[Random Process] 1. 집합론, 확률론

랜덤프로세스라는 과목은 확률론에 기반하며, 확률론은 집합론으로부터 시작된다. 

그럼 집합론의 정의부터 시작하여 확률론까지의 기본 정의에 대해 이야기해보자.

1. Overview of probability

- Probability theory is a branch of mathematics that deal with uncertainty.

- A random experiment is an experiment for which the outcome cannot be predicted with certainty.

- Probability = chance or likelihood that some outcomes will happen.

2. 집합론(Set Theory)

* Definitions

- A set is a collection of well defined objects, called elements

- If A is a set and x is an element of A, we say "x belongs to A"

- In a random experiment, one of several possible results will occur each time. These possible results are called outcomes.

- The collection of all possible outcomes is called the sample space, Denote it by S

- A set of some well defined outcomes is called an event.

- Any event in a subset of the sample space.

* We talk about the probability of the event.

Definition

- countable set : A set containing either finite elts or the same number of elts as natural number.

ㅁBasic operation : on elts :: subset/ superset  A ⊂ B, if x ∈ A -> x ∈ B.

- equality : A = B if they have exactly the same elts.

ㅁ A=b iff A ⊃ B & A ⊂ B

- union : forming a larger collection, A∪B consists of all elts which are in A or in B or in both A and B, We say " either A or B"

ㅁ A∪B = { x | x ∈ A or x ∈ B }

- intersection : collecting the common elts. A∩B consists of all elts which common to both A and B, We say "both A and B"

ㅁ A∩B = { x | x ∈ A and x ∈ B }

- complement : opposite event. 

ㅁ A^c  = { x | x ∈/ A}

- difference : A\B or A-B consists of those elts which are in A but not in B, We say A  but not B".

- disjoint(mutually exclusive) : We say  "A and B disjoint if A ∩ B = ∮

- pairwise disjoint(mutually exclusive) : The events A1, A2, ... An are pairwise disjoint if Ai∩Aj = ∮

Theorem

 1. commutavity ( 교환법칙)

 2. associativity (결합법칙)

 3. distributivity (분배법칙)

 4. de Morgan's Laws

ㅁ Axiomatic foundation of prob. theory

- For each event A in the sample space S, the probability is a function which associates A with a number between zero and one. P(A) ∈ [0, 1]

- Basic principles of prob. 

1) Prob. is a measure of strength like length, area, volume, taking a value between 0 and 1. The impossible should have prob. 0.

2) The prob. should be additive whenever two events are disjoint.

3) In general, consider a good collection B of events which is large enough to contain all useful events including ∮ and S, and is closed under all possible countable set operations. This collection is called a "sigma algebra". Prob. is a set function defined only on this collection. 

** 집합론과 확률론에 관한 간단한 정의들.