INSTRUCTOR’S SOLUTIONS MANUAL AN INTRODUCTION TO MATHEMATICAL STATISTICS AND ITS APPLICATIONS
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MATHEMATICAL STATISTICS
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MATHEMATICAL STATISTICS
Contents
Chapter 2: Probability..................................................................................................................................................1
2.2 Samples Spaces and the Algebra of Sets ..............................................................................
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
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
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
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
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