By von zur Gathen J., Gerhard J.

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A, 124, 1961, pp. 227-239. 2 A. Hall, "On an Experimental Determination of1t," Messeng. Math. 2, 1873, pp. 113·14. I 1. Introduction 19 ) Lord Kelvin, "Ninteenth Century Clouds over the Dynamical Theory of Heat and Light," Phil Mag 6,2, 1901, pp. 1-40. 4 W. S. Gosset, "Probable Error ofa Correlation Coefficient," Biometrika 6, 1908, p. 302. s A. S. Householder, G. E. Forsythe, and H. H. S. S. , 1951. 6 The Rand Corporation, A Million Random Digits with 100,000 Normal Deviates, Free Press Publishers, Glencoe, IL, 1955.

26) -00 Again, the standard deviation cr of a continuous random variable V is the positive square root of the variance of V. 29 is useful for calculating or estimating the variance of a random variable. It is valid for both discrete and continuous random variables. However, the expected value of a random variable may not exist, or may not be finite. If the expected value is not finite, the variance does not exist. If the expected value is finite, the variance exists but may not be finite. We will see examples of the latter in Chapter 7.

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. In fact, provided the sampling is distributed pro rata among the strata (the number of samples in each stratum is proportional to the "size" of the stratum), in no case will stratified sampling reduce the accuracy of a Monte Carlo estimate.