..................................................................................................................................67
3.12 Moment-Generating Functions....................................................................................................................71
Chapter 4: Special Distributions ................................................................................................................................75
4.2 The Poisson Distribution ................................................................................................................................75
4.3 The Normal Distribution ................................................................................................................................80
4.4 The Geometric Distribution............................................................................................................................87
4.5 The Negative Binomial Distribution ..............................................................................................................89
4.6 The Gamma Distribution ................................................................................................................................91
Chapter 5: Estimation ................................................................................................................................................93
5.2 Estimating Parameters: The Method of Maximum Likelihood and Method of Moments .............................93
5.3 Interval Estimation .........................................................................................................................................98
5.4 Properties of Estimators................................................................................................................................102
5.5 Minimum-Variance Estimators: The Cramér-Rao Lower Bound ................................................................105
5.6 Sufficient Estimators ....................................................................................................................................107
5.7 Consistency...................................................................................................................................................109
5.8 Bayesian Estimation .....................................................................................................................................111
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
, ii Contents
Chapter 6: Hypothesis Testing.................................................................................................................................113
6.2 The Decision Rule ........................................................................................................................................113
6.3 Testing Binomial Data - H0: p = po.......................................................................................................................................................................... 114
6.4 Type I and Type II Errors .............................................................................................................................115
6.5 A Notion of Optimality: The Generalized Likelihood Ratio........................................................................119
Chapter 7: Inferences Based on the Normal Distribution ........................................................................................121
7.3 Deriving the Distribution of Y −µ ...............................................................................................................121
S/ n
7.4 Drawing Inferences about µ .........................................................................................................................123
2
7.5 Drawing Inferences about σ ........................................................................................................................127
Chapter 8: Types of Data: A Brief Overview ..........................................................................................................131
8.2 Classifying Data ...........................................................................................................................................131
Chapter 9: Two-Sample Inference ...........................................................................................................................133
9.2 Testing H 0 : µ X = µY .......................................................................................................................................133
9.3 Testing H 0 : σ X2 = σ Y2 —The F Test .................................................................................................................136
9.4 Binomial Data: Testing H 0 : p X = pY ...........................................................................................................138
9.5 Confidence Intervals for the Two-Sample Problem .....................................................................................140
Chapter 10: Goodness-of-Fit Tests ..........................................................................................................................143
10.2 The Multinomial Distribution ...................................................................................................................143
10.3 Goodness-of-Fit Tests: All Parameters Known.........................................................................................145
10.4 Goodness-of-Fit Tests: Parameters Unknown...........................................................................................148
10.5 Contingency Tables...................................................................................................................................154
Chapter 11: Regression............................................................................................................................................159
11.2 The Method of Least Squares....................................................................................................................159
11.3 The Linear Model......................................................................................................................................169
11.4 Covariance and Correlation.......................................................................................................................174
11.5 The Bivariate Normal Distribution............................................................................................................178
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
