工程概率统计

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"Probability and Statistics for Engineering" 这本书是"概率论与数理统计"的教材，适用于Stat 235或同等课程。它提供了一种电子版本，方便学习。由于电子版权的限制，可能有一些第三方内容被省略，但经过编辑审查，这些省略的内容被认为不会实质影响整体学习体验。出版商保留随时因后续权利限制而移除内容的权利。对于定价、旧版信息、当前版改动以及替代格式等有价值的信息，读者可以通过访问www.cengage.com/highered进行搜索，以ISBN号、作者、书名或关键词查找相关领域材料。 "Probability and Statistics for Engineers and Scientists" 是一本专为工程和科学领域的学生设计的概率论与统计学教材。第四版由Anthony Hayter撰写，他来自丹佛大学。这本书旨在帮助工程和科学专业的学生理解并应用概率论和统计学的基本概念、方法和理论。在工程和科学研究中，概率论是研究随机事件发生的可能性和不确定性的重要工具，而统计学则涉及到数据收集、分析、解释和呈现，以支持决策和发现规律。这本书可能会涵盖以下关键知识点： 1. 概率基础：包括概率的定义、概率的公理系统、条件概率、独立事件、贝叶斯定理等。 2. 随机变量：离散和连续随机变量的概念，概率分布，如二项分布、泊松分布、正态分布等。 3. 大数定律和中心极限定理：这些是统计推断的基础，解释了大量独立随机变量的平均行为。 4. 统计推断：包括参数估计（点估计和区间估计）、假设检验（如t检验、卡方检验、F检验等）。 5. 相关性和回归分析：探讨变量间的关联，以及如何用一个变量预测另一个变量。 6. 方差分析(ANOVA)和实验设计：用于比较多个组别的差异，以及优化实验设置。 7. 线性代数和矩阵理论的基础：在处理多元统计问题时，这些工具非常有用。 8. 贝叶斯统计：一种基于先验知识更新概率的统计方法。 9. 随机过程：如布朗运动、泊松过程等，对工程问题（如信号处理、可靠性分析）有重要意义。这本教材可能通过实际案例和工程问题来阐述这些概念，以增强学生的应用能力。同时，它可能还包含了习题和解答，帮助学生巩固学习成果。通过深入学习这门课程，学生将能够理解和运用概率统计工具解决实际工程和科学中的问题。

xiv PREFACE

Acknowledgments

I would like to express my heartfelt thanks to my editor, Molly Taylor, and to all members

of the team that have contributed to this fourth edition with their wonderful talents, wisdom,

experience, and energy. This especially includes Andrew Coppola, Alexander Gontar, Linda

Helcher, Charu Khanna, Jill A. Quinn, Shaylin Walsh Hogan, and Richard Stratton.

Finally, I would also like to thank various reviewers who have helped with the de-

velopment of this book, especially Christopher Scott Brown of the University of South

Alabama for his work on the fourth edition. The reviewers of the ﬁrst edition include Mary

R. Anderson, Arizona State University; Charles E. Antle, The Pennsylvania State University,

University Park; Sant Ram Arora, University of Minnesota—Twin Cities Campus; William

R. Astle, Colorado School of Mines; Lee J. Bain, University of Missouri—Rolla; Douglas

M. Bates, University of Wisconsin—Madison; Rajan Batta, The State University of New

York at Buffalo; Alan C. Bovik, University of Texas at Austin; Don B. Campbell, Western

Illinois University; M. Jeya Chandra, Pennsylvania State University—University Park; Yueh-

Jane Chang, Idaho State University; Chung-Lung Chen, Mississippi State University; Inchan

Choi, Wichita State University; John R. Cook, North Dakota State University; Rianto A.

Djojosugito, South Dakota School of Mines and Technology; Lucien Duckstein, University

of Arizona; Earnest W. Fant, University of Arkansas; Richard F. Feldman, Texas A & M

University; Sam Gutmann, Northeastern University; Carol O’Connor Holloman, University

of Louisville; Chi-Ming Ip, University of Miami; Rasul A. Khan, Cleveland State Univer-

sity; Stojan Kotefski, New Jersey Institute of Technology; Walter S. Kuklinski, University

of Massachusetts—Lowell; S. Kumar, Rochester Institute of Technology; Gang Li, Univer-

sity of North Carolina at Charlotte; Jiye-Chyi Lu, North Carolina State University; Ditlev

Monrad, University of Illinois at Urbana—Champaign; John Morgan, California Polytechnic

State University—Pomona; Paul J. Nahin, University of New Hampshire; Larry Ringer, Texas

A & M University; Paul L. Schillings, Montana State University; Ioannis Stavrakakis, The

University of Vermont; and James J. Swain, University of Alabama—Huntsville.

The reviewers of the second edition were Alexander Dukhovny, San Francisco State Uni-

versity; Marc Genton, Massachusetts Institute of Technology; Diwakar Gupta, University of

Minnesota; Joseph J. Harrington, Harvard University; and Jim Rowland, University of Kansas.

Survey respondents for the third edition include Mostafa S. Aminzadeh, Towson Univer-

sity; Barb Barnet, University of Wisconsin—Platteville; Ronald D. Bennett, Bethel College;

Shannon Brewer, Northeast State Community College; Frank C. Castronova, Lawrence Tech-

nological University; Mike Doviak, Old Dominion University; Natarajan Gautam, Penn State

University; Peggy Hart, Doane College; Wei-Min Huang, Lehigh University; Xiaoming Huo,

Georgia Tech; Bruce N. Janson, University of Colorado at Denver; Scott Jilek, University

of St. Thomas; Michael Kostreva, Clemson University; Paul Kvam, Georgia Tech; David

W. Matolak, Ohio University; Gary C. McDonald, Oakland University; Megan Meece, Uni-

versity of Florida; Luke Miller, University of San Diego; Steve Patch, University of North

Carolina at Asheville; Robi Polikar, Rowan University; Andrew M. Ross, Lehigh University;

Manuel D. Rossetti, University of Arkansas; Robb Sinn, North Georgia College and State

University; Bradley Thiessen, St. Ambrose University; Dolores Tichenor, Tri-State Univer-

sity; Lewis VanBrackle, Kennesaw State University; Jerry Weyand, Cleary University; Ed

Wheeler, University of Tennessee at Martin; Elaine Zanutto, The Wharton School, University

of Pennsylvania; and Kathy Zhong, University of Detroit Mercy.

The reviewers for the fourth edition include Georgiana Baker, University of South

Carolina; Arthur Cohen, Rutgers University; Diane Evans, Rose-Hulman Institute of Technol-

ogy; Piotr Kokoszka, Utah State University; Nikolay Strigul, Stevens Institute of Technology;

and Daniela Szatmari Voicu, Kettering University.

Anthony Hayter

CASE STUDIES

These running case studies continue through the chapters of the book. They

demonstrate how probability and statistical inference can be applied to the important

engineering problem of microelectronic solder joints and the developing ﬁeld

concerning the analysis of internet usage.

(1) Continuing Case Study: Microelectronic Solder Joints

Solder joints are an important component of microelectronic assemblies. Figure CS.1 shows

a cross-section of a typical assembly known as a ﬂip chip in which as series of conductive

bump-shaped solder joints are used to attach a silicon chip to a printed circuit board, which is

known as the substrate. These solder joints provide the conductive path from the silicon ship

to the substrate, and fatigue in the solder joints is responsible for almost all of the mechanical

and electrical failures of the assembly.

The area surrounding the solder joints between the silicon chip and the substrate is ﬁlled

with a substance known as the underﬁll, which is non-conductive epoxy that helps to protect

the solder joints from moisture as well as adding strength to the assembly. The underﬁll

also helps to minimize the stress in the solder joints that arises from the different thermal

expansions of the silicon chip and the substrate. This helps ensure that the connections are

not damaged or broken.

It is important to investigate the reasons behind the development of cracks in the solder

joints which can affect the operation of the assembly. The development of these cracks can

be related to the shapes of the solder joints, which are illustrated in Figure CS.2. While most

of the joints turn out to be barrel shaped, some may have cylinder shapes or hourglass shapes.

In addition to cracks in the solder joints, failures can also be caused by solder extrusions that

connect two adjacent solder joints as shown in Figure CS.3, or which leave only a very small

gap between two adjacent solder joints.

In addition, a critical component of the assembly is the bonding between the solder joint and

the substrate which is achieved through suitable metallization of the substrate. As Figure CS.4

shows, a bond pad is created in the substrate made of copper, which is coated with thin layers

of nickel and gold. The thickness of the gold layer has an important effect on the reliability

of the electrical connection between the solder joint and the substrate.

FIGURE CS.1

Cross section of a typical ﬂip chip

microelectronic assembly

Silicon chip

Printed circuit board

(Substrate)

Solder joint

Epoxy underfill

xvi CASE STUDIES

Barrel shape Cylinder shape Hourglass shape

FIGURE CS.2

Shapes of solder joints in a microelectronic assembly

Extrusion

FIGURE CS.3

An extrusion between solder joints

in a microelectronic assembly

FIGURE CS.4

Diagram of a substrate bond pad in

a microelectronic assembly

Gold layer Nickel layer

Substrate

Copper

Solder joint

Reliability assessment of these microelectronic assemblies is often performed with accel-

erated life tests. Since it is known that temperature changes cause stress in the solder joints that

can ultimately lead to failure, the accelerated life tests often consist of subjecting the assembly

to alternate periods at low and high temperatures. For example, the assembly may be alter-

nately immersed in liquid at −55 degrees Centigrade for ﬁve minutes, and then switched to a

liquid at 125 degrees Centigrade for ﬁve minutes. This cycle is repeated many times until the

assembly eventually fails. These accelerated life tests are designed to mimic (at an accelerated

speed) the conditions to which the assembly will be subjected in its everyday operation, and it

is anticipated that designs which survive the most cycles of the accelerated life test will have

the best reliability in real world applications.

A considerable amount of research is conducted on the fatigue life of the solder joints

together with their optimal production method, and the areas of probability and statistics play

major roles in this research. This case study will be continued through the ﬁrst twelve chapters

of the book to show how probability theory and statistical inference can be applied to this

important engineering problem.

Chapter 1 Probability Theory

For a given production method, it is important to know the probabilities that the solder joints

will be formed according to each of the three shape proﬁles. Additionally, the probability that

cracking occurs in each of the different solder joint shapes is investigated, and the probability

of ﬁnding a given number of cracked solder joints within a random sample of solder joints is

calculated.

Chapter 2 Random Variables

The number of extrusions in an assembly with a large number of solder joints is critical

to the overall lifetime of the assembly. In this chapter the average number of extrusions in

CASE STUDIES xvii

an assembly is investigated, and an assessment is made of the amount of variability in the

number of extrusions. In addition, the total number of extrusions in a batch of 250 assemblies

is considered.

Chapter 3 Discrete Probability Distributions

An analysis is conducted of the number of hourglass shaped solder joints on an assembly

which contains 64 solder joints. A calculation is also performed concerning the amount of

work that is necessary if a researcher intends to examine solder joints one at a time until two

cracked hourglass shaped solder joints are discovered. Finally, the distribution of the number

of solder joints of each shape on an assembly consisting of 16 solder joints is discussed.

Chapter 4 Continuous Probability Distributions

Accelerated life tests are considered and attention is directed to the question of how many

temperature cycles an assembly can survive before it fails. A probability distribution is used to

model this failure time. An investigation is also conducted into how different types of epoxies

employed in the underﬁll can affect the reliability of the assembly.

Chapter 5 The Normal Distribution

The thickness of the gold layer at the top of the bond pad has an important effect on the

reliability of the bond between the solder joint and the substrate. In this chapter these thick-

nesses are modelled with a normal probability distribution, and it is shown how to calculate

the probability that the thickness of the gold layer lies within an optimal range. The average

thickness of the gold layers on an assembly consisting of 16 bond pads is also considered.

Finally, calculations are made concerning the number of hourglass shaped solder joints that

will be produced on an assembly comprised of 512 solder joints.

Chapter 6 Descriptive Statistics

A data set is obtained by measuring the thicknesses of the nickel layers deposited by a new

method on each of the bond pads of an assembly. Summary statistics are calculated for the

data set. Also, a categorical data set is obtained from the frequencies of the solder joint shapes

in a large assembly produced by a particular methodology.

Chapter 7 Statistical Estimation and Sampling Distributions

The data sets given in Chapter 6 are used to obtain estimates of the quantities of interests. The

average amount of nickel deposited by the new method is estimated, and an assessment of

the estimate’s accuracy is made. Also, for the methodology under consideration an estimate

is obtained for the probability that a solder joint will have a barrel shape, and the precision of

this estimate is also considered.

Chapter 8 Inferences on a Population Mean

The data set of nickel layer thicknesses is used to examine whether the new method is applying

nickel according to the desired target value. The information pertinent to this question is

extracted from the data set and is summarized in various ways. Finally, calculations are

performed to estimate how many additional nickel layer thickness measurements are needed

in order to increase the sensitivity of the statistical inferences to a speciﬁed level.

xviii CASE STUDIES

Chapter 9 Comparing Two Population Means

A comparison is made between the original method of applying the nickel layer to the substrate

bond pads and the new method. A data set of nickel layer thicknesses achieved from the original

method is compared with the data set of nickel layer thicknesses achieved with the new method

in order to test whether there is any evidence of a difference between the two methods. Analysis

methods are introduced which allow the researcher to quantify this difference.

Chapter 10 Discrete Data Analysis

A test is performed to determine whether the frequencies of the solder joint shapes observed

in the analysis of a large assembly are consistent with some theoretical probability values.

The data set is also used to calculate a range for one of these probability values. In addition,

an experiment is conducted to investigate which of two epoxy formulations for the underﬁll

results in the smallest failure probability after 2000 temperature cycles of the accelerated life

test, and the analysis of the resulting data set is performed.

Chapter 11 The Analysis of Variance

In this chapter it is shown how a comparison can be made between four different companies

in terms of the thicknesses of the gold layers on the substrate bond pads. A random selection

is made of bond pads on assemblies produced by each of the four companies, and a data set is

formed by measuring the gold layer thicknesses. An analysis of the data set is then performed

which allows the determination of which company has the thinnest gold layers and which

company has the thickest gold layers.

Chapter 12 Simple Linear Regression and Correlation

The researcher is interested in whether the heights of the solder joints have any inﬂuence on the

reliability of the microelectronic assembly. An experiment is conducted whereby assemblies

with different solder joint heights are subjected to an accelerated life test until they fail. The

number of temperature cycles that the assemblies can withstand before failing is measured,

and an analysis is performed to investigate whether there is any evidence that the solder joint

heights have any effect on the reliability of the assembly.

(2) Continuing Case Study: Internet Marketing

An organisation’s website is obviously an important way for it to market itself and to drive its

business. Data are often automatically collected when an individual uses the organisation’s

website, and concern information on how the individual reached the websites, as well as the

actual website activities. The analysis of this data can reveal important information to the

organisation about how effective different website designs are, and how successful its various

online marketing campaigns and methods have been. This case study is included at the end of

each chapter, and the following questions are addressed.

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