亨利·斯塔克与约翰·伍兹合著：工程师必备的概率统计与随机过程第四版

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"《概率、统计与随机过程工程教程》第四版是由Henry Stark教授和John W. Woods教授合著的一本教材，两位作者分别来自伊利诺伊理工学院和伦斯勒理工学院。该书专为工程师设计，旨在提供深入理解概率论、统计学以及随机过程的基础知识，是工程领域的经典参考资料。在本书中，读者可以探索的核心知识点包括但不限于： 1. 概率基础：章节涵盖概率的基本概念，如样本空间、事件、概率的加法原理和乘法规则，以及条件概率和贝叶斯定理。通过实例解析，帮助读者建立起对随机现象的概率描述能力。 2. 随机变量：介绍离散随机变量和连续随机变量，涵盖了期望值、方差、协方差和随机变量的分布，如二项分布、泊松分布和正态分布等。这部分内容对于理解和处理工程中的不确定性至关重要。 3. 概率分布理论：深入研究了诸如均匀分布、指数分布、伽马分布、泊松过程等重要概率分布及其应用，包括极限定理，如中心极限定理和大数定律。 4. 统计推断：讲解如何利用样本数据推断总体参数，包括假设检验、置信区间估计和参数估计，这对于实验设计和数据分析至关重要。 5. 随机过程：讨论了随机过程的定义、分类（如马尔科夫过程、布朗运动等），以及重要的随机过程性质，如平稳性和自相关性，这些在信号处理、通信工程等领域有广泛应用。 6. 随机信号分析：介绍了傅立叶变换、拉普拉斯变换等工具，用于处理随机信号的频域特性，对于电子工程、通信和控制系统设计具有实际指导意义。 7. 应用示例：书中包含大量实际工程问题的案例分析，如信号噪声分析、通信系统的性能评估、质量控制和风险管理等，使读者能够将理论知识与实际工程场景相结合。《概率、统计与随机过程工程教程》第四版不仅是一本教科书，也是一本实用的参考手册，适合在校工程学生、研究人员和工程师进行深入学习和实践，以便更好地理解和应对工程领域中的不确定性挑战。"

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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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