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VARIANCE REDUCTION

VARIANCE REDUCTION. CALCULATIONS ON VARIANCES: SOME BASICS. Let X and Y be random variables. COV=0 if X and Y are independent. COMMON RANDOM NUMBERS. Built for distinguishing among two systems d i = y i – x i Variance reduced by COV(X, Y) Streaming induces MORE Covariance. STREAMING.

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VARIANCE REDUCTION

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  1. VARIANCE REDUCTION

  2. CALCULATIONS ON VARIANCES: SOME BASICS • Let X and Y be random variables COV=0 if X and Y are independent.

  3. COMMON RANDOM NUMBERS • Built for distinguishing among two systems • di = yi – xi • Variance reduced by COV(X, Y) • Streaming induces MORE Covariance

  4. STREAMING • Segregate the random number generation task into streams connected to phenomena Zi=aZi-1 mod m seed1 seed2 Inter-arrival times Service times 1. Change features of the service. 2. Use exact same arrival stream for comparing each service setting.

  5. ANTITHETIC VARIATES • Use Uniforms U1, U2, ... to generate a sample • Use Uniforms 1-U1, 1-U2, ... to generate a second sample • Combine the samples • Extreme values get canceled out • Depends on... • effective streaming • straightforward F-1(U) method of variate generation

  6. spreadsheet...

  7. CONTROL VARIATES • X is your output variable • You seek the Expected Value of X • Y is a random variable • Y is one of the variables that we are generating • We know the Expected Value of Y • Example • X is the total waiting time of a customer • Y is the inter-arrival time before he entered service

  8. ...more CONTROL VARIATES • Xc is a random variable with less Variance and the same Expected Value • pick b to minimize VAR(Xc)

  9. OPTIMAL CONTROL

  10. IMPORTANT CALCULATIONSFusing many results in statistics

  11. ALSO KNOWN AS... • We are regressing X vs. Y • b* is the parameter that a regression package would calculate • r = SQRT[COV(X,Y)2/VAR(X)VAR(Y)] is the correlation coefficient of X and Y • r =1 or -1 implies • Y completely explains X and • VAR(Xc)=0

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