电气工程中的概率统计与随机过程导论

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"Probability, Statistics, and Random Processes For Electrical Engineering" 是一本由Alberto Leon-Garcia编写的关于电气工程中的概率论、统计学和随机过程的教科书，第三版。书中包含了大量的例题和详细解释，适用于入门学习者。非影印版本，配备目录，便于读者查阅和学习。本书涵盖了以下主要知识点： 1. **概率论**：这部分介绍了概率论的基本概念，包括事件的概率、条件概率、独立事件、概率分布等。例如，二项分布、泊松分布、均匀分布、正态分布等常见连续和离散概率分布的定义和特性。此外，还会涉及大数定律和中心极限定理，这些都是概率论在实际问题中应用的基础。 2. **统计学**：统计学部分讲解了数据收集、描述性统计（如均值、中位数、方差、标准差等）和推断性统计（假设检验、置信区间、回归分析等）。书中可能涵盖了参数估计、t检验、卡方检验、F检验以及非参数检验等内容，这些都是工程师解决实际问题时常用的数据分析工具。 3. **随机过程**：随机过程是电气工程中至关重要的概念，特别是对于信号处理、通信系统和控制系统等领域。书中会讲解随机变量序列的相关性和协方差，如马尔可夫过程、Wiener过程（布朗运动）、平稳过程、Karhunen-Loève展开、以及随机微分方程等。这些理论为理解和分析随机系统的动态行为提供了基础。 4. **应用实例**：书中通过丰富的例题和练习题帮助读者将理论知识应用于实际问题，如无线通信中的信道建模、滤波器设计、信号检测和估计等。这有助于读者理解并掌握这些抽象概念的实际意义。 5. **数学工具**：学习这本书需要一定的数学基础，包括线性代数、微积分和复数运算。书中可能会回顾这些基础知识，并将其与概率统计和随机过程相结合，帮助读者建立坚实的数学背景。 6. **学习辅助**：配备目录使读者能够快速定位所需内容，而索引则方便查找特定主题。此外，书中的参考文献和索引为深入研究提供了进一步的资源。《Probability, Statistics, and Random Processes For Electrical Engineering》是电气工程学生或从业者掌握概率、统计和随机过程理论的优秀教材，其详尽的解释和实例将有助于读者巩固理论知识并提升问题解决能力。

Electrical and computer engineers have played a central role in the design of modern

information and communications systems. These highly successful systems work reli-

ably and predictably in highly variable and chaotic environments:

• Wireless communication networks provide voice and data communications to

mobile users in severe interference environments.

• The vast majority of media signals, voice, audio, images, and video are processed

digitally.

• Huge Web server farms deliver vast amounts of highly specific information to

users.

Because of these successes, designers today face even greater challenges. The sys-

tems they build are unprecedented in scale and the chaotic environments in which they

must operate are untrodden terrritory:

• Web information is created and posted at an accelerating rate; future search ap-

plications must become more discerning to extract the required response from a

vast ocean of information.

• Information-age scoundrels hijack computers and exploit these for illicit purpos-

es, so methods are needed to identify and contain these threats.

• Machine learning systems must move beyond browsing and purchasing applica-

tions to real-time monitoring of health and the environment.

• Massively distributed systems in the form of peer-to-peer and grid computing

communities have emerged and changed the nature of media delivery, gaming,

and social interaction; yet we do not understand or know how to control and

manage such systems.

Probability models are one of the tools that enable the designer to make sense

out of the chaos and to successfully build systems that are efficient, reliable, and cost

effective. This book is an introduction to the theory underlying probability models as

well as to the basic techniques used in the development of such models.

Probability Models

in Electrical and

Computer Engineering

CHAPTER

2 Chapter 1 Probability Models in Electrical and Computer Engineering

This chapter introduces probability models and shows how they differ from the

deterministic models that are pervasive in engineering. The key properties of the no-

tion of probability are developed, and various examples from electrical and computer

engineering, where probability models play a key role, are presented. Section 1.6 gives

an overview of the book.

1.1 MATHEMATICAL MODELS AS TOOLS IN ANALYSIS AND DESIGN

The design or modification of any complex system involves the making of choices from

various feasible alternatives. Choices are made on the basis of criteria such as cost, re-

liability, and performance.The quantitative evaluation of these criteria is seldom made

through the actual implementation and experimental evaluation of the alternative con-

figurations. Instead, decisions are made based on estimates that are obtained using

models of the alternatives.

A model is an approximate representation of a physical situation. A model at-

tempts to explain observed behavior using a set of simple and understandable rules.

These rules can be used to predict the outcome of experiments involving the given

physical situation. A useful model explains all relevant aspects of a given situation.

Such models can be used instead of experiments to answer questions regarding the

given situation. Models therefore allow the engineer to avoid the costs of experimenta-

tion, namely, labor, equipment, and time.

Mathematical models are used when the observational phenomenon has measur-

able properties. A mathematical model consists of a set of assumptions about how a

system or physical process works. These assumptions are stated in the form of mathe-

matical relations involving the important parameters and variables of the system. The

conditions under which an experiment involving the system is carried out determine the

“givens” in the mathematical relations, and the solution of these relations allows us to

predict the measurements that would be obtained if the experiment were performed.

Mathematical models are used extensively by engineers in guiding system design

and modification decisions. Intuition and rules of thumb are not always reliable in pre-

dicting the performance of complex and novel systems, and experimentation is not pos-

sible during the initial phases of a system design. Furthermore, the cost of extensive

experimentation in existing systems frequently proves to be prohibitive. The availabil-

ity of adequate models for the components of a complex system combined with a

knowledge of their interactions allows the scientist and engineer to develop an overall

mathematical model for the system. It is then possible to quickly and inexpensively an-

swer questions about the performance of complex systems. Indeed, computer pro-

grams for obtaining the solution of mathematical models form the basis of many

computer-aided analysis and design systems.

In order to be useful, a model must fit the facts of a given situation.Therefore the

process of developing and validating a model necessarily consists of a series of experi-

ments and model modifications as shown in Fig. 1.1. Each experiment investigates a

certain aspect of the phenomenon under investigation and involves the taking of ob-

servations and measurements under a specified set of conditions. The model is used

to predict the outcome of the experiment, and these predictions are compared with

the actual observations that result when the experiment is carried out. If there is a

4 Chapter 1 Probability Models in Electrical and Computer Engineering

“measure” the relevant performance parameters. In general, simulation models are

capable of representing systems in greater detail than mathematical models. Howev-

er, they tend to be less flexible and usually require more computation time than math-

ematical models.

In the following two sections we discuss the two basic types of mathematical

models, deterministic models and probability models.

1.2 DETERMINISTIC MODELS

In deterministic models the conditions under which an experiment is carried out deter-

mine the exact outcome of the experiment. In deterministic mathematical models, the

solution of a set of mathematical equations specifies the exact outcome of the experi-

ment. Circuit theory is an example of a deterministic mathematical model.

Circuit theory models the interconnection of electronic devices by ideal circuits

that consist of discrete components with idealized voltage-current characteristics. The

theory assumes that the interaction between these idealized components is completely

described by Kirchhoff’s voltage and current laws. For example, Ohm’s law states that

the voltage-current characteristic of a resistor is The voltages and currents in

any circuit consisting of an interconnection of batteries and resistors can be found by

solving a system of simultaneous linear equations that is found by applying Kirchhoff’s

laws and Ohm’s law.

If an experiment involving the measurement of a set of voltages is repeated a

number of times under the same conditions, circuit theory predicts that the observa-

tions will always be exactly the same. In practice there will be some variation in the ob-

servations due to measurement errors and uncontrolled factors. Nevertheless, this

deterministic model will be adequate as long as the deviation about the predicted val-

ues remains small.

1.3 PROBABILITY MODELS

Many systems of interest involve phenomena that exhibit unpredictable variation and

randomness. We define a random experiment to be an experiment in which the out-

come varies in an unpredictable fashion when the experiment is repeated under the

same conditions. Deterministic models are not appropriate for random experiments

since they predict the same outcome for each repetition of an experiment. In this sec-

tion we introduce probability models that are intended for random experiments.

As an example of a random experiment, suppose a ball is selected from an urn

containing three identical balls, labeled 0, 1, and 2. The urn is first shaken to random-

ize the position of the balls, and a ball is then selected.The number of the ball is noted,

and the ball is then returned to the urn. The outcome of this experiment is a number

from the set We call the set S of all possible outcomes the sample space.

Figure 1.2 shows the outcomes in 100 repetitions (trials) of a computer simulation of

this urn experiment. It is clear that the outcome of this experiment cannot consistent-

ly be predicted correctly.

S = 50, 1, 26.

I = V>R.

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