1.1 Sets (集合) & 1.2 Probabilistic models (概率模型)

本文为《Introduction to Probability》的读书笔记

SETS 集合

Notation & Concepts

• Universal set, denoted by $\Omega$, which contains all objects that could conceivably be of interest in a particular context . Having specified the context in terms of a universal set $\Omega$, we only consider sets $S$ that are subsets of $\Omega$.
• The complement of a set $S$, is denoted by $S^c$.
• Several sets are said to be disjoint(互不相交) if no two of them have a common element .
• A collection of sets is said to be a partition(分割) of a set $S$ if the sets in the collection are disjoint and their union is $S$.
• De Morgan’s laws (德摩根律)
  
  PROOF
  [Hint: if $x\in (\underset{n}{\cup }{S}_n{)}^c$, then $x\in \underset{n}{\cap }{S}_n^c$]

PROBABILISTIC MODELS 概率模型

A probabilistic model is a mathematical description of an uncertain situation. It must be in accordance with a fundamental framework that we discuss in this section. Its two main ingredients are listed below and are visualized in Fig. 1.2.

Sample space: 样本空间

Sample Spaces and Events 样本空间和事件

• Every probabilistic model involves an underlying process, called the experiment, that will produce exactly one out of several possible outcomes.

It is important to note that in our formulation of a probabilistic model, there is only one experiment. So, three tosses of a coin constitute(组成) a single experiment rather than three experiments.

• The set of all possible outcomes is called the sample space of the experiment, and is denoted by $\Omega$.

Different elements of the sample space should be distinct and mutually exclusive, so that when the experiment is carried out, there is a unique outcome.

• A subset of the sample space, that is, a collection of possible outcomes, is called an event(事件).

Sequential Models 序贯模型

Many experiments have an inherently sequential character; for example, tossing a coin three times. observing the value of a stock on five successive days, or receiving eight successive digits at a communication receiver.

It is then often useful to describe the experiment and the associated sample space by means of a tree-based sequential description(序贯树形图), as in Fig. 1.3.

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die: 骰子

Note that every node of the tree can be identified with an event, namely. the set of all leaves downstream from that node. For example, the node labeled by a 1 can be identified with the event {(1, 1), (1, 2). (1, 3), (1, 4) } that the result of the first roll is 1.

Probability Laws 概率律

The probability law assigns to every event $A$. a number $P(A)$, called the probability of $A$. satisfying the following axioms.

• Inference(推论):

This shows that the probability of the empty event is 0:

Discrete Models 离散模型

离散概率律

Note that we are using here the simpler notation $P(s_i)$ to denote the probability of the event { s i } \{s_i\} { si​} , instead of the more precise P ( { S i } P(\{ S_i\} P({ Si​}.

In the special case where the probabilities $P(s_1)$, … , $P(s_n)$ are all the same (by necessity equal to $1/n$), we obtain the following.

离散均匀概率律 (古典概型)

Continuous Models 连续模型

Probabilistic mo dels with co ntinuous sample spaces differ from their discrete counterparts in that the probabilities of the single-element events may not be sufficient to characterize the probability law. This is illustrated in the following examples, which also indicate how to generalize the uniform probability law to the case of a continuous sample space.

Example 1.4
A wheel of fortune (幸运轮) is continuously calibrated from 0 to 1, so the possible outcomes of an experiment consisting of a single spin are the numbers in the interval $n=[0,1]$. Assuming a fair wheel, it is appropriate to consider all outcomes equally likely, but what is the probability of the event consisting of a single element? It cannot be positive, because then, using the additivi ty axiom, it would follow that events with a sufficiently large number of elements would have probability larger than 1. Therefore, the probability of any event that consists of a single element must be 0.

In this example, it makes sense to assign probability $b-a$ to any subinterval $[a,b]$ of $[0,1]$, and to calculate the probability of a more complicated set by evaluating its “length”.

The legitimacy of using length as a probability law hinges on the fact that the unit interval has an uncountably infinite number of elements.

Properties of Probability Laws

These properties, and other similar ones, can be visualized and verified graphically using Venn diagrams. Note that property $(c)$ can be generalized as follows:

Problem 11 Bonferroni’s inequality (邦费罗尼不等式)
(a) Prove that for any two events $A$ and $B$, we have

(b) Generalize to the case of n events $A_1,A_2,...,A_n$ , by showing that

SOLUTION
We have $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ and $P(A\cup B)\le 1$. which implies part (a). For part (b), we use De Morgan’s law to obtain

我想的是 $1+P(A\cap B)=2P(A\cap B)+P(A\cap B^C)+P(A^C\cap B)+P(A^C\cap B^C)=(P(A\cap B)+P(A\cap B^C))+(P(A\cap B)+P(A^C\cap B))+P(A^C\cap B^C)=P(A)+P(B)+P(A^C\cap B^C)\ge P(A)+P(B)$
$(b)$ 可用科学归纳法

Problem 13. Continuity property of probabilities (概率的连续性)
$(a)$ Let $A_1,A_2,...$. be an infinite sequence of events, which is “monotonically increasing,” meaning that $A_n\subset A_{n+1}$ for every $n$. Let $A={\cup }_{n=1}^{\infty }{A}_n$. Show that $P(A)=li{m}_{n\to \infty }P(A_n)$
$(b)$ Suppose now that the events are “monotonically decreasing,” i.e., $A_{n+1}\subset A_n$ for every $n$. Let $A={\cap }_{n=1}^{\infty }{A}_n$ . Show that $P(A)=li{m}_{n\to +\infty }P(An)$.
$(c)$ Consider a probabilistic model whose sample space is the real line. Show that

SOLUTION
(a)
[Hint: Express the event $A$ as a union of countably many disjoint sets.]
(b)
[Hint: Apply the result of part (a) to the complements of the events.]