Probability Theory

Probability Theory

1. Probability Space

   • ( \(\Omega\) , \(\mathcal{F}\) , P )
   • Sigma-Field F
     • \(\sigma\) - field
     • 3 properties
       • closed under compliments
         • \( if A \in \mathcal{F} \) then  \(A^c \in \mathcal{F} \)
       • closed under unions
       • Contains Null
         •  \( \emptyset  \in \mathcal{F}  \) 
   • Probability Set
     • \(\Omega\)
     • set of all possible outcomes
   • Probability Measure
     • P on ( \(\Omega\) , \(\mathcal{F}\) )
     • two properties
       • Between zero and one
         • P(null set) = 0, P(solution set) = 1
       • Identity
         • if An is collection of disjoint members of F, sum of proabability is sum of untion
         • Given Disjoint events, Sum of probability of each events = Probability of Union
   • 4 Properties, Basic Prob Math works
     • Prob of compliments add up to 1
       • \( P(A^c) = 1 - P(A) \)
     • If B is super set of A then P(B) = P(A) + P( B\A) >= P(A)
     • P( A U B) = P(A) + P(B) - P( A intersect B)
     • Complex union math, proof by induction

2. Conditional Probability

   • Based on total number of events
     • \( \frac{N(A \cap B}{N(B)} \)
   • P(A given B) = P(A intersection B) / P(B)
   • Lemma
     • \( P(A) = P(A \mid B)P(B) + P(A \mid B^c)*P(B^c) \) Question, prove above

3. Independance

   • Def.
     • \( P(A \cap B) = P(A)(B) \)