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Unformatted text preview: 18.05 Spring 2005 Lecture Notes 18.05 Lecture 1 February 2, 2005 Required Textbook DeGroot & Schervish, “Probability and Statistics,” Third Edition Recommended Introduction to Probability Text Feller, Vol. 1 § 1.21.4. Probability, Set Operations. What is probability? • Classical Interpretation: all outcomes have equal probability (coin, dice) • Subjective Interpretation (nature of problem): uses a model, randomness involved (such as weather) – ex. drop of paint falls into a glass of water, model can describe P (hit bottom before sides) – or, P (survival after surgery) “subjective,” estimated by the doctor. • Frequency Interpretation: probability based on history – P (make a free shot) is based on history of shots made. Experiment ↔ has a random outcome. 1. Sample Space set of all possible outcomes. coin: S= { H, T } , die: S= { 1, 2, 3, 4, 5, 6 } two dice: S= { (i, j), i, j=1, 2, ..., 6 } 2. Events any subset of sample space ex. A √ S, A collection of all events. 3. Probability Distribution P : A ↔ [0, 1] Event A √ S, P (A) or Pr(A) probability of A Properties of Probability: 1. ← P ( A ) ← 1 2. P ( S ) = 1 3. For disjoint (mutually exclusive) events A, B (definition ↔ A ∞ B = ≥ ) P(A or B) = P(A) + P(B) this can be written for any number of events. For a sequence of events A 1 , ..., A n , ... all disjoint ( A i ∞ A j = ≥ , i = j): ∈ ∗ ∗ P ( A i ) = P ( A i ) i =1 i =1 which is called “countably additive.” If continuous, can’t talk about P (outcome), need to consider P (set) Example: S = [0 , 1] , < a < b < 1 . P ([ a, b ]) = b − a, P ( a ) = P ( b ) = . 1 Need to group outcomes, not sum up individual points since they all have P = . § 1.3 Events, Set Operations Union of Sets: A ⇒ B = { s ⊂ S : s ⊂ A or s ⊂ B } Intersection: A ∞ B = AB = { s ⊂ S : s ⊂ A and s ⊂ B } c Complement: A = { s ⊂ S : s / ⊂ A } Set Difference: A \ B = A − B = { s ⊂ S : s ⊂ A and s / ⊂ B } = A ∞ B 2 c c Symmetric Difference: ( A ∞ B c ) ⇒ ( B ∞ A ) Summary of Set Operations: 1. Union of Sets: A ⇒ B = { s ⊂ S : s ⊂ A or s ⊂ B } 2. Intersection: A ∞ B = AB = { s ⊂ S : s ⊂ A and s ⊂ B 3. Complement: A c = { s ⊂ S : s / } ⊂ A } c 4. Set Difference: A \ B = A − B = { s ⊂ S : s ⊂ A and s / ⊂ B } = A ∞ B 5. Symmetric Difference: A ⇔ B = { s ⊂ S : ( s ⊂ A and s / ) or ( s ⊂ B and s / ⊂ B ⊂ A ) } = c ) ( A ∞ B c ) ⇒ ( B ∞ A Properties of Set Operations: 1. A B = B A ⇒ ⇒ 2. ( A ⇒ B ) ⇒ C = A ⇒ ( B C ) ⇒ Note that 1. and 2. are also valid for intersections. 3. For mixed operations, associativity matters: ( A ⇒ B ) ∞ C = ( A ∞ C ) ⇒ ( B ∞ C ) think of union as addition and intersection as multiplication: (A+B)C = AC + BC c 4. ( A ⇒ B ) c = A ∞ B c Can be proven by diagram below: Both diagrams give the same shaded area of intersection....
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 Fall '11
 HuanHoang
 Probability, Probability distribution, Probability theory, i=1

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