An Introduction to Probability and Statistical Inference, Third Edition, guides the reader through probability models and statistical methods to develop critical-thinking skills. Written by award-winning author George Roussas, this valuable text introduces a thinking process to help them obtain the best solution to a posed question or situation, and provides a plethora of examples and exercises to illustrate applying statistical methods to different situations.
1. An Introduction to Probability and Statistical Inference - p. iii 2. Cover - p. iii 3. Title Page - p. iii 4. Copyright Page - p. iv 5. Dedication - p. v 6. Contents - p. vii 7. About the author - p. xiii 8. Preface - p. xiii 9. Chapter 1 Some motivating examples and some fundamental concepts - p. 1 10. Chapter 2 The concept of probability and basic results - p. 23 11. 2.1: Definition of probability and some basic results - p. 23 12. Exercises - p. 31 13. 2.2: Distribution of a random variable - p. 34 14. Exercises - p. 38 15. 2.3: Conditional probability and related results - p. 42 16. Exercises - p. 49 17. 2.4: Independent events and related results - p. 56 18. Exercises - p. 60 19. 2.5: Basic concepts and results in counting - p. 64 20. Exercises - p. 71 21. Chapter 3 Numerical characteristics of a random variable, some special random variables - p. 79 22. 3.1: Expectation, variance, and moment generating function of a random variable - p. 79 23. Exercises - p. 85 24. 3.2: Some probability inequalities - p. 93 25. Exercises - p. 94 26. 3.3: Some special random variables - p. 95 27. Exercises - p. 113 28. 3.4: Median and mode of a random variable - p. 128 29. Exercises - p. 133 30. Chapter 4 Joint and conditional p.d.f.'s, conditional expectation and variance, moment generating function, covariance, and correlation coefficient - p. 137 31. 4.1: Joint d.f. and joint p.d.f. of two random variables - p. 138 32. Exercises - p. 143 33. 4.2: Marginal and conditional p.d.f.'s, conditional expectation and variance - p. 147 34. Exercises - p. 154 35. 4.3: Expectation of a function of two r.v.'s, joint and marginal m.g.f.'s, covariance, and correlation coefficient - p. 161 36. Exercises - p. 170 37. 4.4: Some generalizations to k random variables - p. 174 38. Exercises - p. 176 39. 4.5: The Multinomial, the Bivariate Normal, and the Multivariate Normal distributions - p. 177 40. Exercises - p. 185 41. Chapter 5 Independence of random variables and some applications - p. 191 42. 5.1: Independence of random variables and criteria of independence - p. 191 43. Exercises - p. 197 44. 5.2: The reproductive property of certain distributions - p. 203 45. Exercises - p. 208 46. Chapter 6 Transformation of random variables - p. 213 47. 6.1: Transforming a single random variable - p. 213 48. Exercises - p. 217 49. 6.2: Transforming two or more random variables - p. 218 50. Exercises - p. 228 51. 6.3: Linear transformations - p. 233 52. Exercises - p. 237 53. 6.4: The probability integral transform - p. 240 54. Exercises - p. 241 55. 6.5: Order statistics - p. 241 56. Exercises - p. 248 57. Chapter 7 Some modes of convergence of random variables, applications - p. 253 58. 7.1: Convergence in distribution or in probability and their relationship - p. 253 59. Exercises - p. 257 60. 7.2: Some applications of convergence in distribution: WLLN and CLT - p. 259 61. Exercises - p. 269 62. 7.3: Further limit theorems - p. 276 63. Exercises - p. 280 64. Chapter 8 An overview of statistical inference - p. 281 65. Chapter 9 Point estimation - p. 297 66. 9.1: Maximum likelihood estimation: motivation and examples - p. 297 67. Exercises - p. 303 68. 9.2: Some properties of MLE's - p. 311 69. Exercises - p. 318 70. 9.3: Uniformly minimum variance unbiased estimates - p. 319 71. Exercises - p. 325 72. 9.4: Decision-theoretic approach to estimation - p. 328 73. Exercises - p. 334 74. 9.5: Other methods of estimation - p. 336 75. Exercises - p. 338 76. Chapter 10 Confidence intervals and confidence regions - p. 341 77. 10.1: Confidence intervals - p. 341 78. Exercises - p. 345 79. 10.2: Confidence intervals in the presence of nuisance parameters - p. 349 80. Exercises - p. 353 81. 10.3: A confidence region for (μ,σ2) in the N(μ,σ2) distribution - p. 353 82. 10.4: Confidence intervals with approximate confidence coefficient - p. 355 83. Exercises - p. 357 84. Chapter 11 Testing hypotheses - p. 361 85. 11.1: General concepts, formulation of some testing hypotheses - p. 362 86. Exercises - p. 363 87. 11.2: Neyman–Pearson fundamental lemma, Exponential type families, UMP tests for some composite hypotheses - p. 364 88. Exercises - p. 369 89. 11.3: Some applications of Theorem 2 - p. 375 90. Exercises - p. 382 91. Exercises - p. 389 92. 11.4: Likelihood ratio tests - p. 390 93. 11.4.1: Testing hypotheses for the parameters in a single Normal population - p. 393 94. 11.4.2: Comparing the parameters of two Normal populations - p. 399 95. Exercises - p. 406 96. Chapter 12 More about testing hypotheses - p. 413 97. 12.1: Likelihood ratio tests in the Multinomial case and contingency tables - p. 413 98. Exercises - p. 418 99. 12.2: A goodness-of-fit test - p. 420 100. Exercises - p. 422 101. 12.3: Decision-theoretic approach to testing hypotheses - p. 425 102. 12.4: Relationship between testing hypotheses and confidence regions - p. 432 103. Chapter 13 A simple linear regression model - p. 437 104. 13.1: Setting up the model — the principle of least squares - p. 438 105. 13.2: The least squares estimates of β1 and β2 and some of their properties - p. 440 106. 13.3: Normally distributed errors: MLE's of β1,β2, and σ2, some distributional results - p. 449 107. Exercises - p. 455 108. 13.4: Confidence intervals and hypotheses testing problems - p. 458 109. Exercises - p. 461 110. 13.5: Some prediction problems - p. 465 111. Exercises - p. 467 112. 13.6: Proof of Theorem 5 - p. 469 113. 13.7: Concluding remarks - p. 471 114. Chapter 14 Two models of analysis of variance - p. 473 115. 14.1: One-way layout with the same number of observations per cell - p. 474 116. Exercises - p. 483 117. 14.2: A multicomparison method - p. 484 118. Exercises - p. 490 119. 14.3: Two-way layout with one observation per cell - p. 490 120. 14.3.1: The MLE's of the parameters of the model - p. 491 121. 14.3.2: Testing the hypothesis of no row or no column effects - p. 492 122. 14.3.3: Proof of lemmas in Section 14.3 - p. 499 123. Exercises - p. 505 124. Chapter 15 Some topics in nonparametric inference - p. 509 125. 15.1: Some confidence intervals with given approximate confidence coefficient - p. 510 126. 15.2: Confidence intervals for quantiles of a distribution function - p. 512 127. 15.3: The two-sample sign test - p. 514 128. Exercises - p. 518 129. 15.4: The rank sum and the Wilcoxon–Mann–Whitney two-sample tests - p. 519 130. 15.5: Nonparametric curve estimation - p. 529 131. Exercises - p. 537 132. Chapter 16 Appendix - p. 541 133. Tables - p. 551 134. Answers to even-numbered exercises - p. 589 135. Index - p. 635 136. Functions - p. 644
An Introduction to Probability and Statistical Inference, Third Edition, guides the reader through probability models and statistical methods to develop critical-thinking skills. Written by award-winning author George Roussas, this valuable text introduces a thinking process to help them obtain the best solution to a posed question or situation, and provides a plethora of examples and exercises to illustrate applying statistical methods to different situations.
1. An Introduction to Probability and Statistical Inference - p. iii 2. Cover - p. iii 3. Title Page - p. iii 4. Copyright Page - p. iv 5. Dedication - p. v 6. Contents - p. vii 7. About the author - p. xiii 8. Preface - p. xiii 9. Chapter 1 Some motivating examples and some fundamental concepts - p. 1 10. Chapter 2 The concept of probability and basic results - p. 23 11. 2.1: Definition of probability and some basic results - p. 23 12. Exercises - p. 31 13. 2.2: Distribution of a random variable - p. 34 14. Exercises - p. 38 15. 2.3: Conditional probability and related results - p. 42 16. Exercises - p. 49 17. 2.4: Independent events and related results - p. 56 18. Exercises - p. 60 19. 2.5: Basic concepts and results in counting - p. 64 20. Exercises - p. 71 21. Chapter 3 Numerical characteristics of a random variable, some special random variables - p. 79 22. 3.1: Expectation, variance, and moment generating function of a random variable - p. 79 23. Exercises - p. 85 24. 3.2: Some probability inequalities - p. 93 25. Exercises - p. 94 26. 3.3: Some special random variables - p. 95 27. Exercises - p. 113 28. 3.4: Median and mode of a random variable - p. 128 29. Exercises - p. 133 30. Chapter 4 Joint and conditional p.d.f.'s, conditional expectation and variance, moment generating function, covariance, and correlation coefficient - p. 137 31. 4.1: Joint d.f. and joint p.d.f. of two random variables - p. 138 32. Exercises - p. 143 33. 4.2: Marginal and conditional p.d.f.'s, conditional expectation and variance - p. 147 34. Exercises - p. 154 35. 4.3: Expectation of a function of two r.v.'s, joint and marginal m.g.f.'s, covariance, and correlation coefficient - p. 161 36. Exercises - p. 170 37. 4.4: Some generalizations to k random variables - p. 174 38. Exercises - p. 176 39. 4.5: The Multinomial, the Bivariate Normal, and the Multivariate Normal distributions - p. 177 40. Exercises - p. 185 41. Chapter 5 Independence of random variables and some applications - p. 191 42. 5.1: Independence of random variables and criteria of independence - p. 191 43. Exercises - p. 197 44. 5.2: The reproductive property of certain distributions - p. 203 45. Exercises - p. 208 46. Chapter 6 Transformation of random variables - p. 213 47. 6.1: Transforming a single random variable - p. 213 48. Exercises - p. 217 49. 6.2: Transforming two or more random variables - p. 218 50. Exercises - p. 228 51. 6.3: Linear transformations - p. 233 52. Exercises - p. 237 53. 6.4: The probability integral transform - p. 240 54. Exercises - p. 241 55. 6.5: Order statistics - p. 241 56. Exercises - p. 248 57. Chapter 7 Some modes of convergence of random variables, applications - p. 253 58. 7.1: Convergence in distribution or in probability and their relationship - p. 253 59. Exercises - p. 257 60. 7.2: Some applications of convergence in distribution: WLLN and CLT - p. 259 61. Exercises - p. 269 62. 7.3: Further limit theorems - p. 276 63. Exercises - p. 280 64. Chapter 8 An overview of statistical inference - p. 281 65. Chapter 9 Point estimation - p. 297 66. 9.1: Maximum likelihood estimation: motivation and examples - p. 297 67. Exercises - p. 303 68. 9.2: Some properties of MLE's - p. 311 69. Exercises - p. 318 70. 9.3: Uniformly minimum variance unbiased estimates - p. 319 71. Exercises - p. 325 72. 9.4: Decision-theoretic approach to estimation - p. 328 73. Exercises - p. 334 74. 9.5: Other methods of estimation - p. 336 75. Exercises - p. 338 76. Chapter 10 Confidence intervals and confidence regions - p. 341 77. 10.1: Confidence intervals - p. 341 78. Exercises - p. 345 79. 10.2: Confidence intervals in the presence of nuisance parameters - p. 349 80. Exercises - p. 353 81. 10.3: A confidence region for (μ,σ2) in the N(μ,σ2) distribution - p. 353 82. 10.4: Confidence intervals with approximate confidence coefficient - p. 355 83. Exercises - p. 357 84. Chapter 11 Testing hypotheses - p. 361 85. 11.1: General concepts, formulation of some testing hypotheses - p. 362 86. Exercises - p. 363 87. 11.2: Neyman–Pearson fundamental lemma, Exponential type families, UMP tests for some composite hypotheses - p. 364 88. Exercises - p. 369 89. 11.3: Some applications of Theorem 2 - p. 375 90. Exercises - p. 382 91. Exercises - p. 389 92. 11.4: Likelihood ratio tests - p. 390 93. 11.4.1: Testing hypotheses for the parameters in a single Normal population - p. 393 94. 11.4.2: Comparing the parameters of two Normal populations - p. 399 95. Exercises - p. 406 96. Chapter 12 More about testing hypotheses - p. 413 97. 12.1: Likelihood ratio tests in the Multinomial case and contingency tables - p. 413 98. Exercises - p. 418 99. 12.2: A goodness-of-fit test - p. 420 100. Exercises - p. 422 101. 12.3: Decision-theoretic approach to testing hypotheses - p. 425 102. 12.4: Relationship between testing hypotheses and confidence regions - p. 432 103. Chapter 13 A simple linear regression model - p. 437 104. 13.1: Setting up the model — the principle of least squares - p. 438 105. 13.2: The least squares estimates of β1 and β2 and some of their properties - p. 440 106. 13.3: Normally distributed errors: MLE's of β1,β2, and σ2, some distributional results - p. 449 107. Exercises - p. 455 108. 13.4: Confidence intervals and hypotheses testing problems - p. 458 109. Exercises - p. 461 110. 13.5: Some prediction problems - p. 465 111. Exercises - p. 467 112. 13.6: Proof of Theorem 5 - p. 469 113. 13.7: Concluding remarks - p. 471 114. Chapter 14 Two models of analysis of variance - p. 473 115. 14.1: One-way layout with the same number of observations per cell - p. 474 116. Exercises - p. 483 117. 14.2: A multicomparison method - p. 484 118. Exercises - p. 490 119. 14.3: Two-way layout with one observation per cell - p. 490 120. 14.3.1: The MLE's of the parameters of the model - p. 491 121. 14.3.2: Testing the hypothesis of no row or no column effects - p. 492 122. 14.3.3: Proof of lemmas in Section 14.3 - p. 499 123. Exercises - p. 505 124. Chapter 15 Some topics in nonparametric inference - p. 509 125. 15.1: Some confidence intervals with given approximate confidence coefficient - p. 510 126. 15.2: Confidence intervals for quantiles of a distribution function - p. 512 127. 15.3: The two-sample sign test - p. 514 128. Exercises - p. 518 129. 15.4: The rank sum and the Wilcoxon–Mann–Whitney two-sample tests - p. 519 130. 15.5: Nonparametric curve estimation - p. 529 131. Exercises - p. 537 132. Chapter 16 Appendix - p. 541 133. Tables - p. 551 134. Answers to even-numbered exercises - p. 589 135. Index - p. 635 136. Functions - p. 644
