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title: 機率TP2
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# Theory of Probability：<br>Sets and Probabilistic Models
NTNU 機率論

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###### tags: `NTNU` `CSIE` `必修` `Theory of Probability`

## Set 集合
- A **set** is a collection of objects which are the **elements** of the set.
- if $x$ is an element of $S$, we write $x \in S$
- Otherwise denoted by $x \notin S$
- A Set that has no elements is called empty set is denoted by $\emptyset$
- Set Specification
- Countably finite
- Countably infinite
- With a certain property
- Countably Infinite
- $\{k|k/2 \text{ is integer}\}$
- Uncountable
- $\{ x| 0 \leq x \leq 1\}$
- If every element of a set $S$ is also an element of a set $T$, then $S$ is a **subset** of $T$
- Denoted by $S\subset T$ or $T \supset S$
- If $S\subset T$ and $T \subset S$, then the two sets are **equal**
- Denoted by $S=T$
- The universal set, denoted by $\Omega$, which contains all objects of interest in a particular context
- After specifying the context in terms of universal set $\Omega$, we only consider sets $S$ that are subsets of $\Omega$
## Set Operator
- **Complement 差集**
- The **complement** of a set $S$ with respect to the universe $\Omega$, is the set$\{x\in\Omega|x\notin S\}$, namely, the set of all elements that do not belong to $S$, denoted by $S^c$.
- The complement of the universe $\Omega^c = \emptyset$.
- **Union 聯集**
- The **union** of two sets $S$ and $T$ is the set of all elements that belong to $S$ or $T$, denoted by $S\cup T$
$S\cup T = \{x|x\in S$ or $x\in T\}$
- **Intersection 交集**
- The **intersection** of two sets $S$ and $T$ is the set of all elements that belong to both $S$ and $T$, denoted by $S\cap T$
$S\cap T = \{x|x\in S$ and $x\in T\}$

- The **union** or the **intersection** of several( or even infinite)
- $\bigcup\limits_{n=1}^{\infty}S_n = S_1 \cup S_2 \cup ... = \{x | x \in S_n \text{ for some } n\}$
- $\bigcap\limits_{n=1}^{\infty}S_n = S_1 \cap S_2 \cap ... = \{x | x \in S_n \forall n\}$

- **Disjoint**
- Two sets are **disjoint** if their intersection us empty($S\cap T = \emptyset$)
- **Partition**
- A collection of sets is said to be a **partition** of a set $S$ if the sets in the collection are disjoint and their union is $S$
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- Visualization of set operations with Venn diagrams

## The Algebra of Sets 集合代數
- The following equations are the elementary consequences of the definitions of set($S$ and their operations)
- **Commutative 交換律**
- **Associative 結合律**
- **Distributive 分配律**
- Two particular useful properties are given by **De Morgan’s law 迪摩根定律**
- $(\bigcup{S_n})^c = \bigcap{S_n^c}$
- $(\bigcap{S_n})^c = \bigcup{S_n^c}$
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## Probabilistic Models
- A probabilistic model is a mathematical description of an uncertain situation.
- It has to be in accordance with a fundamental framework to be discussed shortly
- Elements of a probabilistic model
- The **sample space 樣本空間**
- The set of all possible outcomes of an experiment
- The **probability law**
- Assign to a set $A$ of possible outcomes( also called an event) a non-negative number $\textbf{P(A)}$ (called the probability $A$)that encodes our knowledge or belief about the collective "likelihood" of he elements of $A$
- The main ingredients of a probabilistic model
*[Axioms]: 公理

## Sample Spaces and Events
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- Examples of the experiment
- single toss of a coin ( finite outcomes)
- Three tosses of two dices (finite outcomes)
- An infinite sequences of tosses of a coin ( infinite outcomes)
- Throwing a dart on a square( infinite outcomes)
- Properties of the sample space
- Elements of the sample space must be **mutually exclusive**
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## Granularity of the Sample Space

## Sequential Probabilistic Models
## Probability Laws for Discrete Model 離散機率模型
- Discrete Probability Law
- If
- Discrete Uniform Probability Law
- If the sample space consists of $n$ possible outcomes which are equally likely( i.e., all single-element event $A$ is given by probability), then the probability of any 補
## An Example of Sample Space and Probability Law
## Properties of Probability Laws
$P(A_1 \cap A_2 \cap ... \cap A_n) \leq [\sum\limits_{i = 1}^{n}P(A_i)] - n + 1$