钟开莱英文版《初等概率论》：基础理论与应用

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《初等概率论》是钟开莱所著的一本英文版教材，第四版内容涵盖了概率论的基础概念和重要理论。该书在教学中扮演着关键角色，适合那些对数学金融有初步了解的学生和专业人士深入学习。以下是对书中部分内容的详细解析： 1. **样本集与集合操作** (Chapter 1): 介绍了样本空间（样本集）的概念，即所有可能的结果构成的基本单元，以及如何通过并、交、差等基本集合运算来处理这些元素。这为理解概率论中的事件和概率分布奠定了基础。 2. **概率定义与示例** (Chapter 2): 通过对实例的分析，定义了概率的基本概念，包括古典概型和几何概型。通过概率的公理，如概率测度的非负性和完备性，解释了概率的合理性。独立事件的概念在这里也得到了深入讨论，这对于理解和计算多个事件同时发生的概率至关重要。 3. **计数法则与组合模型** (Chapter 3): 探讨了计数原理，如乘法原理（加法原理）和排列组合的计算，这在随机变量的抽样和实验设计中非常实用。还介绍了二项式系数，它们在处理有限样本的随机试验时起到核心作用。 4. **随机变量的介绍** (Chapter 4): 随机变量是概率论的核心概念，它代表了随机试验结果的数值表示。章节中讲解了随机变量的定义，如何由实验过程产生，以及不同类型随机变量（离散型和连续型）的分布和期望值的计算。对于实数值随机变量和具有密度函数的随机变量，作者提供了详细的理论和实例。 5. **条件概率和独立性** (Chapter 5): 这一章深入探讨了概率的条件性质，如何通过已知信息更新概率，以及如何判断两个事件是否独立。著名的波利亚抽彩问题（Polya urn scheme）在这里作为实际应用的示例引入，帮助读者理解条件概率的实际意义。这本书不仅适合初级学习者系统学习概率论，还为深入研究数学金融、统计学、数据科学等领域的人士提供了扎实的理论基础。通过阅读和练习书中的例子和习题，读者能够逐步掌握概率理论的核心思想和计算技巧。

6 Set

Figure 3

Several remarks are in order. First, the analogy with arithmetic carries over

to (D

(a + b) × c =(a × c)+(b × c);

but breaks down in (D

(a × b)+c =(a + c) × (b + c).

Of course, the alert reader will have observed that the analogy breaks down

already at an earlier stage, for

A = A ∪ A = A ∩ A;

but the only number a satisfying the relation a + a = a is 0; while there

are exactly two numbers satisfying a × a = a, namely 0 and 1.

Second, you have probably already discovered the use of diagrams to

prove or disprove assertions about sets. It is also a good practice to see the

truth of such formulas as (D

)and(D

) by well-chosen examples. Suppose

then that

A =inexpensivethings,B= really good things,

C = food [edible things].

Then (A∪B)∩C means “(inexpensive or really good) food,” while (A∩C)∪

(B ∩C) means “(inexpensive food) or (really good food).” So they are the

same thing all right. This does not amount to a proof, as one swallow does

not make a summer, but if one is convinced that whatever logical structure

or thinking process involved above in no way depends on the precise nature

of the three things A, B,andC, so much so that they can be anything,

then one has in fact landed a general proof. Now it is interesting that the

same example applied to (D

) somehow does not make it equally obvious

1.3 Various relations 7

(at least to the author). Why? Perhaps because some patterns of logic are

in more common use in our everyday experience than others.

This last remark becomes more signiﬁcant if one notices an obvious

duality between the two distributive laws. Each can be obtained from the

other by switching the two symbols ∪ and ∩. Indeed each can be deduced

from the other by making use of this duality (Exercise 11).

Finally, since (D

) comes less naturally to the intuitive mind, we will

avail ourselves of this opportunity to demonstrate the roundabout method

of identifying sets mentioned above by giving a rigorous proof of the for-

mula. According to this method, we must show: (i) each point on the left

side of (D

) belongs to the right side; (ii) each point on the right side of

) belongs to the left side.

(i) Suppose ω belongs to the left side of (D

), then it belongs either

to A ∩ B or to C.Ifω ∈ A ∩ B,thenω ∈ A, hence ω ∈ A ∪ C;

similarly ω ∈ B ∪ C. Therefore ω belongs to the right side of (D

On the other hand, if ω ∈ C,thenω ∈ A ∪ C and ω ∈ B ∪ C and

we ﬁnish as before.

(ii) Suppose ω belongs to the right side of (D

), then ω may or may

not belong to C, and the trick is to consider these two alternatives.

If ω ∈ C, then it certainly belongs to the left side of (D

). On the

other hand, if ω/∈ C, then since it belongs to A ∪C,itmustbelong

to A; similarly it must belong to B. Hence it belongs to A ∩B,and

so to the left side of (D

). Q.E.D.

1.3. Various relations

The three operations so far deﬁned: complement, union, and intersection

obey two more laws called De Morgan’s laws:

(A ∪ B)

= A

∩ B

;(C

)

(A ∩ B)

= A

∪ B

. (C

)

They are dual in the same sense as (D

)and(D

) are. Let us check these

by our previous example. If A = inexpensive, and B = really good, then

clearly (A ∪ B)

= not inexpensive nor really good, namely high-priced

junk, which is the same as A

∩ B

= inexpensive and not really good.

Similarly we can check (C

Logically, we can deduce either (C

)or(C

) from the other; let us show

it one way. Suppose then (C

) is true, then since A and B are arbitrary

sets we can substitute their complements and get

∪ B

)

=(A

)

∩ (B

)

= A ∩ B (1.3.1)

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CapDan

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