confidence-interval - 96 S n < < X +...

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CDA6530: Performance Models of Computers and Networks Chapter 9:Statistical Analysis of Simulated Data and Confidence Interval
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2 Sample Mean R.v. X: E[X]= θ , Var[X]= σ 2 Q: how to use simulation to derive? Simulate X repeatedly X 1 , , X n are i.i.d., = statistic X Sample mean: ¯ X n X i =1 X i n E [ ¯ X ]= θ Var ( ¯ X )= σ 2 n
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3 Sample Variance σ 2 unknown in simulation Hard to use to measure simulation variance Thus we need to estimate σ 2 Sample variance S 2 : n-1 instead of n is to provide unbiased estimator Var ( ¯ X )= σ 2 n S 2 = P n i =1 ( X i ¯ X ) 2 n 1 E [ S 2 ]= σ 2
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4 Estimate Error Sample mean is a good estimator of θ , but has an error How confidence we are sure that the sample mean is within an acceptable error? From central limit theorm: It means that: ¯ X n ( ¯ X θ ) σ N (0 , 1) n ( ¯ X θ ) S N (0 , 1)
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5 Confidence Interval R.v. Z N(0,1), for 0< α <1, define: P(Z>z α ) = α z 0.025 = 1.96 P(- z 0.025 <Z< z 0.025 ) =1-2 α = 0.95 P ( ¯ X z α / 2 S n < θ < ¯ X + z α / 2 S n ) 1 α P ( ¯ X 1 .
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Unformatted text preview: 96 S n &lt; &lt; X + 1 . 96 S n ) . 95 95% confidence interval ( = 0.05) of an estimate is: ( X 1 . 96 S/ n ) 6 When to stop a simulation? Repeatedly generate data (sample) until 100(1- ) percent confidence interval estimate of is less than I Generate at least 100 data values. Continue generate, until you generated k values such that The 100(1- ) percent confidence interval of estimate is ( X z / 2 S/ k, X + z / 2 S/ k ) 2 z / 2 S/ k &lt; I 7 Fix no. of simulation runs Suppose we only simulate 100 times k=100 What is the 95% confidence interval? ( X z / 2 S/ k, X + z / 2 S/ k ) ( X . 196 S/ k, X + 0 . 196 S/ k ) Example: Generating Expo. Distribution 8...
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This note was uploaded on 01/14/2012 for the course CDA 6530 taught by Professor Zou during the Fall '11 term at University of Central Florida.

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confidence-interval - 96 S n &amp;amp;lt; &amp;amp;lt; X +...

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