Probability Theory

Lewis Warne
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Lewis Warne
Created by Lewis Warne over 1 year ago
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Probability Theory
1 Probability Space

Annotations:

  • ( \(\Omega\) , \(\mathcal{F}\) , P )
1.1 Sigma-Field F

Annotations:

  • \(\sigma\) - field
1.1.1 3 properties
1.1.1.1 closed under compliments

Annotations:

  • \( if A \in \mathcal{F} \) then  \(A^c \in \mathcal{F} \)
1.1.1.2 closed under unions
1.1.1.3 Contains Null

Annotations:

  •  \( \emptyset  \in \mathcal{F}  \) 
1.2 Probability Set

Annotations:

  • \(\Omega\)
1.2.1 set of all possible outcomes
1.3 Probability Measure

Annotations:

  • P on ( \(\Omega\) , \(\mathcal{F}\) )
1.3.1 two properties
1.3.1.1 Between zero and one

Annotations:

  • P(null set) = 0, P(solution set) = 1
1.3.1.2 Identity
1.3.1.2.1 if An is collection of disjoint members of F, sum of proabability is sum of untion
1.3.1.2.2 Given Disjoint events, Sum of probability of each events = Probability of Union
1.4 4 Properties, Basic Prob Math works
1.4.1 Prob of compliments add up to 1

Annotations:

  • \( P(A^c) = 1 - P(A) \)
1.4.2 If B is super set of A then P(B) = P(A) + P( B\A) >= P(A)
1.4.3 P( A U B) = P(A) + P(B) - P( A intersect B)
1.4.4 Complex union math, proof by induction
2 Conditional Probability
2.1 Based on total number of events

Annotations:

  • \( \frac{N(A \cap B}{N(B)} \)
2.2 P(A given B) = P(A intersection B) / P(B)
2.3 Lemma

Annotations:

  • \( P(A) = P(A \mid B)P(B) + P(A \mid B^c)*P(B^c) \) Question, prove above
3 Independance
3.1 Def.

Annotations:

  • \( P(A \cap B) = P(A)(B) \)

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