Probability Day 2

# Probability Day 2 - Probability and Statistics Part 2 More...

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Probability and Statistics Part 2. More Probability, Statistics and their Application Chang-han Rhee Stanford University Sep 20, 2011 / CME001 1

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Outline Statistics Estimation Concepts Estimation Strategies More Probability Expectation and Conditional Expectation Interchange of Limit Transforms Simulation Monte Carlo Method Rare Event Simulation Further Reference Classes at Stanford Books 2
Outline Statistics Estimation Concepts Estimation Strategies More Probability Expectation and Conditional Expectation Interchange of Limit Transforms Simulation Monte Carlo Method Rare Event Simulation Further Reference Classes at Stanford Books 3

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Probability and Statistics Probability Statistics Model Data 4
Estimation Making best guess of an unknown parameter out of sample data. eg. Average height of west african giraffe 5

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Estimator An estimator (statistic) is a rule of estimation: ˆ θ n = g ( X 1 , . . . , X n ) 6
Quality of an Estimator I Bias E ˆ θ - θ I Variance var ( ˆ θ ) I Mean Square Error (MSE) E [ ˆ θ - θ ] 2 = ( bias ) 2 + ( var ) 7

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Confidence Interval Consider the sample mean estimator ˆ θ = 1 n S n . From the CLT, S n - n E X 1 n D σ N ( 0 , 1 ) Rearranging terms, (note: this is not a rigorous argument) 1 n S n D E X 1 + σ n N ( 0 , 1 ) 8
Outline Statistics Estimation Concepts Estimation Strategies More Probability Expectation and Conditional Expectation Interchange of Limit Transforms Simulation Monte Carlo Method Rare Event Simulation Further Reference Classes at Stanford Books 9

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Maximum Likelihood Estimation Finding most likely explanation. ˆ θ n = arg max θ f ( x 1 , x 2 , . . . , x n | θ ) = f ( x 1 | θ ) · f ( x 2 | θ ) · f ( x n | θ ) I Gold Standard: Gueranteed to be I Often computationally challenging 10
Method of Moments Matching the sample moment and the parametric moments.

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