Probability Day 1 - Probability and Statistics Part 1 Probability Concepts and Limit Theorems Chang-han Rhee Stanford University CME001 1 Outline

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Unformatted text preview: Probability and Statistics Part 1. Probability Concepts and Limit Theorems Chang-han Rhee Stanford University Sep 19, 2011 / CME001 1 Outline Probability Concepts Probability Space Random Variables Expectation Conditional Probability and Expectation Limit Theorems Modes of Convergence Law of Large Numbers Central Limit Theorem 2 Outline Probability Concepts Probability Space Random Variables Expectation Conditional Probability and Expectation Limit Theorems Modes of Convergence Law of Large Numbers Central Limit Theorem 3 Probability of an Event in a random experiment Relative frequency of an event, when repeating a random experiment. e.g. coin flip, dice roll, roulette 4 Sample Space Set of all possible outcomes. I Single coin flip Ω = { H , T } I Two coin flips Ω = { ( H , H ) , ( H , T ) , ( T , H ) , ( T , T ) } I Single dice roll Ω = { 1 , 2 , 3 , 4 , 5 , 6 } I Two dice rolls Ω = { ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 1 , 4 ) , ( 1 , 5 ) , ( 1 , 6 ) ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 3 ) , ( 2 , 4 ) , ( 2 , 5 ) , ( 2 , 6 ) ( 3 , 1 ) , ( 3 , 2 ) , ( 3 , 3 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) ( 4 , 1 ) , ( 4 , 2 ) , ( 4 , 3 ) , ( 4 , 4 ) , ( 4 , 5 ) , ( 4 , 6 ) ( 5 , 1 ) , ( 5 , 2 ) , ( 5 , 3 ) , ( 5 , 4 ) , ( 5 , 5 ) , ( 5 , 6 ) ( 6 , 1 ) , ( 6 , 2 ) , ( 6 , 3 ) , ( 6 , 4 ) , ( 6 , 5 ) , ( 6 , 6 ) } 5 Event Subset of sample space. I Single coin flip : The event that the coin lands head A = { H } I Two coin flips : The event that the first coin lands head A = { ( H , H ) , ( H , T ) } I Single dice roll : The event that the dice falls on an odd number A = { 1 , 3 , 5 } I Two dice roll : The event that the sum is 4 A = { ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 1 ) } 6 Ω = { ( H , H ) , ( H , T ) , ( T , H ) , ( T , T ) } Sample Space Event: first coin lands on head Outcome: both coin lands on tail 7 Probability Definition A set function P is called a probability if I ≤ P ( A ) ≤ 1 for each event A I P (Ω) = 1 ( Unitarity ) I For each sequence A 1 , A 2 , . . . of mutually disjoint events P ( ∞ ∪ 1 A i ) = ∞ ∑ 1 P ( A i ) ( Countable Additivity ) 8 Back to Examples I Fair Coin P ( ∅ ) = P ( { H } ) = 1 / 2 P ( { T } ) = 1 / 2 P ( { H , T } ) = 1 I Biased Coin ( p ∈ [ , 1 ] ) P ( ∅ ) = P ( { H } ) = p P ( { T } ) = 1- p P ( { H , T } ) = 1 9 Outline Probability Concepts Probability Space Random Variables Expectation Conditional Probability and Expectation Limit Theorems Modes of Convergence Law of Large Numbers Central Limit Theorem 10 Random Variables...
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This note was uploaded on 10/01/2011 for the course EE 221 taught by Professor Ee221a during the Spring '08 term at University of California, Berkeley.

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Probability Day 1 - Probability and Statistics Part 1 Probability Concepts and Limit Theorems Chang-han Rhee Stanford University CME001 1 Outline

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