Probability Theory

Annotations:

• ( \(\Omega\) , \(\mathcal{F}\) , P )

1. Sigma-Field F

   Annotations:

   • \(\sigma\) - field
   1. 3 properties
      1. closed under compliments

         Annotations:

         • \( if A \in \mathcal{F} \) then  \(A^c \in \mathcal{F} \)
      2. closed under unions
      3. Contains Null

         Annotations:

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

   Annotations:

   • \(\Omega\)
   1. set of all possible outcomes
3. Probability Measure

   Annotations:

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

         Annotations:

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

      Annotations:

      • \( P(A^c) = 1 - P(A) \)
   2. If B is super set of A then P(B) = P(A) + P( B\A) >= P(A)
   3. P( A U B) = P(A) + P(B) - P( A intersect B)
   4. Complex union math, proof by induction

1. Based on total number of events

   Annotations:

   • \( \frac{N(A \cap B}{N(B)} \)
2. P(A given B) = P(A intersection B) / P(B)
3. Lemma

   Annotations:

   • \( P(A) = P(A \mid B)P(B) + P(A \mid B^c)*P(B^c) \) Question, prove above

1. Def.

   Annotations:

   • \( P(A \cap B) = P(A)(B) \)