[Random Process] 확률론 2

Defn
Let B be a collection of subsets of a sample space S, then B is called a σ-algebra or σ-field iff it satified the following three properties.
 1) ∮ ∈ B
 2) If A ∈ B, then A^c∈B (B is closed under complementation)
 3) If A1, A2, .. ∈ B then union of all Ai ∈ B (closed under countable unions)

Defn(Koliogorov)
Given a sample space S and an associated σ-algebra B, a set function P defined on  B is called a probability iff it satisfies the following
 1) P(A) ≥ 0 for all A ∈ B (nonnegativity)
 2) total prob : P(S) = 1
 3) If A1, A2, ... are mutually exclusive events, then P(∪Ai) = ∑P(Ai) (countable additivity)
// U(Ai), ∑P(Ai) 는 모든 Ai에 대해 연산을 함을 뜻한다.
// 앞으로 구할 확률의 성질들은 모두 이 세가지 성질로부터 나온다.

3. Calculus of prob // 모두 앞에서 이야기한 확률의 기본 성질 3가지로부터 증명가능하다.
Thm 3.1
 1) P(∮)=0
 2) P(A)≤1
 3) P(A^c) = 1-P(A)

Thm 3.2
 For any A, B
 1) P(A^c∩B) = P(B) - P(A∩B)
 2) P(A∪B) = P(A) + P(B) - P(A∩B)
 3) A⊂B implies P(A) ≤ P(B)

ㅁ Some inequalities
1) Booles ineq.
 P(A∪B) ≤P(A) + P(B)
2) generalization of 1)
 P(A1∪A2∪ ... ∪An) ≤ ∑P(Ai)
3) Bonferroni's ineq
 P(A∩B) ≥ P(A) + P(B) - 1
4) generalization of 2)
 P(A1∩A2∩ ... ∩An) ≥ ∑P(Ai) - 1

5) union of three events

 P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C)

6) generalization of 5)

 P(A1∪A2∪...∪An) = P1 - P2 + ...(-1)^(n-1)Pn