概率论与数理统计 2 Probability（概率） (上篇)

概率论与数理统计 2 Probability（概率） (上篇)"/>

概率论与数理统计 2 Probability（概率） (上篇)

概率论_2.1_2.2

• 2.1 Sample Spaces and Events（样本空间及事件）
• • The Sample Space of an Experiment
  • Events
  • Some relations from Set theory（集合论中的一些关系）
• 2.2 Axioms（公理）, interpretations（解释）, and Properties of Probability（概率的性质）
• • Interpreting Probability
  • More Probability Properties
  • Determining Probabilities Systematically
  • Equally Likely Outcomes(同样可能的结果)

2.1 Sample Spaces and Events（样本空间及事件）

An experiment(试验) is any activity or process whose outcome is subject to uncertainty（结果具有不确定性）.

The Sample Space of an Experiment

The sample space of an experiment, denoted by S S S, is the set of all possible outcomes of that experiment.

Events

An event is any collection (subset) of outcomes contained in the sample space S. An event is simple（基本的） if it consists of exactly one outcome and compound（复合的） if it consists of more than one outcome.

When an experiment is performed, a particular event A is said to occur if the resulting experimental outcome is contained in A.

In general, exactly one simple event will occur, but many compound events will occur simultaneously.(基本事件只发生一个，但复合事件可同时发生多个)

Some relations from Set theory（集合论中的一些关系）

An event is just a set, so relationships and results from elementary set theory can be used to study events

1. The complement(补集) of an event A, denoted by A’ , is the set of all outcomes in S that are not contained in A.
2. The union(并集) of two events A and B, denoted by A ∪ \cup ∪ B and read “A or B,” is the event consisting of all outcomes that are either in A or in B or in both events (so that the union includes outcomes for which both A and B occur as well as outcomes for which exactly one occurs)—that is, all outcomes in at least one of the events.
3. The intersection(交集) of two events A and B, denoted by A ∩ \cap ∩ B and read “A and B,” is the event consisting of all outcomes that are in both A and B.

Sometimes A and B have no outcomes in common, so that the intersection of A and B contains no outcomes.

Let ∅ \emptyset ∅ denote the null event (the event consisting of no outcomes whatsoever(任何)).
When A ∩ \cap ∩ B = ∅ \emptyset ∅, A and B are said to be mutually exclusive(互斥的) or disjoint(不相连的) events.

The operations of union and intersection can be extended to more than two events

Given events A 1 A_1 A1​, A 2 A_2 A2​, A 3 A_3 A3​,…, these events are said to be mutually exclusive (or pairwise disjoint,成对不相连) if no two events have any outcomes in common

A pictorial representation of events and manipulations with events（事件和对事件的处理的图形表示） is obtained by using Venn diagrams（维恩图）. To construct a Venn diagram, draw a rectangle（矩形） whose interior will represent the sample space S. Then any event A is represented as the interior of a closed curve（闭合曲线） (often a circle) contained in S.

2.2 Axioms（公理）, interpretations（解释）, and Properties of Probability（概率的性质）

Given an experiment and a sample space S, the objective（目标） of probability is to assign to each event A a number P(A), called the probability of the event A, which will give a precise measure of the chance that A will occur. To ensure that the probability assignments will be consistent with our intuitive notions of probability（直观的概率概念）, all assignments should satisfy the following axioms (basic properties) of probability.

AXIOM 1:

For any event A, P(A) ≥ \ge ≥ 0.

AXIOM 2:

P(S) = 1.

AXIOM 3:

If A 1 A_1 A1​, A 2 A_2 A2​, A 3 A_3 A3​,… is an infinite collection of disjoint events(无数不相连的事件的集合), then

P ( A 1 ∪ A 2 ∪ A 3 ∪ . . . ) = ∑ i = 1 ∞ P ( A i ) P(A_1 \cup A_2 \cup A_3\cup...)= \sum_{i=1}^{\infty} P(A_i) P(A1​∪A2​∪A3​∪...)=i=1∑∞​P(Ai​)

P( ∅ \emptyset ∅) = 0 where ∅ \empty ∅ is the null event (the event containing no outcomes whatsoever). This in turn implies that the property contained in Axiom 3 is valid(有效的) for a finite collection of disjoint events.

Interpreting Probability

Consider an experiment that can be repeatedly performed in an identical and independent fashion（可以以相同和独立的方式重复进行）, and let A be an event consisting of a fixed set of outcomes of the experiment（由一组固定的实验结果组成的事件）. Simple examples of such repeatable experiments include the tacktossing and die-tossing experiments（抛粘和模抛实验） previously discussed. If the experiment is performed n times, on some of the replications the event A will occur (the outcome will be in the set A), and on others, A will not occur. Let n(A) denote the number of replications on which A does occur. Then the ratio n(A)/n is called the relative frequency（相对频率） of occurrence of the event A in the sequence of n replications.

More generally, empirical evidence（经验证据）, based on the results of many such repeatable experiments, indicates that any relative frequency of this sort will stabilize（趋于稳定） as the number of replications n increases. That is, as n gets arbitrarily large, n(A)/n approaches a limiting value referred to as the limiting (or long-run，长期) relative frequency（极限相对频率） of the event A. The objective interpretation of probability（对概率的客观解释） identifies this limiting relative frequency with P(A)

This relative frequency interpretation of probability（对概率的相对频率解释） is said to be objective because it rests on a property of the experiment（取决于实验的性质） rather than on any particular individual concerned with the experiment（与实验有关的任何特定个体）.

Because the objective interpretation of probability is based on the notion of limiting frequency（以限制频率的概念为基础的）, its applicability is limited to experimental situations that are repeatable（适用性仅限于可重复的实验情况）. Yet the language of probability is often used in connection with situations that are inherently unrepeatable（本质上不可重复）. Examples include: “The chances are good for a peace agreement”

More Probability Properties

For any event A, P(A) + P(A’) = 1, from which P(A) = 1 - P(A’).

When you are having difficulty calculating P(A) directly, think of determining P(A’).

For any event A, P(A) ≤ \leq ≤ 1.

For any two events A and B,
P(A ∪ \cup ∪ B) = P(A) + P(B) - P(A ∩ \cap ∩ B)

The addition rule for a triple union probability is similar to the foregoing(前面的) rule.

For any three events A, B, and C,

P ( A ∪ B ∪ C ) = P ( A ) + P ( B ) + P ( C ) − P ( A ∩ B ) − P ( A ∩ C ) − P ( B ∩ C ) + P ( A ∩ B ∩ C ) P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C)- P(B \cap C) + P(A \cap B \cap C) P(A∪B∪C)=P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C)

Determining Probabilities Systematically

Consider a sample space that is either finite or “countably infinite(可数无限的)” (the latter means that outcomes can be listed in an infinite sequence(结果可以以无限序列列出), so there is a first outcome, a second outcome, a third outcome, and so on.

Let E 1 E_1 E1​, E 2 E_2 E2​, E 3 E_3 E3​,… denote the corresponding simple events, each consisting of a single outcome. A sensible strategy for probability computation(一种合理的概率计算策略) is to first determine each simple event probability, with the requirement that ∑ P ( E i ) = 1 \sum_{}^{} P(E_i) = 1 ∑​P(Ei​)=1

Then the probability of any compound event A is computed by adding together the P( E i E_i Ei​)’s for all E i E_i Ei​’s in A:

P ( A ) = ∑ a l l E i ′ s i n A P ( E i ) P(A) = \sum_{all \hspace{1mm} E_i's \hspace{1mm} in \hspace{1mm} A}^{} P(E_i) P(A)=allEi′​sinA∑​P(Ei​)

Equally Likely Outcomes(同样可能的结果)

In many experiments consisting of N outcomes, it is reasonable to assign equal probabilities to all N simple events.

1 = ∑ i = 1 N P ( E i ) = ∑ i = 1 N p = p ⋅ N s o p = 1 N 1 = \sum_{i=1}^{N} P(E_i) = \sum_{i=1}^{N} p= p \cdot N\hspace{1cm}so\hspace{1mm}p=\frac{1}{N} 1=i=1∑N​P(Ei​)=i=1∑N​p=p⋅Nsop=N1​

That is, if there are N equally likely outcomes, the probability for each is 1/N.

Now consider an event A, with N(A) denoting the number of outcomes contained in A. Then

P ( A ) = ∑ E i i n A P ( E i ) = ∑ E i i n A 1 N = N ( A ) N P(A)= \sum_{E_i \hspace{1mm} in \hspace{1mm} A}^{} P(E_i) =\sum_{E_i \hspace{1mm} in \hspace{1mm} A}^{} \frac{1}{N}=\frac{N(A)}{N} P(A)=Ei​inA∑​P(Ei​)=Ei​inA∑​N1​=NN(A)​

Thus when outcomes are equally likely, computing probabilities reduces to counting(计算概率就简化为计数): determine both the number of outcomes N(A) in A and the number of outcomes N in S, and form their ratio.