哈佛大学概率论教材：《Introduction to Probability》Blitzstein & Hwang

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"哈佛大学的Joseph K. Blitzstein和Jessica Hwang合著的《Introduction to Probability》是一本概率论入门的经典教材，以其深入浅出的讲解受到推崇。此书还提供了VitalSource® ebook版本，允许读者在线或离线阅读，并支持在各种设备上同步笔记和高亮内容。" 在《Introduction to Probability》这本书中，作者Joseph K. Blitzstein和Jessica Hwang引导读者进入概率论的世界，这是一个在统计科学和信息技术中至关重要的领域。概率论是理解和解决随机现象的关键工具，它不仅在数学、统计学和计算机科学中有基础性地位，而且在经济学、生物学、物理学、工程学以及数据科学等领域都发挥着重要作用。该教材可能涵盖以下核心概念： 1. **概率基础**：书中首先会介绍概率的基本概念，如样本空间、事件、概率的定义及其性质，包括概率的加法法则和乘法法则，以及条件概率。 2. **概率分布**：书中将详细讲解常见的离散概率分布，如二项分布、几何分布、泊松分布，以及连续概率分布，如均匀分布、正态分布（高斯分布）和指数分布。这些分布是理解和建模随机变量的基础。 3. **随机变量**：随机变量是概率论中的重要概念，书中会解释如何处理离散和连续随机变量，包括它们的期望值（均值）、方差和其他统计量。 4. **大数定律和中心极限定理**：这两个定理是概率论的核心，大数定律描述了独立随机事件平均结果的稳定性，而中心极限定理解释了为什么很多自然现象的样本均值接近正态分布。 5. **依赖关系**：书中会讨论随机变量之间的相关性和独立性，包括联合分布、边缘分布和条件分布，以及协方差和相关系数。 6. **贝叶斯统计**：作为概率论的一个分支，贝叶斯统计提供了一种更新先验信念以形成后验概率的框架，这对于统计推断和决策至关重要。 7. **模拟与蒙特卡洛方法**：这些现代概率分析工具可以帮助解决复杂问题，书中可能涉及如何使用随机数生成和重复试验来近似计算概率和期望值。 8. **概率模型**：通过实例和案例研究，书中的章节可能展示如何构建和应用概率模型来解决实际问题，如赌博、疾病检测、排队理论等。为了充分利用VitalSource® ebook，读者需要创建或登录VitalSource Bookshelf账户，兑换提供的代码以获取在线访问权限。下载BookShelf应用程序后，可以在PC、Mac、iOS设备或Android设备上离线阅读，并保持笔记和高亮内容的同步。这种数字化学习体验使得学习过程更加便捷和互动，有助于提高学习效率。

xiv Preface

3. Dual teaching of concepts and strategies. Our intent is that in reading this

book, students will learn not only the concepts of probability, but also

a set of problem-solving strategies that are widely ap p l i cab l e outside of

probability. In the worked examples, we explain each step of the soluti on

but also comment on how we knew t o take the approach we did. Often we

present multiple solutions to the same probl em.

We explicitly identify and name important strategies such as symmetry

and pattern recognition, and we proactively dispel common misunder-

standings, which are marked with t h e

h (biohazard) symbol.

4. Practice problems. The book contains about 600 exercises of varying dif-

ﬁculty. The exercises are intended to rein for ce understanding of the ma-

terial and strengthen problem-solving skills instead of req u i r i n g repetitive

calculations. Some are strategic practice problems, grouped by theme to

facilitate practice of a particular topic, while others are mixed practice,

in which several earlier topics may need to be synthesi ze d. About 250

exercises have detailed online solution s for practice and self-study.

5. Simulation, Monte Carlo, and R. Many probability problems are too dif-

ﬁcult to solve exactly, and in any case it is important to be able to check

one’s answer. We introduce techniques for exploring probability via sim-

ulation, and show that often a few lines of R code suﬃce to create a

simulation for a seemingly complicated p r ob l em.

6. Focu s on real-world relevance and statistical thinking. Examp l es and ex-

ercises in this book have a clear real-world motivation, with a particu-

lar focus on building a strong foundation for further study of statistical

inference and modeling. We preview important statist i cal id eas such as

sampling, simulation, Bayesian inference, and Markov chain Monte Carlo;

other applicat i on areas include genetics, medicine, computer science, and

information theory. Our choice of examples and exercises is intended to

highlight the power, applicability, and beauty of probabilistic thinking.

Acknowledgments

We thank our colleagues, th e Stat 110 teaching assi st ants, and several thousand Stat

110 stu d ents for their comments and ideas related to the course and the book. In

particular, we than k Alvin Siu , Angela Fan, Anji Tang, Car ol y n Stein, David Jones,

David Rosengarten, David Watson, Johannes Ruf, Kari Lock, Keli Liu , Kevin Bartz,

Lazhi Wang, Mar t i n Lysy, Michele Zemplenyi, Peng Ding, Rob Phillips, Sam Fisher,

Sebastian Chi u , Soﬁa Hou, Theresa Gebert, Valeria Espinosa, Viktoriia Liublinska,

Viviana Garcia, William Chen, and Xander Marcus for their f ee d back.

We especially thank Bo J i an g, Raj Bhuptani, Shira M i t chell, and th e anonymous

reviewers for their detailed comments on d r af t versions of the book, and Andrew

Probability and counti n g

Luck. Coincidence. Randomness. Uncertainty. Risk. Doubt. Fortune. Chance.

You’ve probably heard these words countless times, but chances are that they were

used in a vague, casual way. Unfortunately, despite its ubiquity in science and ev-

eryday life, probability can be deep l y counterintuitive. If we rely on intuitions of

doubtful validity, we run a serious risk of making inaccurate predictions or over-

conﬁdent decisions. The goal of this book i s to introduce probability as a logical

framework for quantifying uncertainty and randomness in a principled way. We’ll

also aim to strengthen intuition, both when our initial guesses coincide with logical

reasoning and when we’re not so lucky.

1.1 Why study probability?

Mathematics is the logic of certainty; probabi l i ty is the logic of uncertainty. Prob-

ability is extremely useful in a wid e variety of ﬁelds, since it provides tools for

understanding and explaining variation, separatin g si gn al from noise, and modeling

complex phenomena. To give just a small sample from a continually growing list of

applications:

1. Statistics: Probability is the foun dat i on and language for statistics, en-

abling many powerful met h ods for using data to learn about the world.

2. Physics: Einstein famously said “G od does not play dice with the uni-

verse”, b u t current understandi n g of quantum physics heavily involves

probability at the most fundamental level of nature. Statistical mechanics

is another major branch of physics that is built on probabil i ty.

3. Biology: Genetics is deeply intertwined with probability, both in the in-

heritance of genes and in modelin g random mutations.

4. Computer science: Randomized algorithms make random choices while

they are run, and in many important applications they are simpler and

more eﬃcient than any currently known determinist i c alternatives. Proba-

bility also plays an essenti al role in studying the performance of algorithms,

and in machine learning and artiﬁcial intelligence.

2 Introduction to Probability

5. Meteorology: Weather forecasts are (or should be) computed and expressed

in terms of probability.

6. Gambling: Many of the earl i est investigations of probability were aimed

at answering questions about gambling and games of chance.

7. Finance: At the risk of redundancy with the previous example, it should

be pointed out that probability is central in quantitative ﬁnance. Model i n g

stock prices over time an d determining “fair” prices for ﬁnancial instr u -

ments are based heavily on probability.

8. Political science: In recent years, political science has become more and

more quantitative and statistical. For example, Nate Silver’s successes in

predicting election results, such as in the 2008 an d 2012 U.S. presidential

elections, were achieved using probability models to make sense of polls

and to drive simulations (see Sil ver [25]).

9. Medicine: The development of randomized clinical trials, in which patients

are randomly assigned to receive treatment or placebo, has transformed

medical r es ear ch in recent years. As the b i ost at i st i ci an David Harrington

remarked, “Some have conjectured that it could be the most signiﬁcant

advance in scientiﬁc medicine in the twentieth century. . . . In one of the

delightful ironies of modern science, the randomized trial ‘adjusts’ for both

observed and unobserved heterogeneity in a controlled experiment by in-

troducing chance variation into the study design.” [17]

10. Life: Life is uncertain, and probability is the logic of uncertainty. While it

isn’t practical to carry out a for mal probability calculation for every deci-

sion made in life, thinki n g hard about probability can help us avert some

common fallacies, shed light on coinciden ces, and make better p r ed i ct i on s.

Probability provides procedures for principled problem-solving, but it can also pro-

duce pitfalls and paradoxes. For exampl e, we’ll see in this chapter that even Got-

tfried Wilhelm von Leibniz and Sir Isaac Newton, the two people who independently

discove re d calculus in the 17th century, were not immu ne to basic errors in prob-

ability. Throughout this book, we will use the following strategies to help avoid

potential pitfalls.

1. Simulation: A beautiful aspect of probability is that it is often possible to

study problems vi a simulation. Rather than endlessly debating an answer

with someone who di sagr ees with you, you can run a simulation and see

empirically wh o is right. Each chap t er in this book ends with a section

that gives examples of how to do calculations and simulations in R, a free

statistical computing environment.

2. Biohazards: Stu d y i n g common mistakes is important for gaining a stronger

understanding of what is and is not valid reasonin g in probabil i ty. In this

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