This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View Full
Document
 Fall '11
 HuanHoang

Click to edit the document details