**Introduction **

We want to brieﬂy justify why there should be another probability book, when so many others are available. Motivation: Students from majors in the mathematical sciences and in other areas will be more engaged with the material if they are studying problems that are relevant to them. While testing drafts of the book in the classroom, the students who used this book were asked to contribute questions. As a result, many of the exercises in this text began with questions motivated by the students’ own interests. Example- and exercise-oriented approach: Our book serves as a student’s ﬁrst introduction to probability theory, so we devote signiﬁcant attention to a wealth of exercises and examples. We encourage students to practice their skills by solving lots of questions. Our exercises are split into practice, extensions, and advanced types of questions. We recommend assigning a small number of questions to students on a daily basis. This promotes more interactive discussion in class between the students and the instructor. It consistently empowers the students to try their hand at some problems of their own. It reduces stress and “cramming” at exam time, as the students consistently develop their understanding during the course. It also provides a ﬁrm foundation for the students’ long term understanding of probability. The exercises, theorems, deﬁnitions, and remarks are all numbered in one list (instead of numbered separately) because we believe that they all should be used in tandem to understand the chapter material.

**Content **

**Part 1 Randomness 1**

1 Outcomes, Events, and Sample Spaces 3

2 Probability 17

3 Independent Events 37

4 Conditional Probability

5 Bayes’ Theorem

6 Review of Randomness

**Part 2 Discrete Random Variables 77**

7 Discrete Versus Continuous Random Variables 78

8 Probability Mass Functions and CDFs 87

9 Independence and Conditioning 102

10 Expected Values of Discrete Random Variables 120

11 Expected Values of Sums of Random Variables 130

12 Variance of Discrete Random Variables 142

13 Review of Discrete Random Variables 161

**Part 3 Named Discrete Random Variables 171**

14 Bernoulli Random Variables 173

15 Binomial Random Variables 182

16 Geometric Random Variables 197

17 Negative Binomial Random Variables 214

18 Poisson Random Variables 224

19 Hypergeometric Random Variables 241

20 Discrete Uniform Random Variables 254

21 Review of Named Discrete Random Variables 259

**Part 4 Counting 271**

22 Introduction to Counting 272

23 Two Case Studies in Counting 293

**Part 5**** Continuous Random Variables 299**

24 Continuous Random Variables and PDFs 300

25 Joint Densities 319

26 Independent Continuous Random Variables 333

27 Conditional Distributions 345

28 Expected Values of Continuous Random Variables 355

29 Variance of Continuous Random Variables 366

30 Review of Continuous Random Variables 384

**Part 6 **** Named Continuous Random Variables 389**

31 Continuous Uniform Random Variables 391

32 Exponential Random Variables 412

33 Gamma Random Variables 428

34 Beta Random Variables 438

35 Normal Random Variables 445

36 Sums of Independent Normal Random Variables 470

37 Central Limit Theorem 483

38 Review of Named Continuous Random Variables 505

**Part 7 Additional Topics 515**

39 Variance of Sums; Covariance; Correlation 516

40 Conditional Expectation 541

41 Markov and Chebyshev Inequalities 554

42 Order Statistics

43 Moment Generating Functions 582

44 Transformations of One or Two Random Variables 596

45 Review Questions for All Chapters 613

Tags: #by Mark Daniel Ward And Ellen Gundlach 1st Edition #by Mark Daniel Ward And Ellen Gundlach pdf 2017 #Introduction To Probability #Introduction To Probability Mark Ward pdf #W.h. Freeman