INSTRUCTOR’S SOLUTIONS
MANUAL
AN INTRODUCTION TO
MATHEMATICAL STATISTICS
AND ITS A PPLICATIONS
FIFTH E DITION
Richard J. Larsen
Vanderbilt University
Morris L. Marx
University of West Florida
,
, Contents
Chapter 2: Probability ................................................................................................................................................. 1
Samples Spaces and the Algebra of Sets ......................................................................................................... 1
The Probability Function ................................................................................................................................. 5
Conditional Probability. ................................................................................................................................... 7
Independence ................................................................................................................................................. 13
Combinatorics ................................................................................................................................................ 17
Combinatorial Probability .............................................................................................................................. 23
Chapter 3: Random Variables ................................................................................................................................... 27
Binomial and Hypergeometric Probabilities .................................................................................................. 27
Discrete Random Variables ........................................................................................................................... 34
Continuous Random Variables ...................................................................................................................... 37
Expected Values............................................................................................................................................. 39
The Variance .................................................................................................................................................. 45
Joint Densities ................................................................................................................................................ 49
Transforming and Combining Random Variables ......................................................................................... 58
Further Properties of the Mean and Variance ................................................................................................ 60
Order Statistics ........................................................................................................................................... 64
Conditional Densities ................................................................................................................................. 67
Moment-Generating Functions. .................................................................................................................. 71
Chapter 4: Special Distributions ............................................................................................................................... 75
The Poisson Distribution................................................................................................................................ 75
The Normal Distribution ................................................................................................................................ 80
The Geometric Distribution ........................................................................................................................... 87
The Negative Binomial Distribution .............................................................................................................. 89
The Gamma Distribution ............................................................................................................................... 91
Chapter 5: Estimation ................................................................................................................................................ 93
Estimating Parameters: The Method of Maximum Likelihood and Method of Moments ............................. 93
Interval Estimation ......................................................................................................................................... 98
Properties of Estimators. .............................................................................................................................. 102
Minimum-Variance Estimators: The Cramér-Rao Lower Bound ................................................................ 105
Sufficient Estimators.................................................................................................................................... 107
Consistency .................................................................................................................................................. 109
Bayesian Estimation..................................................................................................................................... 111
, ii Contents
Chapter 6: Hypothesis Testing ................................................................................................................................ 113
The Decision Rule........................................................................................................................................ 113
Testing Binomial Data - H0: p = po .......................................................................................................................................................................... 114
Type I and Type II Errors ............................................................................................................................ 115
A Notion of Optimality: The Generalized Likelihood Ratio ....................................................................... 119
Chapter 7: Inferences Based on the Normal Distribution ....................................................................................... 121
Deriving the Distribution of Y −µ ...............................................................................................................121
S/ n
Drawing Inferences about µ ............................................................................................................................... 123
Drawing Inferences about σ2 ........................................................................................................................ 127
Chapter 8: Types of Data: A Brief Overview.......................................................................................................... 131
8.2 Classifying Data ........................................................................................................................................... 131
Chapter 9: Two-Sample Inference .......................................................................................................................... 133
Testing H0 : µ X = µ Y .................................................................................................................................................... 133
Testing H0 :σ2 = σ2 —The F Test................................................................................................................. 136
X Y
Binomial Data: Testing H0 : pX = pY .................................................................................................................................................................................. 138
Confidence Intervals for the Two-Sample Problem..................................................................................... 140
Chapter 10: Goodness-of-Fit Tests ......................................................................................................................... 143
The Multinomial Distribution ............................................................................................................................... 143
Goodness-of-Fit Tests: All Parameters Known ..................................................................................................... 145
Goodness-of-Fit Tests: Parameters Unknown ....................................................................................................... 148
Contingency Tables ............................................................................................................................................... 154
Chapter 11: Regression ........................................................................................................................................... 159
The Method of Least Squares ................................................................................................................................ 159
The Linear Model .................................................................................................................................................. 169
Covariance and Correlation ................................................................................................................................... 174
The Bivariate Normal Distribution........................................................................................................................ 178
MANUAL
AN INTRODUCTION TO
MATHEMATICAL STATISTICS
AND ITS A PPLICATIONS
FIFTH E DITION
Richard J. Larsen
Vanderbilt University
Morris L. Marx
University of West Florida
,
, Contents
Chapter 2: Probability ................................................................................................................................................. 1
Samples Spaces and the Algebra of Sets ......................................................................................................... 1
The Probability Function ................................................................................................................................. 5
Conditional Probability. ................................................................................................................................... 7
Independence ................................................................................................................................................. 13
Combinatorics ................................................................................................................................................ 17
Combinatorial Probability .............................................................................................................................. 23
Chapter 3: Random Variables ................................................................................................................................... 27
Binomial and Hypergeometric Probabilities .................................................................................................. 27
Discrete Random Variables ........................................................................................................................... 34
Continuous Random Variables ...................................................................................................................... 37
Expected Values............................................................................................................................................. 39
The Variance .................................................................................................................................................. 45
Joint Densities ................................................................................................................................................ 49
Transforming and Combining Random Variables ......................................................................................... 58
Further Properties of the Mean and Variance ................................................................................................ 60
Order Statistics ........................................................................................................................................... 64
Conditional Densities ................................................................................................................................. 67
Moment-Generating Functions. .................................................................................................................. 71
Chapter 4: Special Distributions ............................................................................................................................... 75
The Poisson Distribution................................................................................................................................ 75
The Normal Distribution ................................................................................................................................ 80
The Geometric Distribution ........................................................................................................................... 87
The Negative Binomial Distribution .............................................................................................................. 89
The Gamma Distribution ............................................................................................................................... 91
Chapter 5: Estimation ................................................................................................................................................ 93
Estimating Parameters: The Method of Maximum Likelihood and Method of Moments ............................. 93
Interval Estimation ......................................................................................................................................... 98
Properties of Estimators. .............................................................................................................................. 102
Minimum-Variance Estimators: The Cramér-Rao Lower Bound ................................................................ 105
Sufficient Estimators.................................................................................................................................... 107
Consistency .................................................................................................................................................. 109
Bayesian Estimation..................................................................................................................................... 111
, ii Contents
Chapter 6: Hypothesis Testing ................................................................................................................................ 113
The Decision Rule........................................................................................................................................ 113
Testing Binomial Data - H0: p = po .......................................................................................................................................................................... 114
Type I and Type II Errors ............................................................................................................................ 115
A Notion of Optimality: The Generalized Likelihood Ratio ....................................................................... 119
Chapter 7: Inferences Based on the Normal Distribution ....................................................................................... 121
Deriving the Distribution of Y −µ ...............................................................................................................121
S/ n
Drawing Inferences about µ ............................................................................................................................... 123
Drawing Inferences about σ2 ........................................................................................................................ 127
Chapter 8: Types of Data: A Brief Overview.......................................................................................................... 131
8.2 Classifying Data ........................................................................................................................................... 131
Chapter 9: Two-Sample Inference .......................................................................................................................... 133
Testing H0 : µ X = µ Y .................................................................................................................................................... 133
Testing H0 :σ2 = σ2 —The F Test................................................................................................................. 136
X Y
Binomial Data: Testing H0 : pX = pY .................................................................................................................................................................................. 138
Confidence Intervals for the Two-Sample Problem..................................................................................... 140
Chapter 10: Goodness-of-Fit Tests ......................................................................................................................... 143
The Multinomial Distribution ............................................................................................................................... 143
Goodness-of-Fit Tests: All Parameters Known ..................................................................................................... 145
Goodness-of-Fit Tests: Parameters Unknown ....................................................................................................... 148
Contingency Tables ............................................................................................................................................... 154
Chapter 11: Regression ........................................................................................................................................... 159
The Method of Least Squares ................................................................................................................................ 159
The Linear Model .................................................................................................................................................. 169
Covariance and Correlation ................................................................................................................................... 174
The Bivariate Normal Distribution........................................................................................................................ 178
