概率论与统计学：不确定性的科学——塔巴克第二版

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"《概率与统计：不确定性的科学》第二版，由迈克尔·J·埃文斯和杰弗里·S·罗森塔尔合著，这是一本大学教科书，英文版，主要涵盖了概率论和统计学的基本概念和理论。" 在概率论和统计学这一科学领域中，不确定性是核心概念。这本书的第一部分，"Probability Models"（概率模型），深入探讨了如何用数学方式来量化和理解不确定性。作者首先解释了为何我们需要概率理论，这是因为现实生活中充满了不确定性，我们需要一种工具来预测未来可能发生的事件及其可能性。 1.1.1章节 "Probability: A Measure of Uncertainty" 揭示了概率作为衡量不确定性的工具，它是通过概率值来描述事件发生的可能性。概率的取值范围在0到1之间，其中0表示不可能发生，1表示必然发生。 1.2章节 "Probability Models" 引入了概率模型的概念，这些模型是基于特定规则建立的概率系统，用于模拟真实世界的随机现象。1.2.1部分 "Venn Diagrams and Subsets" 使用维恩图和子集的概念帮助理解概率模型的构建。 1.3章节 "Properties of Probability Models" 讨论了概率模型的基本性质，如概率的非负性、概率总和为1以及特定事件的概率等。 1.4章节 "Uniform Probability on Finite Spaces" 讲述了在有限空间上均匀分布的概率，这是最简单的概率模型之一，1.4.1部分 "Combinatorial Principles" 进一步介绍了组合原则，如排列组合等，它们在计算概率时非常关键。 1.5章节 "Conditional Probability and Independence" 阐述了条件概率和事件独立性的概念。1.5.1 "Conditional Probability" 是在已知某些信息的情况下，对事件发生的概率进行修正。1.5.2 "Independence of Events" 则解释了两个事件相互独立的含义，即一个事件的发生不会影响另一个事件的概率。 1.6章节 "Continuity of P" 讨论了概率的连续性，这对于理解连续概率分布至关重要。 1.7章节 "Further Proofs (Advanced)" 提供了更高级的证明方法，适合对概率理论有深入研究的读者。第二部分 "Random Variables and Distributions"（随机变量与分布）深入到了统计学的核心。2.1章节 "Random Variables" 介绍了随机变量的概念，它是用来描述随机试验结果的数学对象。2.2章节 "Distributions of Random Variables" 讨论了随机变量的分布，即随机变量可能取值的概率分布。 2.3章节 "Discrete Distributions" 关注离散分布，2.3.1部分 "Important Discrete Distributions" 探讨了一些常见的离散分布，如二项分布、泊松分布和几何分布等。2.4章节 "Continuous Distributions" 转向连续分布，2.4.1部分 "Important Absolutely Continuous Distributions" 强调了均匀分布、正态分布等重要的连续分布。2.5章节 "Cumulative Distribution Functions" 解释了累积分布函数（CDF），它给出了随机变量小于或等于某一值的概率，并且其特性在后续的章节中得到了讨论。这些内容构成了概率论与统计学的基础，对于理解和应用这些理论解决实际问题具有重要意义。无论是工程、科学、金融还是其他领域，理解和掌握这些概念都是至关重要的。

Chapter 1: Probability Models 3

that week, leading to a larger probability that you will have to share the jackpot with

other winners even if you do win — so it is probably not in your favor to buy a lottery

ticket even then.

Suppose instead that a “friend” offers you a bet. He has three cards, one red on

both sides, one black on both sides, and one red on one side and black on the other.

He mixes the three cards in a hat, picks one at random, and places it ﬂat on the table

with only one side showing. Suppose that one side is red. He then offers to bet his $4

against your $3 that the other side of the card is also red.

At ﬁrst you might think it sounds like the probability that the other side is also red is

50%; thus a good bet. However, after mastering conditional probability (Section 1.5),

you will know that, conditional on one side being red, the conditional probability that

the other side is also red is equal to 2/3. So, by the theory of expected values (Chap-

ter 3), you will know that you should not accept your “friend’s” bet.

Finally, suppose your “friend” suggests that you ﬂip a coin one thousand times.

Your “friend” says that if the coin comes up heads at least six hundred times, then he

will pay you $100; otherwise, you have to pay him just $1.

At ﬁrst you might think that, while 500 heads is the most likely, there is still a

reasonable chance that 600 heads will appear — at least good enough to justify accept-

ing your friend’s $100 to $1 bet. However, after mastering the laws of large numbers

(Chapter 4), you will know that as the number of coin ﬂipsgetslarge,itbecomesmore

and more likely that the number of heads is very close to half of the total number of

coin ﬂips. In fact, in this case, there is less than one chance in ten billion of getting

more than 600 heads! Therefore, you should not accept this bet, either.

As these examples show, a good understanding of probability theory will allow you

to correctly assess probabilities in everyday situations, which will in turn allow you to

make wiser decisions. It might even save you money!

Probability theory also plays a key role in many important applications of science

and technology. For example, the design of a nuclear reactor must be such that the

escape of radioactivity into the environment is an extremely rare event. Of course, we

would like to say that it is categorically impossible for this to ever happen, but reac-

tors are complicated systems, built up from many interconnected subsystems, each of

which we know will fail to function properly at some time. Furthermore, we can never

deﬁnitely say that a natural event like an earthquake cannot occur that would damage

the reactor sufﬁciently to allow an emission. The best we can do is try to quantify our

uncertainty concerning the failures of reactor components or the occurrence of natural

events that would lead to such an event. This is where probability enters the picture.

Using probability as a tool to deal with the uncertainties, the reactor can be designed to

ensure that an unacceptable emission has an extremely small probability — say, once

in a billion years — of occurring.

The gambling and nuclear reactor examples deal essentially with the concept of

risk — the risk of losing money, the risk of being exposed to an injurious level of

radioactivity, etc. In fact, we are exposed to risk all the time. When we ride in a car,

or take an airplane ﬂight,orevenwalkdownthestreet,weareexposedtorisk. We

know that the risk of injury in such circumstances is never zero, yet we still engage in

these activities. This is because we intuitively realize that the probability of an accident

occurring is extremely low.

4 Section 1.2: Probability Models

So we are using probability every day in our lives to assess risk. As the problems

we face, individually or collectively, become more complicated, we need to reﬁne and

develop our rough, intuitive ideas about probability to form a clear and precise ap-

proach. This is why probability theory has been developed as a subject. In fact, the

insurance industry has been developed to help us cope with risk. Probability is the

tool used to determine what you pay to reduce your risk or to compensate you or your

family in case of a personal injury.

Summary of Section 1.1

• Probability theory provides us with a precise understanding of uncertainty.

• This understanding can help us make predictions, make better decisions, assess

risk, and even make money.

DISCUSSION TOPICS

1.1.1 Do you think that tomorrow’s weather and next week’s stock prices are “really”

random, or is this just a convenient way to discuss and analyze them?

1.1.2 Do you think it is possible for probabilities to depend on who is observing them,

or at what time?

1.1.3 Do you ﬁnd it surprising that probability theory was not discussed as a mathe-

matical subject until the seventeenth century? Why or why not?

1.1.4 In what ways is probability important for such subjects as physics, computer

science, and ﬁnance? Explain.

1.1.5 What are examples from your own life where thinking about probabilities did

save — or could have saved — you money or helped you to make a better decision?

(Listasmanyasyoucan.)

1.1.6 Probabilities are often depicted in popular movies and television programs. List

as many examples as you can. Do you think the probabilities were portrayed there in a

“reasonable” way?

1.2 Probability Models

Aformaldeﬁnition of probability begins with a sample space, often written S. This

sample space is any set that lists all possible outcomes (or, responses) of some unknown

experiment or situation. For example, perhaps

S ={rain, snow, clear}

when predicting tomorrow’s weather. Or perhaps S is the set of all positive real num-

bers, when predicting next week’s stock price. The point is, S can be any set at all,

even an inﬁnite set. We usually write s for an element of S, so that s ∈ S. Note that S

describes only those things that we are interested in; if we are studying weather, then

rain and snow are in S, but tomorrow’s stock prices are not.

Chapter 1: Probability Models 5

A probability model also requires a collection of events, which are subsets of S

to which probabilities can be assigned. For the above weather example, the subsets

{rain}, {snow}, {rain, snow}, {rain, clear}, {rain, snow, clear}, and even the empty set

∅=

{}

, are all examples of subsets of S that could be events. Note that here the comma

means “or”; thus, {rain, snow} is the event that it will rain or snow. We will generally

assume that all subsets of S are events. (In fact, in complicated situations there are

some technical restrictions on what subsets can or cannot be events, according to the

mathematical subject of measure theory. But we will not concern ourselves with such

technicalities here.)

Finally, and most importantly, a probability model requires a probability measure,

usually written P. This probability measure must assign, to each event A, a probability

P(A). We require the following properties:

1. P(A) is always a nonnegative real number, between 0 and 1 inclusive.

2. P(∅) = 0, i.e., if A is the empty set ∅, then P(A) = 0.

3. P(S) = 1, i.e., if A is the entire sample space S, then P(A) = 1.

4. P is (countably) additive, meaning that if A

, A

,... is a ﬁnite or countable

sequence of disjoint events, then

P(A

∪ A

∪···) = P(A

) + P( A

) +···. (1.2.1)

The ﬁrst of these properties says that we shall measure all probabilities on a scale

from 0 to 1, where 0 means impossible and 1 (or 100%) means certain. The second

property says the probability that nothing happens is 0; in other words, it is impossible

that no outcome will occur. The third property says the probability that something

happens is 1; in other words, it is certain that some outcome must occur.

The fourth property is the most subtle. It says that we can calculate probabilities

of complicated events by adding up the probabilities of smaller events, provided those

smaller events are disjoint and together contain the entire complicated event. Note that

events are disjoint if they contain no outcomes in common. For example, {rain} and

{snow, clear} are disjoint, whereas {rain} and {rain, clear} are not disjoint. (We are

assuming for simplicity that it cannot both rain and snow tomorrow.) Thus, we should

have P({rain}) + P({snow, clear}) = P({rain, snow, clear}),butdonot expect to

have P({rain}) + P({rain, clear}) = P({rain, rain, clear}) (the latter being the same

as P({rain, clear})).

We now formalize the deﬁnition of a probability model.

Deﬁnition 1.2.1 A probability model consists of a nonempty set called the sample

space S; a collection of events that are subsets of S; and a probability measure P

assigning a probability between 0 and 1 to each event, with P(∅) = 0andP(S) = 1

and with P additive as in (1.2.1).

6 Section 1.2: Probability Models

EXAMPLE 1.2.1

Consider again the weather example, with S ={rain, snow, clear}. Suppose that the

probability of rain is 40%, the probability of snow is 15%, and the probability of a

clear day is 45%. We can express this as P({rain}) = 0.40, P({snow}) = 0.15, and

P({clear}) = 0.45.

For this example, of course P(∅) = 0, i.e., it is impossible that nothing will happen

tomorrow. Also P({rain, snow, clear}) = 1, because we are assuming that exactly

one of rain, snow, or clear must occur tomorrow. (To be more realistic, we might say

that we are predicting the weather at exactly 11:00 A.M. tomorrow.) Now, what is the

probability that it will rain or snow tomorrow? Well, by the additivity property, we see

that

P({rain, snow}) = P({rain}) + P({snow}) = 0.40 + 0.15 = 0.55.

We thus conclude that, as expected, there is a 55% chance of rain or snow tomorrow.

EXAMPLE 1.2.2

Suppose your candidate has a 60% chance of winning an election in progress. Then

S ={win, lose},withP(win) = 0.6andP(lose) = 0.4. Note that P(win)+P(lose) =

EXAMPLE 1.2.3

Suppose we ﬂip a fair coin, which can come up either heads (H)ortails(T ) with equal

probability. Then S ={H, T }, with P(H) = P(T ) = 0.5. Of course, P(H)+P(T ) =

EXAMPLE 1.2.4

Suppose we ﬂip three fair coins in a row and keep track of the sequence of heads and

tails that result. Then

S ={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

Furthermore, each of these eight outcomes is equally likely. Thus, P(HHH) = 1/8,

P(TTT) = 1/8, etc. Also, the probability that the ﬁrst coin is heads and the second

coin is tails, but the third coin can be anything, is equal to the sum of the probabilities

of the events HTH and HTT, i.e., P(HTH) + P(HTT) = 1/8 + 1/8 = 1/4.

EXAMPLE 1.2.5

Suppose we ﬂip three fair coins in a row but care only about the number of heads

that result. Then S ={0, 1, 2, 3}. However, the probabilities of these four outcomes

are not all equally likely; we will see later that in fact P(0) = P(3) = 1/8, while

P(1) = P(2) = 3/8.

We note that it is possible to deﬁne probability models on more complicated (e.g.,

uncountably inﬁnite) sample spaces as well.

EXAMPLE 1.2.6

Suppose that S = [0, 1] is the unit interval. We can deﬁne a probability measure P on

S by saying that

P([a, b]) = b −a , whenever 0 ≤ a ≤ b ≤ 1. (1.2.2)

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