By Schaaf W.L.

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The standard deviation thus improves by one over the square root of the number of samples. For large n the estimates of the mean frequently approach a Normal or Gaussian distribution. [2;. l is the mean and 0' is the standard deviation of the distribution. Here Y is the probability of finding a sample value from a normally distributed popUlation within X of the mean. 73% of the time the estimate will be within three standard deviations of the correct value. 5 Introduction to Variance Reduction Techniques As indicated above, the purpose of a Monte Carlo calculation is usually to obtain an estimate of the expected value of a random variable.

L)var = (sumsq(i)/dj - (sumf(i)/dj)**2) totals=totals+sumf (i) /dj; tvarl=tvar1+var; tvar2=tvar2+var/dj 200 CCNI'INUE WRITE (8, 16)totals,DSQRT(tvarl),DSQRT(tvar2) 16 ~T(' sum:', flO . 6,' stdevof distr:', flO. 6) var=O . OdO ! 6,' stdev of distr: ',fl0 . 7,' stdevof sum:',eI4 . 6) REIURN END For many random variables of interest, the range over which a Monte Carlo estimate varies within a stratum can be less than that for the total sample space. Since the variance of the variable depends upon this range, and reducing the range reduces the variance, stratification generally reduces the variance.

100))nstrata=100 CALL stats delta=(b-a)/nstrata ! delta x value for each stratun delta2=(b-a) ! delta x for IIDStratified calculatioo 00 100 j=l,nsamples dK=fltrn () ! lber in (0, 1) n=linstrata*dK ! 1late the strata associated with dK dK=a+(b-a)*dK; score=delta*f(dK) ! calculate score at dK CALL statlp ! store statistics for score score=delta2*f (dK) ! calculate score for IIDStratified case n=nstrata+1 ! store IIDStratified scores in last strata CALL statlp ! store statistics for IIDStratified case 100 CCNl'INUE n=nstrata CALL statend !