Foreword to the Online Edition

This book was a product of RAND's computing power (and patience). The tables of random numbers in the book have become a standard reference in engineering and econometrics textbooks and have been widely used in gaming and simulations that employ Monte Carlo trials. Still the largest known source of random digits and normal deviates, the work is routinely used by statisticians, physicists, polltakers, market analysts, lottery administrators, and quality control engineers.

A humorous sidelight: The New York Public Library originally indexed this book under the heading "Psychology."

Acknowledgments

The following persons participated in the production, testing, and preparation for publication of the tables of random digits and random normal deviates: Paul Armer, E. C. Bower, Mrs. Bernice Brown, G. W. Brown, Walter Frantz, J. J. Goodpasture, W. F. Gunning, Cecil Hastings, Olaf Helmer, M. L. Juncosa, J. D. Madden, A. M. Mood, R. T. Nash, J. D. Williams. These tables were prepared in connection with analyses done for the United States Air Force.

Introduction

Early in the course of research at The RAND Corporation a demand arose for random numbers; these were needed to solve problems of various kinds by experimental probability procedures, which have come to be called Monte Carlo methods. Many of the applications required a large supply of random digits or normal deviates of high quality, and the tables presented here were produced to meet those requirements. The numbers have been used extensively by research workers at RAND, and by many others, in the solution of a wide range of problems during the past seven years.

One distinguishing feature of the digit table is its size. On numerous RAND problems the largest existing table of Kendall and Smith, 1939, would have had to be used many times over, with the consequent dangers of introducing unwanted correlations. The feasibility of working with as large a table as the present one resulted from developments in computing machinery which made possible the solving of very complicated distribution problems in a reasonable time by Monte Carlo methods. The tables were constructed primarily for use with punched card machines. With the high-speed electronic computers recently developed, the storage of such tables is usually not practical and, in fact, much larger tables than the present one are often required; these machines have caused research workers to turn to pseudo-random numbers which are computed by simple arithmetic processes directly by the machine as needed. These developments are summarized in Juncosa, 1953; Meyer, Gephart, and Rasmussen, 1954; and Moshman, 1954, where other references may be found. The Monte Carlo Method, 1951; Curtiss, 1949; Kahn and Marshall, 1953; and Kahn, 1956, discuss the uses and applications of the Monte Carlo methods and give references to other applications.

Production of the Random Digits

The random digits in this book were produced by rerandomization of a basic table generated by an electronic roulette wheel. Briefly, a random frequency pulse source, providing on the average about 100,000 pulses per second, was gated about once per second by a constant frequency pulse. Pulse standardization circuits passed the pulses through a 5-place binary counter. In principle the machine was a 32-place roulette wheel which made, on the average, about 3000 revolutions per trial and produced one number per second. A binary-to-decimal converter was used which converted 20 of the 32 numbers (the other twelve were discarded) and retained only the final digit of two-digit numbers; this final digit was fed into an IBM punch to produce finally a punched card table of random digits.

Production from the original machine showed statistically significant biases, and the engineers had to make several modifications and refinements of the circuits before production of apparently satisfactory numbers was achieved. The basic table of a million digits was then produced during May and June of 1947. This table was subjected to fairly exhaustive tests and it was found that it still contained small but statistically significant biases. For example, the following table[1] shows the results of three tests (described later) on two blocks of 125,000 digits:

Block 1 Block 2 χ2 Probability χ2 Probability Frequency (9 d.f.a) 6.0 .74 21.0 .02 Odd-even (1 d.f) 3.0 .09 7.0 <.0l Serial (81 d.f.) 78.7 .55 105.6 .03 aThe letters "d.f." (degrees of freedom) identify a parameter associated with the test. A discussion of the test may be found in any textbook on statistics.

Block 1 was produced immediately after a careful tune-up of the machine; Block 2 was produced after one month of continuous operation without adjustment. Apparently the machine had been running down despite the fact that periodic electronic checks indicated it had remained in good order.

The table was regarded as reasonably satisfactory because the deviations from expectations in the various tests were all very small—the largest being less than 2 per cent—and no further effort was made to generate better numbers with the machine. However, the table was transformed by adding pairs of digits modulo 10 in order to improve the distribution of the digits. There were 20,000 punched cards with 50 digits per card; each digit on a given card was added modulo 10 to the corresponding digit of the preceding card to yield a rerandomized digit. It is this transformed table which is published here and which is the subject of the tests described below.

The transformation was expected to, and did, improve the distribution in view of a limit theorem to the effect that sums of random variables modulo 1 have the uniform distribution over the unit interval as their limiting distribution. (See Horton and Smith, 1949, for a version of this theorem for discrete variates.)

These tables were reproduced by photo-offset from pages printed by the IBM model 856 Cardatype. Because of the very nature of the tables, it did not seem necessary to proofread every page of the final manuscript in order to catch random errors of the Cardatype. All pages were scanned for systematic errors, every twentieth page was proofread (starting with page 10 for both the digits and deviates), and every fortieth page (starting with page 5 for both the digits and deviates) was summed and the totals checked against sums obtained from the cards.[2]

Tests on the Random Digits

Frequency Tests. The table was divided into 1000 blocks of 1000 digits each and the frequency of each digit was recorded for each block. Then for each block a goodness-of-fit χ2 was computed with 9 d.f. These 1000 values of χ2 provided an empirical fit to the χ2 distribution (with 9 d.f.); to test the fit, a goodness-of-fit χ2 was computer using 50 class intervals, each of which was expected to contain 2 per cent of the values. (The number of intervals was chosen in accordance with the result of Wald and Mann, 1942.) The value of the test χ2 was 54.6 which, for 49 d.f., corresponded to about the 0.45 probability level.

To examine further the frequencies, the digits were tallied in 20 blocks of 50,000 digits each. The results are shown in Table 1 together with the goodness-of-fit χ2 for each block. On the total frequencies the χ2 (13.316) for 9 d.f. has been partitioned into three components as follows:

χ2 d.f. Probability Odd versus even digits 1.37 1 0.25 Within groups of odd digits 7.90 4 0.10 Within groups of even digits 4.04 4 0.40

Block No. 0 1 2 3 4 5 6 7 8 9 χ2 1 4923 5013 4916 4951 5109 4993 5055 5080 4986 4974 7.556 2 4870 4956 5080 5097 5066 5034 4902 4974 5012 5009 10.132 3 5065 5014 5034 5057 4902 5061 4942 4946 4960 5019 6.078 4 5009 5053 4966 4891 5031 4895 5037 5062 5170 4886 15.004 5 5033 4982 5180 5074 4892 4992 5011 5005 4959 4872 13.846 6 4976 4993 4932 5039 4965 5034 4943 4932 5116 5070 7.076 7 5011 5152 4990 5047 4974 5107 4869 4925 5023 4902 14.116 8 5003 5092 5163 4936 5020 5069 4914 4943 4914 4946 13.051 9 4860 4899 5138 4959 5089 5047 5030 5039 5002 4937 13.410 10 4998 4957 4964 5124 4909 4995 5053 4946 4995 5059 7.212 11 4948 5048 5041 5077 5051 5004 5024 4886 4917 5004 7.142 12 4958 4993 5064 4987 5041 4984 4991 4987 5113 4882 6.992 13 4968 4961 5029 5038 5022 5023 5010 4988 4936 5025 2.162 14 5110 4923 5025 4975 5095 5051 5035 4962 4942 4882 10.172 15 5094 4962 4945 4891 5014 5002 5038 5023 5179 4852 16.261 16 4957 5035 5051 5021 5036 4927 5022 4988 4910 5053 4.856 17 5088 4989 5042 4948 4999 5028 5037 4893 5004 4972 5.347 18 4970 5034 4996 5008 5049 5016 4954 4989 4970 5014 1.625 19 4998 4981 4984 5107 4874 4980 5057 5020 4978 5021 6.584 20 4963 5013 5101 5084 4956 4972 5018 4971 5021 4901 6.584 Total 99802 100050 100641 100311 100094 100214 99942 99559 100107 99280 13.316

Of the 200 frequencies recorded in Table 1, 59 (29.5 per cent) deviate from 5000 by more than σ (= 30√5 = 67.08), and 8 (4 per cent) deviate from 5000 by more than 2σ. Of the twenty χ2 values in Table 1, eight exceed the 50 per cent value (8.34), two fall below the 10 per cent value (4.17), and two exceed the 90 per cent value (14.7).

Poker Tests.Sets of 5 digits in blocks of 5000 digits were taken to be poker hands and were classified as:

Class Symbol Expected Frequency Per Block Busts abcde 302.4 Pairs aabcd 504 Two pairs aabbc 108 Threes aaabc 72 Full house aaabb 9 Fours aaaab {4.5} Five aaaaa {0.1}

There were 200 sets of 1000 poker hands in the table, and for each set a goodness-of-fit χ2 was computed with 5 d.f. (the fours and fives were combined). The manner in which these 200 values fit the χ2 distribution is shown in Table 2.

Table 2. Distribution of Chi-square Values Probability Values of χ2 Expected Frequency Observed Frequency P > .90 0 – 1.60 20 22 .90 >P > .80 1.61 – 2.35 20 19 .80 > P > .70 2.36 – 3.00 20 22 .70 > P > .60 3.01 – 3.70 20 19 .60 > P > .50 3.71 – 4.35 20 20 .50 > P > .40 4.36 – 5.20 20 29 .40 > P > .30 5.21 – 6.10 20 22 .30 > P > .20 6.11 – 7.30 20 15 .20 > P > .10 7.31 – 9.20 20 15 P < .10 9.21 or more 20 17 Total 200 200

The goodness-of-fit test gives:

χ2 = 7.7 for 9 d.f., P = 0.55.

The combined frequencies of poker hands in the whole table are shown in Table 3. The largest difference between expected and observed frequencies (for threes) is about 2.25 times its standard deviation, which is roughly at about the 9 or 10 per cent probability level (looking merely at the largest of five independent normal observations).

Table 3. Poker Test on The Million Digits (200,000 Poker Hands) Classes Expected Frequency Observed Frequency Busts (abcde) 60,480 60,479 Pairs (aabcd) 100,800 100,570 Two pairs (aabbc) 21,600 21,572 Threes (aaabc) 14,400 14,659 Full house (aaabb) 1,800 1,788 Fours (aaaab) {900} {914} Fives (aaaaa) {20} {18} Total 200,000 200,000

The goodness-of-fit test gives:

χ2 = 5.5 for 5 d.f., P = 0.35.

Also, the frequencies of poker hands were computed for each of ten blocks of 100,000 digits and the mean and standard deviation was computed from the ten values for each kind of hand. The results are shown in Table 4.

Table 4. Mean and Standard Deviation of Frequencies in Seven Classes of Poker Hands Classes Theoretical Mean Actual Mean Theoretical Std. Dev. Actual Std. Dev. Busts 6048 6047.9 64.9 60.3 Pairs 10080 10057.0 70.7 78.4 Two pairs 2160 2157.2 43.9 45.8 Threes 1440 1465.9 36.9 26.6 Full house 180 178.8 13.4 8.9 Fours 90 91.4 9.5 11.5 Fives 2 1.8 1.4 1.9

Serial and Run Tests. Some further tests were made on the first block of 50,000 digits to look particularly for any evidence of serial association among the digits. The serial test classified every successive pair of digits by each digit of the pair in a ten-by-ten table. The frequencies of the different pairs are given in Table 5, where the first digit of the pair is shown in the left column of the table and the second digit is shown at the top. Thus there were 510 cases in which a zero followed a one. The frequency χ2 for the row (or column) totals is 7.56, which is about the 0.60 probability level for 9 d.f.

Table 5. Frequencies of Ordered Pairs of Digits Second Digit First Digit 0 1 2 3 4 5 6 7 8 9 Total 0 508 456 509 507 502 489 471 504 488 489 4923 1 510 514 474 514 504 481 496 486 507 527 5013 2 451 523 493 484 502 466 514 506 493 484 4916 3 500 472 476 466 513 478 540 513 530 463 4951 4 513 561 481 485 526 513 485 510 524 511 5109 5 475 490 527 507 493 481 489 512 465 554 4993 6 494 486 491 483 525 504 530 539 513 490 5055 7 508 512 454 498 550 533 516 504 485 520 5080 8 463 503 475 514 520 544 514 491 520 442 4986 9 501 496 536 493 474 504 500 515 461 494 4974 Total 4923 5013 4916 4951 5109 4993 5055 5080 4986 4974 50000

Table 5 can be tested by a criterion originally due to Kendall and Smith, 1939, and revised by Good, 1953. Assuming all pairs equally likely we get a normalized sum of squared deviations of 107.8. However, this statistic does not have a χ2-distribution. On the other hand, it is the sum of the error variation and twice the row (or column) variation, where under the assumption of perfect randomness, the error variation is asymptotically distributed like χ2 with 81 degrees of freedom. We take the error variation as our test criterion. This gives a χ2 of 107.8 – 2(7.56) ~ 92.7, which is about the 0.18 level for 81 degrees of freedom.

Finally, in the same block of 50,000 digits all runs were counted with the obviously satisfactory results shown in Table 6.

Table 6. Run Test Length of Run Expected Frequency Observed Frequency r = 1 40500 40410 r = 2 4050 4055 r = 3 405 421 r = 4 40.5 48 r ≥ 5 4.5 5

Normal Deviates

Half of the random digit table was used to produce 100,000 standard normal deviates by solving for x in Equation (1),

where D is a five-digit number from the table and

is the cumulative standard normal distribution. The Bureau of Standards tables of F(x) were used (1948).

The deviates were determined by the five-digit numbers on the left-hand half of every page of the digit table. The deviates in the first column correspond page by page with the five-figure digits in the first column of the first 200 pages of the digit table; the deviates in the second column correspond page by page with the first column of the second 200 pages of the digit table. Similarly, the third and fourth columns of deviates were derived from the second column of five-figure digits, etc.

A χ2 test of the fit of the entire table of deviates to the normal distribution was performed using 400 class intervals (Mann and Wald, 1942) with roughly 250 expected in each. The χ2 value was found to be 346.4, which for 399 d.f. indicates a very close fit; the probability of a larger value of χ2 is about 0.97. The detailed data for this test are given in Table 7.

Table 7: Goodness-of-fit Test for Normal Deviates Class Limits Expected Number Observed Number (Obs.- Exp.)2/Exp. Left Tail Right Tail Left Tail Right Tail .000 239.364 229 267 .449 3.191 .006 239.355 253 240 .778 .002 .012 239.338 225 232 .859 .225 .018 279.195 275 286 .063 .166 .025 239.271 243 245 .058 .137 .031 239.227 232 255 .218 1.040 .037 239.173 285 221 8.781 1.381 .043 278.958 284 277 .091 .014 .050 239.029 225 244 .823 .103 .056 238.948 225 218 .814 1.836 .062 238.860 260 231 1.871 .259 .068 278.546 225 287 10.293 .257 .075 238.638 253 211 .864 3.201 .081 238.522 224 224 .884 .884 .087 278.118 254 290 2.091 .508 .094 238.242 212 222 2.891 1.107 .100 238.098 225 250 .721 .595 .106 277.590 297 272 1.357 .113 .113 237.760 217 235 1.813 .032 .119 237.590 232 224 .132 .777 .125 237.413 230 266 .231 3.442 .131 276.744 262 272 .786 .081 .138 236.998 244 226 .207 .510 .144 236.792 235 250 .014 .737 .150 275.989 271 275 .090 .004 .157 236.320 246 273 .397 5.693 .163 275.415 267 279 .257 .047 .170 235.810 250 221 .854 .930 .176 235.561 233 233 .028 .028 .182 274.495 265 291 .328 .992 .189 234.994 230 236 .106 .004 .195 234.718 250 233 .995 .013 .201 273.481 263 283 .402 .331 .208 234.095 232 228 .019 .159 .214 272.731 302 292 3.141 1.361 .221 233.435 243 224 .392 .381 .227 233.117 223 241 .439 .267 .233 271.557 276 286 .073 .768 .240 232.401 234 248 .011 1.047 .246 270.701 292 302 1.676 3.619 .253 231.649 229 225 .030 .191 .259 269.802 274 253 .065 1.046 .266 230.859 251 227 1.757 .065 .272 268.860 269 271 .000 .017 .279 230.034 226 233 .071 .038 .285 267.876 266 274 .013 .140 .292 229.173 235 221 .148 .291 .298 266.850 248 238 1.332 3.119 .305 266.282 249 283 1.122 1.050 .312 227.779 240 216 .656 .609 .318 265.193 244 274 1.694 .292 .325 226.830 213 234 .843 .227 .331 264.065 284 284 1.505 1.505 .338 263.440 282 269 1.308 .117 .345 225.302 230 214 .098 .567 .351 262.251 299 265 5.150 .029 .358 261.596 267 279 .112 1.158 .365 223.693 229 217 .126 .200 .371 260.347 275 279 .825 1.336 .378 259.659 266 253 .155 .171 .385 222.009 224 225 .018 .040 .391 258.353 266 247 .226 .499 .398 257.633 250 244 .226 .721 .405 256.905 268 269 .479 .569 .412 256.164 260 283 .057 2.811 .419 255.415 243 250 .603 .115 .426 254.654 284 269 3.382 .808 .433 217.661 230 208 .699 .429 .439 253.215 265 259 .548 .132 .446 252.425 251 250 .008 .023 .453 251.626 257 235 .115 1.099 .460 250.817 264 249 .693 .013 .467 249.999 256 231 .144 1.444 .474 249.170 249 271 .000 1.913 .481 248.333 244 268 .076 1.558 .488 247.486 233 274 .848 2.841 .495 246.630 239 246 .236 .002 .502 280.803 276 264 .082 1.005 .510 244.765 275 262 3.735 1.214 .517 243.881 249 254 .107 .420 .524 242.988 231 250 .591 .202 .531 242.087 223 242 1.505 .000 .538 275.555 288 256 .562 1.388 .546 240.126 229 228 .516 .612 .553 239.199 246 244 .193 .096 .560 272.224 282 281 .351 .283 .568 237.183 235 252 .020 .926 .575 236.231 255 229 1.491 .221 .582 268.802 255 273 .709 .066 .590 234.163 239 211 .100 2.291 .597 266.419 281 276 .798 .345 .605 232.062 218 220 .852 .627 .612 263.999 270 249 .136 .852 .620 262.692 261 285 .011 1.894 .628 228.776 253 207 2.565 2.073 .635 260.216 272 260 .534 .000 .643 258.881 256 265 .032 .145 .651 225.417 233 230 .255 .093 .658 256.352 263 270 .172 .727 .666 254.989 272 248 1.135 .192 .674 253.618 266 263 .605 .347 .682 252.239 245 235 .208 1.178 .690 250.849 255 274 .069 2.137 .698 249.453 239 240 .438 .358 .706 248.048 228 252 1.620 .063 .714 246.635 248 243 .008 .054 .722 245.215 230 254 .944 .315 .730 243.787 235 246 .317 .020 .738 272.544 296 240 2.019 3.886 .747 240.729 231 230 .393 .478 .755 239.279 240 250 .002 .480 .763 267.449 269 266 .009 .008 .772 236.177 229 256 .218 1.664 .780 263.944 292 283 2.982 1.376 .789 233.048 241 215 .271 1.398 .797 260.410 246 276 .797 .933 .806 258.526 239 227 1.475 3.844 .815 228.216 228 203 .000 2.786 .823 254.953 235 236 1.562 1.409 .832 253.050 262 250 .317 .037 .841 251.143 262 258 .469 .187 .850 249.228 241 261 .272 .556 .859 247.309 245 225 .022 2.012 .868 245.385 237 247 .287 .011 .877 270.387 253 270 1.118 .001 .887 241.307 220 245 1.881 .057 .896 239.369 237 231 .023 .293 .905 263.687 267 281 .042 1.137 .915 235.266 241 211 .140 2.503 .924 259.121 226 236 4.234 2.063 .934 256.713 266 252 .336 .087 .944 254.300 235 250 1.465 .073 .954 251.886 268 235 1.031 1.132 .964 249.469 272 244 2.035 .120 .974 247.051 238 249 .332 .015 .984 244.633 243 239 .011 .130 .994 242.212 230 231 .616 .519 1.004 263.640 274 260 .407 .050 1.015 237.132 250 250 .698 .698 1.025 258.052 267 270 .310 .553 1.036 255.128 256 241 .003 .782 1.047 252.207 219 254 4.372 .013 1.058 249.289 238 245 .511 .074 1.069 246.374 258 244 .549 .023 1.080 243.465 253 235 .373 .294 1.091 262.286 271 256 .290 .151 1.103 237.399 229 215 .297 2.113 1.114 255.683 267 265 .501 .340 1.126 252.252 280 275 3.052 2.051 1.138 248.830 243 227 .137 1.915 1.150 245.421 231 271 .847 2.666 1.162 242.022 256 234 .807 .266 1.174 258.370 270 263 .524 .083 1.187 254.414 235 256 1.481 .010 1.200 250.476 260 247 .362 .048 1.213 246.557 271 260 2.423 .733 1.226 242.659 255 227 .628 1.010 1.239 256.990 256 242 .004 .874 1.253 252.521 251 262 .009 .356 1.267 248.081 257 225 .321 2.147 1.281 243.672 244 220 .000 2.300 1.295 256.219 257 232 .002 2.289 1.310 251.234 273 215 1.886 5.226 1.325 246.291 244 241 .021 .114 1.340 257.309 249 285 .268 2.980 1.356 251.786 268 266 1.044 .802 1.372 246.320 240 229 .162 1.218 1.388 255.788 265 229 .332 2.805 1.405 249.751 252 278 .020 3.195 1.422 243.787 280 237 5.379 .189 1.439 251.707 268 267 1.055 .929 1.457 245.192 240 237 .110 .274 1.475 251.846 247 257 .093 .105 1.494 257.489 259 268 .009 .429 1.514 249.809 238 256 .558 .153 1.534 242.262 233 238 .354 .075 1.554 257.931 247 237 .463 1.699 1.576 249.141 242 263 .205 .771 1.598 251.267 239 265 .599 .751 1.621 242.072 236 249 .152 .198 1.644 252.941 252 222 .004 3.785 1.669 252.098 226 281 2.702 3.313 1.695 250.295 254 224 .055 2.762 1.722 247.560 263 276 .963 3.267 1.750 252.118 247 217 .104 4.892 1.780 246.755 265 265 1.349 1.349 1.811 255.166 255 257 .000 .013 1.845 246.473 246 226 .001 1.701 1.880 249.853 249 259 .003 .335 1.918 249.912 266 237 1.036 .667 1.959 252.136 248 238 .068 .793 2.004 249.874 249 251 .003 .005 2.053 252.080 239 273 .679 1.736 2.108 251.207 248 217 .041 4.658 2.170 249.038 238 251 .489 .015 2.241 250.376 242 251 .280 .002 2.326 250.143 226 240 2.330 .411 2.432 249.585 248 241 .010 .295 2.575 251.174 262 256 .467 .093 2.807 249.526 254 264 .080 .840 Total 50124 49876 158.967 187.450

A more refined test of the fit in the tails was made on the deviates exceeding 2.326 in absolute value. Eighty intervals (Mann and Wald, 1942) were used, each with an expectation of approximately 25. The χ2 value was 76.26, with 80 d.f.; the probability of a larger value is about 0.61. The details of this test are given in Table 8.

Table 8: Goodness-of-fit in 1 Per Cent Tails Class Limits Expected Number Observed Number (Obs.- Exp.)2/Exp. Left Tail Right Tail Left + Right Tails 2.326 26.366 29 22 .986 2.336 25.757 21 20 2.165 2.346 25.159 26 30 .960 2.356 24.574 17 24 2.348 2.366 23.999 19 21 1.416 2.376 25.748 24 22 .664 2.387 25.082 20 21 1.694 2.398 24.428 20 21 1.284 2.409 23.790 20 24 .606 2.420 25.240 30 35 4.672 2.432 26.524 32 36 4.516 2.445 23.748 23 26 .237 2.457 24.948 30 23 1.175 2.470 25.986 25 30 .657 2.484 25.098 23 14 5.083 2.498 24.236 20 23 .803 2.512 25.037 29 24 .670 2.527 25.682 28 25 .227 2.543 24.657 22 24 .304 2.559 23.669 16 16 4.970 2.575 26.868 28 25 .178 2.594 24.261 21 26 .563 2.612 25.652 24 18 2.389 2.632 24.336 24 31 1.829 2.652 25.321 21 23 .950 2.674 24.926 26 26 .093 2.697 24.412 29 32 3.221 2.721 25.624 31 26 1.133 2.748 24.639 31 24 1.659 2.776 25.135 27 25 .139 2.807 25.164 26 39 7.635 2.841 24.759 24 21 .594 2.878 25.087 25 28 .339 2.920 25.144 26 35 3.893 2.968 24.731 19 24 1.350 3.023 25.063 29 18 2.609 3.090 25.160 35 22 4.245 3.175 25.002 20 32 2.959 3.291 24.939 24 15 3.996 3.481 24.977 26 30 1.052 Total 1000.928 990 1001 76.263

The only tests made on the squares of the deviates consisted in computing sums of k squares and comparing the distribution of the sums with the χ2 distribution with k d.f., employing again the standard goodness-of-fit test. This was done for k = 25, 50, 100, 300, with the following results:

k Number of Sums Number of Intervals (i) χ2 with i – 1 Degrees of Freedom Probability of a Larger χ2 25 4000 100 92.92 0.66 50 2000 100 92.45 0.67 100 1000 50 57.75 0.19 300 333 34 38.70 0.23

The fourth column gives the goodness-of-fit χ2 value for the fit to the χ2 distribution with k degrees of freedom. Intervals of approximately equal probability were used in all cases.

Use of the Tables

The lines of the digit table are numbered from 00000 to 19999. In any use of the table, one should first find a random starting position. A common procedure for doing this is to open the book to an unselected page of the digit table and blindly choose a five-digit number; this number with the first digit reduced modulo 2 determines the starting line; the two digits to the right of the initially selected five- digit number are reduced modulo 50 to determine the starting column in the starting line. To guard against the tendency of books to open repeatedly at the same page and the natural tendency of a person to choose a number toward the center of the page: every five-digit number used to determine a starting position should be marked and not used a second time for this purpose.

The digit table is also used to find a random starting position in the deviate table: Select a five-digit number as before; the first four digits give the starting line (the lines being numbered from 0000 to 9999) and the fifth digit gives the starting position in the line.

Ordinarily, the table is read in the same direction as a book is read; however, the size of the table may be effectively increased by varying the direction in which it is read. Thus, one may read columns instead of lines, may read the table backward, may read lines forward but pages from bottom to top, etc. Of course, care must be taken in using these devices to avoid introducing correlations when the table is used more than once on the same problem.

To obtain a random permutation of the integers 1, 2, . . . , n, select a random starting position; use the five-digit number containing the starting position and the following n – 1 five-digit numbers; put the integers in the same order as these n five-digit numbers. In case of ties among the five-digit numbers, use additional columns to the right to make six or more digit numbers. The same procedure is used to obtain a random permutation of n objects, some of which are indistinguishable, by merely numbering the objects arbitrarily from 1 to n.

To obtain random observations from any distribution G(x), use Eq. (1), substitute G(x) for F(x), and employ as many digits in D as required for the desired accuracy of the observations. Of course the negative exponent of 10 in Eq. (1) must be equal to the number of digits in D. If G(x) has a discontinuity at x 0 , define it to be continuous on the right and take the solution of Eq. (1) to be x 0 when the left side of Eq. (1) falls between G(x 0 -) and G(x 0 ). For example, if

G(x) = 1 – e–x

and one is content with three-figure accuracy, then the three-digit number 082 determines an observation from a population distributed by G(x) as follows:

.0825 = 1 – e–x, x = .086.

A technique suggested by von Neumann, called the "rejection method," enables one to substitute for the solution of Eq. (1) a stochastic process involving a much simpler computation; this technique is discussed in Kahn, 1956.

To obtain pairs of normal deviates with given correlation ρ, use pairs (x, y) of independent deviates from the table and transform them to

Thus for ρ = –.6, for example, if (.732, –1.205) are two deviates from the table, then

(.732, –1.403)

is a pair of deviates from a normal population with the desired correlation.[3]

In general, to obtain a random observation from a bivariate population with distribution G(x,y), one uses a marginal distribution on one variate, say, G 1 (x), and the conditional distribution, say, G 2 (y/x), on the other. Two random numbers determine the observation: one determines x by employing G 1 (x) in Eq. (1), and the other determines y by employing G 2 (y/x) in Eq. (1). Thus, if a probability density is uniform (and equal to two) over the triangle bounded by x = 0, y = 0, x + y = 1 and is zero elsewhere, then

and two four-digit random numbers, 5402 and 1770, determine the observation (.3220, .1200). The direct generalization of this procedure will determine observations from multivariate populations.

Complete tables available for download at the top of the page ⤴

issues

References

1. Kendall, M. G., and B. B. Smith, Random Sampling Numbers, Cambridge University Press, 1939.

2. Juncosa, M. L., Random Number Generation on the BRL High-Speed Computing Machines, Ballistic Research Laboratories Report No. 855, Aberdeen Proving Ground, Maryland, 1953.

3. Meyer, H. A., L. S. Gephart, and N. L. Rasmussen, On the Generation and Testing of Random Digits, WADC Technical Report 54-55, Wright-Patterson Air Force Base, Ohio, 1954.

4. Moshman, Jack, "Generation of Pseudo-random Numbers on a Decimal Calculator," J. Assoc. Computing Machinery, Vol. 1, 1954, p. 88.

5. The Monte Carlo Method (Proceedings of a Symposium held in 1949), National Bureau of Standards Report AMS 12, Government Printing Office, Washington 25, D.C., 1951.

6. Curtiss, J. H., "Sampling Methods Applied to Differential and Difference Equations," Seminar on Scientific Computation, International Business Machines Corp., New York, 1949.

7. Kahn, H., and A. W. Marshall, "Methods of Reducing Sample Size in Monte Carlo Computations," J. Operations Res. Soc. of Amer., Vol. 1, 1953, pp. 263–278.

8. Kahn, H., Applications of Monte Carlo, The RAND Corporation, RM-1237, 1956.

9. Horton, H. B., and R. T. Smith, "A Direct Method for Producing Random Digits in Any Number System," Ann. Math. Statistics, Vol. 20, 1949, pp. 82–90.

10. Mann, H. B., and A. Wald, "On the Choice of the Number of Class Intervals in the Application of the Chi-Square Test," Ann. Math. Statistics, Vol. 13, 1942, pp. 306–317.

11. Tables of Probability Functions, 2d ed., U.S. Government Printing Office, Washington, D.C., 1948.

12. Good, I.J., "The Serial Test for Sampling Numbers and Other Tests for Randomness," Proc. Camb. Phil. Soc., Vol. 49, 1953, pp. 276–284.

Notes