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Sigma Alpha

xโˆˆAโŠ‚ฮฉโˆˆฮฑโŠ‚2ฮฉx \in A \subset \Omega \in \alpha \subset 2^{\Omega}

ฮฑ\alpha is Sigma Alpha if and only if it is CUT

(ฮฉ,ย ฮฑ)(\Omega,~\alpha) is the measurable space.

PP, ฮฑโ†’[0,ย 1]\alpha \rightarrow [0,~1] and CA (Countably Additive)

P(โˆชk=1โˆžAk)=โˆ‘k=1โˆžP(Ak)P(\cup_{k=1}^{\infty} A_k) = \sum\limits_{k=1}^{\infty}P(A_k) if A1โˆฉAj=โˆ…,ย โˆ€iโ‰ j,ย P(ฮฉ)=1A_1 \cap A_j = \emptyset,~\forall i \neq j,~P(\Omega) = 1

(P,ย ฮฑ,ย ฮฉ)(P,~\alpha,~\Omega) is the probability space.

AA and BB are mutually exclusive.

AโˆฉB=โˆ…A \cap B = \emptyset

AA and BB are independent

P(AโˆฉB)=P(A)P(B)P(A \cap B) = P(A) P(B)

P(AโˆชB)=P(A)+P(B)โˆ’P(AโˆฉB)P (A \cup B) = P (A) + P(B) - P (A \cap B)

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