Understanding Probability: Sample Spaces, Events, Measures, and Random Variables, Study notes of Probability and Statistics

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Review of Probability Theory

Zahra Koochak and Jeremy Irvin

Elements of Probability

Elements of Probability

Sample Space Ω {HH, HT , TH, TT }

Elements of Probability

Sample Space Ω {HH, HT , TH, TT } Event A ⊆ Ω

Elements of Probability

Sample Space Ω {HH, HT , TH, TT } Event A ⊆ Ω {HH, HT }, Ω

Elements of Probability

Sample Space Ω {HH, HT , TH, TT } Event A ⊆ Ω {HH, HT }, Ω Event Space F

Elements of Probability

Sample Space Ω {HH, HT , TH, TT } Event A ⊆ Ω {HH, HT }, Ω Event Space F Probability Measure P : F → R P(A) ≥ 0 ∀A ∈ F

Elements of Probability

Sample Space Ω {HH, HT , TH, TT } Event A ⊆ Ω {HH, HT }, Ω Event Space F Probability Measure P : F → R P(A) ≥ 0 ∀A ∈ F P(Ω) = 1

Conditional Probability and Independence

Conditional Probability and Independence

Let B be any event such that P(B) 6 = 0.

Conditional Probability and Independence

Let B be any event such that P(B) 6 = 0. P(A|B) := P P(A(∩BB)) A ⊥ B if and only if P(A ∩ B) = P(A)P(B)

Conditional Probability and Independence

Let B be any event such that P(B) 6 = 0. P(A|B) := P P(A(∩BB)) A ⊥ B if and only if P(A ∩ B) = P(A)P(B) A ⊥ B if and only if P(A|B) = P(A∩B) P(B) =^ P(A)P(B) P(B) =^ P(A)

Random Variables (RV)

ω 0 = HHHTHTTHTT

Random Variables (RV)

ω 0 = HHHTHTTHTT A RV is X : Ω → R