概率论与随机过程在电气工程中的应用

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"《概率与随机过程》是一本专为电气工程课程中的本科高年级或研究生低年级学生设计的概率论和随机过程教材。本书强调数学作为工具的实际应用，而非仅仅欣赏其理论美感。书中并未涉及代数、波雷尔域或测度论等深奥的数学理论，而是集中于提供解决实际问题所需的技能和工具。" 本书的核心知识点包括： 1. **概率论基础**：概率论是理解随机现象的基础。书中会涵盖基本的概率定义，如概率空间、事件的概率、条件概率、独立事件、乘法定理以及贝叶斯定理。这些概念是工程师分析不确定性系统时的必备知识。 2. **随机变量**：随机变量是概率论中的关键元素，它将概率分布与实值变量联系起来。书中会讨论离散随机变量（如伯努利随机变量、二项分布、泊松分布）和连续随机变量（如均匀分布、正态分布、指数分布），以及它们的期望值、方差和其他统计特性。 3. **概率分布函数与密度函数**：理解分布函数和密度函数对于计算随机变量的性质至关重要。书中将详细介绍如何使用分布函数和密度函数来描述随机变量的行为，并进行概率计算。 4. **大数定律与中心极限定理**：这两个定理是概率论中最重要的结果，它们提供了在大量独立随机变量下平均行为的稳定性。大数定律解释了随着试验次数增加，样本均值趋于期望值的现象，而中心极限定理则说明了独立同分布随机变量的和接近正态分布。 5. **随机过程**：随机过程是时间序列中随机变量的集合，常用于描述动态系统中的随机变化。书中可能涵盖常见随机过程，如泊松过程、布朗运动、马尔可夫链和高斯过程，以及它们在信号处理、通信工程和控制理论中的应用。 6. **统计推断与估计理论**：这部分内容可能涉及参数估计、置信区间、假设检验等，这些都是基于观测数据对模型参数进行推断的关键技术。 7. **随机过程的平稳性和 Ergodic性**：理解随机过程的平稳性和Ergodic性对于分析长期行为和平均性质至关重要。这些特性在系统建模和预测中起到关键作用。 8. **滤波理论**：在随机过程中，滤波理论涉及如何从噪声中提取有用信号，如卡尔曼滤波器，这是现代控制系统和信号处理中的重要工具。 9. **应用实例**：书中可能包含大量的工程应用示例，如通信系统的误码率分析、随机信号的谱分析、随机振动的模拟和随机网络性能评估等，以帮助学生将理论知识应用于实践。《概率与随机过程》旨在通过提供实用的数学工具和技巧，使学生能够有效地解决工程中的随机性问题，而不仅仅是理论上的探讨。书中力求在理论深度和易读性之间找到平衡，确保学生既能欣赏到数学的美学，又能掌握解决实际问题的能力。

2.2 Axioms of Probability 13

AXIOM 2.3b: For an inﬁnite number of mutually exclusive sets, A

, i =

1, 2, 3, ... , A

∩A

= Ø for all i = j,



∞



i=1



∞



i=1

Pr(A

). (2.2)

It should be noted that Axiom 2.3a and Corollary 2.1 could be viewed as special

cases of Axiom 2.3b. So, a more concise development could be obtained by starting

with Axioms 2.1, 2.2, and 2.3b. This may be more pleasing to some, but we believe

the approach given here is a little easier to follow for the student learning the

material for the ﬁrst time.

The preceding axioms do not tell us directly how to deal with the probability of

the union of two sets that are not mutually exclusive. This can be determined from

these axioms as follows.

THEOREM 2.1: For any sets A and B (not necessarily mutually exclusive),

Pr(A ∪ B) = Pr(A) +Pr(B) − Pr(A ∩ B). (2.3)

PROOF: We give a visual proof of this important result using the Venn diagram

shown in Figure 2.1. To aid the student in the type of reasoning needed to complete

proofs of this type, it is helpful to think of a pile of sand lying in the sample space

shown in Figure 2.1. The probability of the event A would then be analogous to

the mass of that subset of the sand pile that is above the region A and likewise for

the probability of the event B. For the union of the two events, if we simply added

the mass of the sand above A to the mass of the sand above B, we would double

count that region that is common to both sets. Hence, it is necessary to subtract the

A ∩ B

Figure 2.1 Venn diagram for proof of Theorem 2.1.

14 Chapter 2 Introduction to Probability Theory

probability of A ∩B. We freely admit that this proof is not rigorous. It is possible to

prove Theorem 2.1 without having to call on our sand analogy or even the use of

Venn diagrams. The logic of the proof will closely follow what we have done here.

The reader is led through that proof in Exercise 2.2. 

Many other fundamental results can also be obtained from the basic axioms of

probability. A few simple ones are presented here. More will be developed later

in this chapter and in subsequent chapters. As with Theorem 2.1, it might help the

student to visualize these proofs by drawing a Venn diagram.

THEOREM 2.2: Pr(

A) = 1 − Pr(A).

PROOF:

1 = Pr(S) = Pr(A ∪

A) (by Axiom 2.2)

= Pr(A) + Pr(

A) (by Axiom 2.3a)

∴ Pr(

A) = 1 − Pr(A).



THEOREM 2.3: If A ⊂ B, then Pr(A) ≤Pr(B).

PROOF: See Exercise 2.4. 

2.3 Assigning Probabilities

In the previous section, probability was deﬁned as a measure of the likelihood

of an event or events that satisfy the three Axioms 2.1–2.3. How probabilities are

assigned to particular events was not speciﬁed. Mathematically, any assignment

that satisﬁes the given axioms is acceptable. Practically speaking, we would like to

assign probabilities to events in such a way that the probability assignment actually

represents the likelihood of occurrence of that event. Two techniques are typically

used for this purpose and are described in the following paragraphs.

In many experiments, it is possible to specify all of the outcomes of the exper-

iment in terms of some fundamental outcomes, which we refer to as atomic

outcomes. These are the most basic events that cannot be decomposed into simpler

events. From these atomic outcomes, we can build more complicated and more

interesting events. Quite often we can justify assigning equal probabilities to all

atomic outcomes in an experiment. In that case, if there are M mutually exclusive

exhaustive atomic events, then each one should be assigned a probability of 1/M.

Not only does this make perfect common sense, it also satisﬁes the mathematical

16 Chapter 2 Introduction to Probability Theory

EXAMPLE 2.8: In Example 2.3, a pair of dice were rolled. In this exper-

iment, the most basic outcomes are the 36 different combinations of

the six atomic outcomes of the previous example. Again, each of these

atomic outcomes is assigned a probability of 1/36. Next, suppose we

want to ﬁnd the probability of the event A ={sum of two dice =5}. Then,

Pr(A) = Pr((1,4) ∪ (2, 3) ∪(3,2) ∪ (4, 1))

= Pr(1, 4) + Pr(2, 3) +Pr(3, 2) +Pr(4, 1) (by Corollary 2.1)

= 1/36 + 1/36 + 1/36 + 1/36 (by probability assignment)

= 1/9.

EXAMPLE 2.9: In this example we will use the MATLAB com-

mand rand to simulate the ﬂipping of coins and the rolling of

dice. The command rand(m,n) creates a matrix of m rows and n

columns, where each element of the matrix is a randomly selected

number equally likely to fall anywhere in the interval (0,1). By rounding this

number to the nearest integer, we can create a randomly selected number

equally likely to be 0 or 1. This can be used to simulate the ﬂipping of a coin

if we interpret 0 as “tails” and 1 as “heads” or vice versa. Similarly, if we

multiply rand(1) by 6 and round up to the nearest integer, we will get one

of the numbers 1, 2, ... , 6 with equal probability. This can be used to simulate

the rolling of a die. Try running the following script in MATLAB.

% Simulation of coin flipping and die tossing.

coin_flip=round(rand(1)) % Simulate flip of a coin.

die_toss=ceil(6*rand(1)) % Simulate toss of one die.

dice_toss=ceil(6*rand(1,2)) % Simulate toss of two dice.

You should ﬁnd that each time you run this script, you get different (random)

looking results. With any MATLAB command, if you want more information

on what the command does, type help followed by the command name at the

MATLAB prompt for detailed information on that command. For example,

to get help on the rand command, type help rand.

Care must be taken when using the classical approach to assigning probabil-

ities. If we deﬁne the set of atomic outcomes incorrectly, unsatisfactory results

2.3 Assigning Probabilities 17

may occur. In Example 2.8, we may be tempted to deﬁne the set of atomic outcomes

as the different sums that can occur on the two die faces. If we assign equally likely

probability to each of these outcomes, then we arrive at the assignment

Pr(sum = 2) = Pr(sum = 3) =···=Pr(sum = 12) = 1/11. (2.5)

Anyone with experience in games involving dice knows that the likelihood of

rollinga2ismuch lower than the likelihood of rolling a 7. The problem here is that

the atomic events we have assigned are not the most basic outcomes and can be

decomposed into simpler outcomes, as demonstrated in Example 2.8.

This is not the only problem encountered in the classical approach. Suppose

we consider an experiment that consists of measuring the height of an arbitrarily

chosen student in your class and rounding that measurement to the nearest inch.

The atomic outcomes of this experiment would consist of all the heights of the

students in your class. However, it would not be reasonable to assign an equal

probability to each height. Those heights corresponding to very tall or very short

students would be expected to be less probable than those heights corresponding

to a medium height. So, how then do we assign probabilities to these events? The

problems associated with the classical approach to assigning probabilities can be

overcome by using the relative frequency approach.

The relative frequency approach requires that the experiment we are concerned

with be repeatable, in which case, the probability of an event, A, can be assigned by

repeating the experiment a large number of times and observing how many times

the event A actually occurs. If we let n be the number of times the experiment is

repeatedand n

bethe number oftimes the event A is observed, then the probability

of the event A can be assigned according to

Pr(A) = lim

n→∞

. (2.6)

This approach to assigning probability is based on experimental results and thus

has a more practical ﬂavor to it. It is left as an exercise for the reader (seeExercise 2.6)

to conﬁrm that this method does indeed satisfy the axioms of probability and is

thereby mathematically correct as well.

EXAMPLE 2.10: Consider the dice rolling experiment of Exam-

ples 2.3 and 2.8. We will use the relative frequency approach to

assign the probability of the event, A = {sum of two dice = 5}. We

simulated the tossing of two dice using the following MATLAB

code. The results of this dice tossing simulation are shown in Table 2.1.

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