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本书从概率论的基础开始，通过例子与习题的旁征博引，引进了大量近代统计处理的新技术和一些国内同类教材中不能见而广为使用的分布。其内容包括工科概率论入门、经典统计和现代统计的基础，又加进了不少近代统计中数据处理的实用方法和思想，例如：Bootstrap再抽样法、刀切（Jackknife）估计、EM算法、Logistic回归、稳健（Robust）回归、Markov链、Monte Carlo方法等。它的统计内容与国内流行的教材相比，理论较深，模型较多，案例的涉及面要广，理论的应用面要丰富，统计思想的阐述与算法更为具体。

本书可作为工科、管理类学科专业本科生、研究生的教材或参考书，也可供教师、工程技术人员自学之用。

出版说明

序

1 Probability Theory

1.1 Set Theory

1.2 Basics of Probability Theory

1.2.1 Axiomatic Foundations

1.2.2 The Calculus of Probabilities

1.2.3 Counting

1.2.4 Enumerating Outcomes

1.3 Conditional Probability and Independence

1.4 Random Variables

1.5 Distribution Functions

1.6 Density and Mass Functions

1.7 Exercises

1.8 Miscellanea

2 Transformations and Expectations

2.1 Distributions of Functions of a Random Variable

2.2 Expected Values

2.3 Moments and Moment Generating Functions

2.4 Differentiating Under an Integral Sign

2.5 Exercises

2.6 Miscellanea

3 Common Families of Distributions

3.1 Introduction

3.2 Discrete Distributions

3.3 Continuous Distributions

3.4 Exponential Families

3.5 Location and Scale Families

3.6 Inequalities and Identities

3.6.1 Probability Inequalities

3.6.2 Identities

3.7 Exercises

3.8 Miscellanea

4 Multiple Random Variables

4.1 Joint and Marginal Distributions

4.2 Conditional Distributions and Independence

4.3 Bivariate Transformations

4.4 Hierarchical Models and Mixture Distributions

4.5 Covariance and Correlation

4.6 Multivariate Distributions

4.7 Inequalities

4.7.1 Numerical Inequalities

4.7.2 Functional Inequalities

4.8 Exercises

4.9 Miscellanea

5 Properties of a Random Sample

5.1 Basic Concepts of Random Samples

5.2 Sums of Random Variables from a Random Sample

5.3 Sampling from the Normal Distribution

5.3.1 Properties of the Sample Mean and Variance

5.3.2 The Derived Distributions: Student's t and Snedecor's F

5.4 Order Statistics

5.5 Convergence Concepts

5.5.1 Convergence in Probability

5.5.2 Almost Sure Convergence

5.5.3 Convergence in Distribution

5.5.4 The Delta Method

5.6 Generating a Random Sample

5.6.1 Direct Methods

5.6.2 Indirect Methods

5.6.3 The Accept/Reject Algorithm

5.7 Exercises

5.8 Miscellanea

6 Principles of Data Reduction

6.1 Introduction

6.2 The Sufficiency Principle

6.2.1 Sufficient Statistics

6.2.2 Minimal Sufficient Statistics

6.2.3 Ancillary Statistics

6.2.4 Sufficient, Ancillary, and Complete Statistics

……

7 Point Estimation

8 Hypothesis Testing

8.1 Introduction

9 Interval Estimation

10 Asymptotic Evaluations

11 Analysis of Variance and Regression

12 Regression Models

Appendix: Computer Algebra

Table of Common Distributions

References

Author Index

Subject Index

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If E|g(X)|=∞, we say that Eg(X) does not exist. (Ross 1988 refers to this as the "law of the unconscious statistician." We do not find this amusing.)

The point here is that normality comes from sums of "small" (finite variance), independent disturbances.

Definition 1.2.4 Given a sample sapce S and an association sigma algebra B, a probability function is a function P with domain B that satisifies

1. P(A) >= 0 for all A in B.

2. P(S) = 1.

3. If A1, A2, ... in B are parwise disjoint, then P(Union(Ai)) = sum(P(Ai))

The three properties given in Definition 1.2.4 are usually referred to as the Axioms of Probability (or the Kolmogorov Axioms, after A. Kolmogorov, one of the fathers of probability theory).

The probability definition deals with one probability structure, but the statistis definition deals with an entire family.

The property of consistency is concerned with the asymptotic accuracy of an estimator: Does it converge to the parameter that it is estimating? In this section we look at a related property, efficiency, which is concerned with the asymptotic variance of an estimator.

"I've wasted time enough," said Lestrade rising. "I believe in hard work and not in sitting by the fire spinning fine theories."

书籍介绍

本书从概率论的基础开始，通过例子与习题的旁征博引，引进了大量近代统计处理的新技术和一些国内同类教材中不能见而广为使用的分布。其内容包括工科概率论入门、经典统计和现代统计的基础，又加进了不少近代统计中数据处理的实用方法和思想，例如：Bootstrap再抽样法、刀切（Jackknife）估计、EM算法、Logistic回归、稳健（Robust）回归、Markov链、Monte Carlo方法等。它的统计内容与国内流行的教材相比，理论较深，模型较多，案例的涉及面要广，理论的应用面要丰富，统计思想的阐述与算法更为具体。

本书可作为工科、管理类学科专业本科生、研究生的教材或参考书，也可供教师、工程技术人员自学之用。