control variable variance reduction the estimation of performance measures in monte carlo simulation can be made more efficient by utilizing known information about the simulation model variance reduction techniques include antithetic variablescontrol variablesimportance samplingconditional monte carloand stratified samplingseefor example[ we shall only deal with control variables and importance sampling here obtained suppose is the output of simulation experiment random variable ye are correlated from the same simulation runis called control variable for if and is known the use of control variables (negatively or positivelyand the expectation of for variance reduction is based on the following theorem we leave its proof to exercise theorem control variable estimation en be let yn be the output of independent simulation runs and let ek the corresponding control variableswith ey known let % ,ye be the correlation coefficient between each yk and yk for each the estimator  xh ek yk = ( ( is an unbiased estimator for ey the minimal variance of (cis var ( ( % ,yevar yn ( which is obtained for % ,ye vary/vary + from ( we see thatby using the optimal in ( )the variance of the control variate estimator is factor % ,ye smaller than the variance of the crude monte carlo is highly correlated with ya significant variance reduction can be estimator thusif achieved the optimal is usually unknownbut it can be easily estimated from the sample ek )covariance matrix of {(yk in the next examplewe estimate the multiple integral in example using control variables example (monte carlo integration (cont )the random variable | for the | / ( ) / is positively correlated with the random variable iid var( we can use it as same choice of ( as ey control variable to estimate the expectation of the following python program is based on theorem it imports the crude monte carlo sampling code from example