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VARIANCE REDUCTION

This article explores fundamental concepts in variance reduction calculations relevant for independent random variables X and Y. It discusses common techniques such as common random numbers and antithetic variates to minimize variance and improve simulation accuracy. The text highlights splitting tasks into streams and employing control variates to correlate outputs effectively. Detailed explanations on generating random samples are provided, ensuring a comprehensive understanding of effective streaming methods and statistical significance in comparisons.

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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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