工程视角下的概率统计与随机过程详解第四版

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《概率统计与随机过程：工程师指南》第四版是一本专为工程领域读者编写的权威教材，由Henry Stark和John W. Woods两位作者合著。本书基于他们深厚的专业背景，涵盖了概率论、统计学以及随机过程的核心概念，旨在为工程师提供坚实的理论基础和实际应用技巧。该书的核心知识点包括： 1. **概率论基础**：介绍了概率的基本概念，如概率空间、事件、概率分布、独立性和条件概率等。它通过实例演示了如何计算和理解随机变量的概率性质，为后续随机过程的讨论打下坚实基础。 2. **统计方法**：涵盖了抽样理论、参数估计、假设检验和回归分析等内容。这部分强调了如何从数据中提取信息，通过统计推断来支持决策制定。 3. **随机过程理论**：重点探讨了随机过程的不同类型，如离散时间随机过程（如马尔可夫链）、连续时间随机过程（如布朗运动），以及随机信号处理的基础，如平稳随机过程和随机过程的傅立叶变换。 4. **应用示例与工程实践**：书中提供了大量的实际工程问题和案例研究，将理论知识与工程领域的具体问题相结合，帮助读者理解如何在通信、控制系统、信号处理等领域运用概率统计和随机过程的知识。 5. **最新研究成果**：作为第四版，本书可能包含了最新的研究成果和技术发展，反映了概率统计和随机过程领域的前沿动态。 6. **作者贡献**：Henry Stark以其对黑暗时期到光亮的研究经历（可能指的是二战期间的经历）为灵感，体现了他对科学严谨性的追求。John W. Woods则贡献了他的编辑视角和专业知识，确保了内容的准确性和易理解性。《概率统计与随机过程：工程师指南》不仅是一本深入学习概率统计的教科书，也是一本实用的工具书，适合研究生、工程师以及对这些主题感兴趣的从业者参考阅读。版权信息显示，该书由Pearson Education出版，享有广泛的全球发行网络，确保了读者能够获取到最新的教育资源。

Sec. 1.2. THE DIFFERENT KINDS OF PROBABILITY 3

daughter Jane swim in the local pond, if Frank reports that Bill thought that he might

have seen an alligator in it. This despite the fact that no one has ever reported seeing

an alligator in this pond, and countless people have enjoyed swimming in it without ever

having been bitten by an alligator. To give this example some credibility, assume that the

pond is in Florida. Here Joe is ignoring the statistics and reacting to, what is essentially,

a rumor. Why? Possibly because the cost to Joe “just-in-case” there is an alligator in the

pond would be too high [1-3].

People buying lottery tickets intuitively believe that certain number combinations like

month/day/year of their grandson’s birthday are more likely to win than say, 06–06–06.

How many people will bet even odds that a coin that, heretofore has behaved “fairly,” that

is, in an unbiased fashion, will come up heads on the next toss, if in the last seven tosses it

has come up heads? Many of us share the belief that the coin has some sort of memory and

that, after seven heads, that coin must “make things right” by coming up with more tails.

A mathematical theory dealing with intuitive probability was developed by

B. O. Koopman [1-4]. However, we shall not discuss this subject in this book.

Probability as the Ratio of Favorable to Total Outcomes

(Classical Theory)

In this approach, which is not experimental, the probability of an event is computed a priori

†

by counting the number of ways n

that E can occur and forming the ratio n

/n,where

n is the number of all possible outcomes, that is, the number of all alternatives to E plus

. An important notion here is that all outcomes are equally likely. Since equally likely

is really a way of saying equally probable, the reasoning is somewhat circular. Suppose we

throw a pair of unbiased six-sided dice

‡

and ask what is the probability of getting a 7. We

partition the outcome space into 36 equally likely outcomes as shown in Table 1.2-1, where

each entry is the sum of the numbers on the two dice.

Table 1.2-1 Outcomes of Throwing

Two Dice

1st die

2nd die 123 4 5 6

1 234 5 6 7

345 6 7 8

456 7 8 9

567 8 910

678 91011

7 8 9 10 11 12

†

A priori means relating to reasoning from self-evident propositions or prior experience. The related

phrase, a posteriori means relating to reasoning from observed facts.

‡

We will always assume that our dice have six sides.

4 Chapter 1 Introduction to Probability

The total number of outcomes is 36 if we keep the dice distinct. The number of ways

of getting a 7 is n

= 6. Hence

P [getting a 7] =

Example 1.2-1

(toss a fair coin twice) The possible outcomes are HH, HT, TH, and TT. The probability

of getting at least one tail T is computed as follows: With E denoting the event of getting

at least one tail, the event E is the set of outcomes

E = {HT, TH, TT}.

Thus, event E occurs whenever the outcome is HT or TH or TT. The number of elements

in E is n

= 3; the number of all outcomes N , is four. Hence

P [at least one T] =

Note that since no physical experimentation is involved, there is no problem in postulating

an ideal “fair coin.” Eﬀectively, in classical probability every experiment is considered

“fair.”

The classical theory suﬀers from at least two signiﬁcant problems: (1) It cannot deal

with outcomes that are not equally likely; and (2) it cannot handle an inﬁnite number

of outcomes, that is when n = ∞. Nevertheless, in those problems where it is impractical

to actually determine the outcome probabilities by experimentation and where, because of

symmetry considerations, one can indeed argue equally likely outcomes, the classical theory

is useful.

Historically, the classical approach was the predecessor of Richard Von Mises’ [1-6]

relative frequency approach developed in the 1930s, which we consider next.

Probability as a Measure of Frequency of Occurrence

The relative frequency approach to deﬁning the probability of an event E is to perform

an experiment n times. The number of times that E appears is denoted by n

.Thenitis

tempting to deﬁne the probability of E occurring by

P [E] = lim

n→∞

. (1.2-1)

Quite clearly since n

≤ n we must have 0 ≤ P [E] ≤ 1. One diﬃculty with this approach

is that we can never perform the experiment an inﬁnite number of times, so we can only

estimate P [E] from a ﬁnite number of trials. Secondly, we postulate that n

/n approaches

a limit as n goes to inﬁnity. But consider ﬂipping a fair coin 1000 times. The likelihood

of getting exactly 500 heads is very small; in fact, if we ﬂipped the coin 10,000 times, the

likelihood of getting exactly 5000 heads is even smaller. As n →∞, the event of observing

Sec. 1.2. THE DIFFERENT KINDS OF PROBABILITY 5

exactly n/2 heads becomes vanishingly small. Yet our intuition demands that P[head] =

for a fair coin. Suppose we choose a δ>0; then we shall ﬁnd experimentally that if the coin

is truly fair, the number of times that



−



>δ, (1.2-2)

as n becomes large, becomes very small. Thus, although it is very unlikely that at any stage

of this experiment, especially when n is large, n

/n is exactly

, this ratio will nevertheless

hover around

, and the number of times it will make signiﬁcant excursion away from the

vicinity of

according to Equation 1.2-2, becomes very small indeed.

Despite these problems with the relative frequency deﬁnition of probability, the relative

frequency concept is essential in applying probability theory to the physical world.

Example 1.2-2

(random.org ) An Internet source of random numbers is RANDOM.ORG, which was founded

by a professor in the School of Computer Science and Statistics at Trinity College, Dublin,

Ireland. It calculates random digits as a function of atmospheric noise and has passed

many statistical tests for true randomness. Using one of the site’s free services, we have

downloaded 10,000 random numbers, each taking on values from 1 to 100 equally likely. We

have written the MATLAB function RelativeFrequencies() that takes this ﬁle of random

numbers and plots the ratio n

/n as a function of the trial number n =1,...,10, 000. We

can choose the event E to be the occurrence of any one of the 100 numbers. For example for

= {occurrence of number 5},thenumbern

counts the number of times 5 has occurred

among the 10,000 numbers up to position n. A resulting output plot is shown in Figure 1.2-

1, where we see a general tendency toward convergence to the ideal value of 0.01 = 1/100

for 100 equally likely numbers. An output plot for another number choice 23 is shown in

Figure 1.2-2 again showing a general tendency to converge to the ideal value here of 0.01. In

both cases though, we note that the convergence is not exact at any value of n, but rather

just convergence to a small neighborhood of the ideal value.

This program is available at this book’s website.

Probability Based on an Axiomatic Theory

The axiomatic approach is followed in most modern textbooks on the subject. To develop it

we must introduce certain ideas, especially those of a random experiment, a sample space,

and an event. Brieﬂy stated, a random experiment is simply an experiment in which the

outcomes are nondeterministic, that is, more than one outcome can occur each time the

experiment is run. Hence the word random in random experiment.Thesample space is the

set of all outcomes of the random experiment. An event is a subset of the sample space that

satisﬁes certain constraints. For example, we want to be able to calculate the probability for

each event. Also in the case of noncountable or continuous sample spaces, there are certain

technical restrictions on what subsets can be called events. An event with only one outcome

will be called a singleton or elementary event. These notions will be made more precise in

Sections 1.4 and 1.5.

Sec. 1.3. MISUSES, MISCALCULATIONS, AND PARADOXES IN PROBABILITY 7

1.3 MISUSES, MISCALCULATIONS, AND PARADOXES IN PROBABILITY

The misuse of probability and statistics in everyday life is quite common. Many of the

misuses are illustrated by the following examples. Consider a defendant in a murder trial

who pleads not guilty to murdering his wife. The defendant has on numerous occasions

beaten his wife. His lawyer argues that, yes, the defendant has beaten his wife but that

among men who do so, the probability that one of them will actually murder his wife is

only 0.001, that is, only one in a thousand. Let us assume that this statement is true. It

is meant to sway the jury by implying that the fact of beating one’s wife is no indicator

of murdering one’s wife. Unfortunately, unless the members of the jury have taken a good

course in probability, they might not be aware that a far more signiﬁcant question is the

following: Given that a battered wife is murdered, what is the probability that the husband is

the murderer ? Statistics show that this probability is, in fact, greater than one-half.

In the 1996 presidential race, Senator Bob Dole’s age became an issue. His opponents

claimed that a 72-year-old white male has a 27 percent risk of dying in the next ﬁve years.

Thus it was argued, were Bob Dole elected, the probability that he would fail to survive his

term was greater than one-in-four. The trouble with this argument is that the probability

of survival, as computed, was not conditioned on additional pertinent facts. As it happens,

if a 72-year-old male is still in the workforce and, additionally, happens to be rich, then

taking these additional facts into consideration, the average 73-year-old (the age at which

Dole would have assumed the presidency) has only a one-in-eight chance of dying in the

next four years [1-3].

Misuse of probability appears frequently in predicting life elsewhere in the universe.

In his book Probability 1 (Harcourt Brace & Company, 1998), Amir Aczel assures us

that we can be certain that alien life forms are out there just waiting to be discovered.

However, in a cogent review of Aczel’s book, John Durant of London’s Imperial College

writes,

Statistics are extremely powerful and important, and Aczel is a very clear and capable

exponent of them. But statistics cannot substitute for empirical knowledge about the

way the universe behaves. We now have no plausible way of arriving at robust estimates

about the way the universe behaves. We now have no plausible way of arriving at

robust estimates for the probability of life arriving spontaneously when the conditions

are right. So, until we either discover extraterrestrial life or understand far more about

how at least one form of life—terrestrial life—ﬁrst appeared, we can do little more

than guess at the likelihood that life exists elsewhere in the universe. And as long as

we’re guessing, we should not dress up our interesting speculations as mathematical

certainties.

The computation of probabilities based on relative frequency can lead to paradoxes. An

excellent example is found in [1-3]. We repeat the example here:

In a sample of American women between the ages of 35 and 50, 4 out of 100 develop

breast cancer within a year. Does Mrs. Smith, a 49-year-old American woman, there-

fore have a 4% chance of getting breast cancer in the next year? There is no answer.

Suppose that in a sample of women between the ages of 45 and 90—a class to which

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