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XR232USB RANDOM DATA SAMPLE AND TEST REPORTS
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Date: 20140725
HW: 20130725
SW: 20140720
Baud: 115.200 bps
Filename: xr232utest.dat
Size: 11468800 Bytes ( = 11200 KB pursuant to DIEHARD recommendation)
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RESULTS FROM ENTITLE ENTROPY-CHECK (http://www.fourmilab.ch/random)
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Entropy = 7.999985 bits per byte.
Optimum compression would reduce the size
of this 11468800 byte file by 0 percent.
Chi square distribution for 11468800 samples is 235.83, and randomly
would exceed this value 79.99 percent of the times.
Arithmetic mean value of data bytes is 127.5015 (127.5 = random).
Monte Carlo value for Pi is 3.141571966 (error 0.00 percent).
Serial correlation coefficient is 0.000119 (totally uncorrelated = 0.0).
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RESULTS FROM DIEHARD (http://www.stat.fsu.edu/pub/diehard)
------------------------------------------------------------------------
NOTE: Most of the tests in DIEHARD return a p-value, which
should be uniform on [0,1) if the input file contains truly
independent random bits. Those p-values are obtained by
p=F(X), where F is the assumed distribution of the sample
random variable X---often normal. But that assumed F is just
an asymptotic approximation, for which the fit will be worst
in the tails. Thus you should not be surprised with
occasional p-values near 0 or 1, such as .0012 or .9983.
When a bit stream really FAILS BIG, you will get p's of 0 or
1 to six or more places. By all means, do not, as a
Statistician might, think that a p < .025 or p> .975 means
that the RNG has "failed the test at the .05 level". Such
p's happen among the hundreds that DIEHARD produces, even
with good RNG's. So keep in mind that " p happens".
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: This is the BIRTHDAY SPACINGS TEST ::
:: Choose m birthdays in a year of n days. List the spacings ::
:: between the birthdays. If j is the number of values that ::
:: occur more than once in that list, then j is asymptotically ::
:: Poisson distributed with mean m^3/(4n). Experience shows n ::
:: must be quite large, say n>=2^18, for comparing the results ::
:: to the Poisson distribution with that mean. This test uses ::
:: n=2^24 and m=2^9, so that the underlying distribution for j ::
:: is taken to be Poisson with lambda=2^27/(2^26)=2. A sample ::
:: of 500 j's is taken, and a chi-square goodness of fit test ::
:: provides a p value. The first test uses bits 1-24 (counting ::
:: from the left) from integers in the specified file. ::
:: Then the file is closed and reopened. Next, bits 2-25 are ::
:: used to provide birthdays, then 3-26 and so on to bits 9-32. ::
:: Each set of bits provides a p-value, and the nine p-values ::
:: provide a sample for a KSTEST. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
BIRTHDAY SPACINGS TEST, M= 512 N=2**24 LAMBDA= 2.0000
Results for xr232utest.dat
For a sample of size 500: mean
xr232utest.dat using bits 1 to 24 1.938
duplicate number number
spacings observed expected
0 72. 67.668
1 138. 135.335
2 136. 135.335
3 90. 90.224
4 38. 45.112
5 21. 18.045
6 to INF 5. 8.282
Chisquare with 6 d.o.f. = 3.24 p-value= .221731
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
xr232utest.dat using bits 2 to 25 1.860
duplicate number number
spacings observed expected
0 93. 67.668
1 125. 135.335
2 143. 135.335
3 72. 90.224
4 44. 45.112
5 13. 18.045
6 to INF 10. 8.282
Chisquare with 6 d.o.f. = 16.18 p-value= .987190
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
xr232utest.dat using bits 3 to 26 1.998
duplicate number number
spacings observed expected
0 75. 67.668
1 135. 135.335
2 120. 135.335
3 98. 90.224
4 41. 45.112
5 22. 18.045
6 to INF 9. 8.282
Chisquare with 6 d.o.f. = 4.51 p-value= .391640
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
xr232utest.dat using bits 4 to 27 1.994
duplicate number number
spacings observed expected
0 70. 67.668
1 132. 135.335
2 142. 135.335
3 81. 90.224
4 50. 45.112
5 17. 18.045
6 to INF 8. 8.282
Chisquare with 6 d.o.f. = 2.03 p-value= .083405
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
xr232utest.dat using bits 5 to 28 1.978
duplicate number number
spacings observed expected
0 64. 67.668
1 143. 135.335
2 139. 135.335
3 81. 90.224
4 45. 45.112
5 23. 18.045
6 to INF 5. 8.282
Chisquare with 6 d.o.f. = 4.34 p-value= .368770
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
xr232utest.dat using bits 6 to 29 1.956
duplicate number number
spacings observed expected
0 69. 67.668
1 132. 135.335
2 147. 135.335
3 91. 90.224
4 42. 45.112
5 10. 18.045
6 to INF 9. 8.282
Chisquare with 6 d.o.f. = 4.98 p-value= .454125
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
xr232utest.dat using bits 7 to 30 1.944
duplicate number number
spacings observed expected
0 81. 67.668
1 126. 135.335
2 150. 135.335
3 73. 90.224
4 35. 45.112
5 26. 18.045
6 to INF 9. 8.282
Chisquare with 6 d.o.f. = 13.98 p-value= .970182
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
xr232utest.dat using bits 8 to 31 1.966
duplicate number number
spacings observed expected
0 70. 67.668
1 142. 135.335
2 136. 135.335
3 79. 90.224
4 49. 45.112
5 14. 18.045
6 to INF 10. 8.282
Chisquare with 6 d.o.f. = 3.41 p-value= .243602
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
xr232utest.dat using bits 9 to 32 2.042
duplicate number number
spacings observed expected
0 67. 67.668
1 133. 135.335
2 133. 135.335
3 91. 90.224
4 43. 45.112
5 25. 18.045
6 to INF 8. 8.282
Chisquare with 6 d.o.f. = 2.88 p-value= .176644
:::::::::::::::::::::::::::::::::::::::::
The 9 p-values were
.221731 .987190 .391640 .083405 .368770
.454125 .970182 .243602 .176644
A KSTEST for the 9 p-values yields .723713
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: THE OVERLAPPING 5-PERMUTATION TEST ::
:: This is the OPERM5 test. It looks at a sequence of one mill- ::
:: ion 32-bit random integers. Each set of five consecutive ::
:: integers can be in one of 120 states, for the 5! possible or- ::
:: derings of five numbers. Thus the 5th, 6th, 7th,...numbers ::
:: each provide a state. As many thousands of state transitions ::
:: are observed, cumulative counts are made of the number of ::
:: occurences of each state. Then the quadratic form in the ::
:: weak inverse of the 120x120 covariance matrix yields a test ::
:: equivalent to the likelihood ratio test that the 120 cell ::
:: counts came from the specified (asymptotically) normal dis- ::
:: tribution with the specified 120x120 covariance matrix (with ::
:: rank 99). This version uses 1,000,000 integers, twice. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
OPERM5 test for file xr232utest.dat
For a sample of 1,000,000 consecutive 5-tuples,
chisquare for 99 degrees of freedom= 65.764; p-value= .004109
OPERM5 test for file xr232utest.dat
For a sample of 1,000,000 consecutive 5-tuples,
chisquare for 99 degrees of freedom=176.297; p-value= .999997
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: This is the BINARY RANK TEST for 31x31 matrices. The leftmost ::
:: 31 bits of 31 random integers from the test sequence are used ::
:: to form a 31x31 binary matrix over the field {0,1}. The rank ::
:: is determined. That rank can be from 0 to 31, but ranks< 28 ::
:: are rare, and their counts are pooled with those for rank 28. ::
:: Ranks are found for 40,000 such random matrices and a chisqua-::
:: re test is performed on counts for ranks 31,30,29 and <=28. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Binary rank test for xr232utest.dat
Rank test for 31x31 binary matrices:
rows from leftmost 31 bits of each 32-bit integer
rank observed expected (o-e)^2/e sum
28 199 211.4 .729394 .729
29 5045 5134.0 1.543204 2.273
30 23169 23103.0 .188279 2.461
31 11587 11551.5 .108948 2.570
chisquare= 2.570 for 3 d. of f.; p-value= .591656
--------------------------------------------------------------
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: This is the BINARY RANK TEST for 32x32 matrices. A random 32x ::
:: 32 binary matrix is formed, each row a 32-bit random integer. ::
:: The rank is determined. That rank can be from 0 to 32, ranks ::
:: less than 29 are rare, and their counts are pooled with those ::
:: for rank 29. Ranks are found for 40,000 such random matrices ::
:: and a chisquare test is performed on counts for ranks 32,31, ::
:: 30 and <=29. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Binary rank test for xr232utest.dat
Rank test for 32x32 binary matrices:
rows from leftmost 32 bits of each 32-bit integer
rank observed expected (o-e)^2/e sum
29 199 211.4 .729394 .729
30 5140 5134.0 .006988 .736
31 23181 23103.0 .263025 .999
32 11480 11551.5 .442863 1.442
chisquare= 1.442 for 3 d. of f.; p-value= .419230
--------------------------------------------------------------
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: This is the BINARY RANK TEST for 6x8 matrices. From each of ::
:: six random 32-bit integers from the generator under test, a ::
:: specified byte is chosen, and the resulting six bytes form a ::
:: 6x8 binary matrix whose rank is determined. That rank can be ::
:: from 0 to 6, but ranks 0,1,2,3 are rare; their counts are ::
:: pooled with those for rank 4. Ranks are found for 100,000 ::
:: random matrices, and a chi-square test is performed on ::
:: counts for ranks 6,5 and <=4. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Binary Rank Test for xr232utest.dat
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 1 to 8
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 897 944.3 2.369 2.369
r =5 21917 21743.9 1.378 3.747
r =6 77186 77311.8 .205 3.952
p=1-exp(-SUM/2)= .86139
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 2 to 9
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 900 944.3 2.078 2.078
r =5 21739 21743.9 .001 2.079
r =6 77361 77311.8 .031 2.111
p=1-exp(-SUM/2)= .65194
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 3 to 10
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 865 944.3 6.660 6.660
r =5 21826 21743.9 .310 6.970
r =6 77309 77311.8 .000 6.970
p=1-exp(-SUM/2)= .96934
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 4 to 11
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 953 944.3 .080 .080
r =5 21675 21743.9 .218 .298
r =6 77372 77311.8 .047 .345
p=1-exp(-SUM/2)= .15858
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 5 to 12
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 938 944.3 .042 .042
r =5 21828 21743.9 .325 .367
r =6 77234 77311.8 .078 .446
p=1-exp(-SUM/2)= .19973
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 6 to 13
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 886 944.3 3.600 3.600
r =5 21647 21743.9 .432 4.031
r =6 77467 77311.8 .312 4.343
p=1-exp(-SUM/2)= .88599
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 7 to 14
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 903 944.3 1.806 1.806
r =5 21628 21743.9 .618 2.424
r =6 77469 77311.8 .320 2.744
p=1-exp(-SUM/2)= .74638
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 8 to 15
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 944 944.3 .000 .000
r =5 21848 21743.9 .498 .498
r =6 77208 77311.8 .139 .638
p=1-exp(-SUM/2)= .27307
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 9 to 16
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 983 944.3 1.586 1.586
r =5 21901 21743.9 1.135 2.721
r =6 77116 77311.8 .496 3.217
p=1-exp(-SUM/2)= .79980
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 10 to 17
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 955 944.3 .121 .121
r =5 21676 21743.9 .212 .333
r =6 77369 77311.8 .042 .376
p=1-exp(-SUM/2)= .17120
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 11 to 18
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 979 944.3 1.275 1.275
r =5 21750 21743.9 .002 1.277
r =6 77271 77311.8 .022 1.298
p=1-exp(-SUM/2)= .47750
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 12 to 19
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 944 944.3 .000 .000
r =5 21502 21743.9 2.691 2.691
r =6 77554 77311.8 .759 3.450
p=1-exp(-SUM/2)= .82182
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 13 to 20
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 967 944.3 .546 .546
r =5 21439 21743.9 4.275 4.821
r =6 77594 77311.8 1.030 5.851
p=1-exp(-SUM/2)= .94636
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 14 to 21
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 933 944.3 .135 .135
r =5 21658 21743.9 .339 .475
r =6 77409 77311.8 .122 .597
p=1-exp(-SUM/2)= .25799
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 15 to 22
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 890 944.3 3.123 3.123
r =5 21885 21743.9 .916 4.038
r =6 77225 77311.8 .097 4.136
p=1-exp(-SUM/2)= .87354
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 16 to 23
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 958 944.3 .199 .199
r =5 21649 21743.9 .414 .613
r =6 77393 77311.8 .085 .698
p=1-exp(-SUM/2)= .29467
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 17 to 24
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 933 944.3 .135 .135
r =5 21810 21743.9 .201 .336
r =6 77257 77311.8 .039 .375
p=1-exp(-SUM/2)= .17099
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 18 to 25
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 872 944.3 5.536 5.536
r =5 21822 21743.9 .281 5.816
r =6 77306 77311.8 .000 5.817
p=1-exp(-SUM/2)= .94544
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 19 to 26
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 915 944.3 .909 .909
r =5 21827 21743.9 .318 1.227
r =6 77258 77311.8 .037 1.264
p=1-exp(-SUM/2)= .46854
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 20 to 27
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 951 944.3 .048 .048
r =5 21502 21743.9 2.691 2.739
r =6 77547 77311.8 .716 3.454
p=1-exp(-SUM/2)= .82220
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 21 to 28
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 993 944.3 2.511 2.511
r =5 21593 21743.9 1.047 3.559
r =6 77414 77311.8 .135 3.694
p=1-exp(-SUM/2)= .84227
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 22 to 29
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 959 944.3 .229 .229
r =5 21908 21743.9 1.238 1.467
r =6 77133 77311.8 .414 1.881
p=1-exp(-SUM/2)= .60952
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 23 to 30
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 958 944.3 .199 .199
r =5 21772 21743.9 .036 .235
r =6 77270 77311.8 .023 .258
p=1-exp(-SUM/2)= .12087
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 24 to 31
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 919 944.3 .678 .678
r =5 21705 21743.9 .070 .748
r =6 77376 77311.8 .053 .801
p=1-exp(-SUM/2)= .32995
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG xr232utest.dat
b-rank test for bits 25 to 32
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 980 944.3 1.350 1.350
r =5 21643 21743.9 .468 1.818
r =6 77377 77311.8 .055 1.873
p=1-exp(-SUM/2)= .60796
TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices
These should be 25 uniform [0,1] random variables:
.861385 .651942 .969342 .158579 .199735
.885988 .746376 .273071 .799800 .171204
.477501 .821823 .946364 .257995 .873538
.294673 .170987 .945436 .468535 .822197
.842272 .609524 .120869 .329952 .607956
brank test summary for xr232utest.dat
The KS test for those 25 supposed UNI's yields
KS p-value= .764672
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: THE BITSTREAM TEST ::
:: The file under test is viewed as a stream of bits. Call them ::
:: b1,b2,... . Consider an alphabet with two "letters", 0 and 1 ::
:: and think of the stream of bits as a succession of 20-letter ::
:: "words", overlapping. Thus the first word is b1b2...b20, the ::
:: second is b2b3...b21, and so on. The bitstream test counts ::
:: the number of missing 20-letter (20-bit) words in a string of ::
:: 2^21 overlapping 20-letter words. There are 2^20 possible 20 ::
:: letter words. For a truly random string of 2^21+19 bits, the ::
:: number of missing words j should be (very close to) normally ::
:: distributed with mean 141,909 and sigma 428. Thus ::
:: (j-141909)/428 should be a standard normal variate (z score) ::
:: that leads to a uniform [0,1) p value. The test is repeated ::
:: twenty times. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
THE OVERLAPPING 20-tuples BITSTREAM TEST, 20 BITS PER WORD, N words
This test uses N=2^21 and samples the bitstream 20 times.
No. missing words should average 141909. with sigma=428.
---------------------------------------------------------
tst no 1: 141438 missing words, -1.10 sigmas from mean, p-value= .13540
tst no 2: 141951 missing words, .10 sigmas from mean, p-value= .53878
tst no 3: 141766 missing words, -.33 sigmas from mean, p-value= .36886
tst no 4: 141277 missing words, -1.48 sigmas from mean, p-value= .06978
tst no 5: 142431 missing words, 1.22 sigmas from mean, p-value= .88855
tst no 6: 141380 missing words, -1.24 sigmas from mean, p-value= .10809
tst no 7: 141701 missing words, -.49 sigmas from mean, p-value= .31322
tst no 8: 142102 missing words, .45 sigmas from mean, p-value= .67371
tst no 9: 141322 missing words, -1.37 sigmas from mean, p-value= .08499
tst no 10: 142455 missing words, 1.27 sigmas from mean, p-value= .89883
tst no 11: 141253 missing words, -1.53 sigmas from mean, p-value= .06258
tst no 12: 141379 missing words, -1.24 sigmas from mean, p-value= .10766
tst no 13: 141544 missing words, -.85 sigmas from mean, p-value= .19667
tst no 14: 141484 missing words, -.99 sigmas from mean, p-value= .16017
tst no 15: 141587 missing words, -.75 sigmas from mean, p-value= .22569
tst no 16: 142117 missing words, .49 sigmas from mean, p-value= .68624
tst no 17: 141553 missing words, -.83 sigmas from mean, p-value= .20255
tst no 18: 141943 missing words, .08 sigmas from mean, p-value= .53135
tst no 19: 142157 missing words, .58 sigmas from mean, p-value= .71859
tst no 20: 142759 missing words, 1.99 sigmas from mean, p-value= .97644
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: The tests OPSO, OQSO and DNA ::
:: OPSO means Overlapping-Pairs-Sparse-Occupancy ::
:: The OPSO test considers 2-letter words from an alphabet of ::
:: 1024 letters. Each letter is determined by a specified ten ::
:: bits from a 32-bit integer in the sequence to be tested. OPSO ::
:: generates 2^21 (overlapping) 2-letter words (from 2^21+1 ::
:: "keystrokes") and counts the number of missing words---that ::
:: is 2-letter words which do not appear in the entire sequence. ::
:: That count should be very close to normally distributed with ::
:: mean 141,909, sigma 290. Thus (missingwrds-141909)/290 should ::
:: be a standard normal variable. The OPSO test takes 32 bits at ::
:: a time from the test file and uses a designated set of ten ::
:: consecutive bits. It then restarts the file for the next de- ::
:: signated 10 bits, and so on. ::
:: ::
:: OQSO means Overlapping-Quadruples-Sparse-Occupancy ::
:: The test OQSO is similar, except that it considers 4-letter ::
:: words from an alphabet of 32 letters, each letter determined ::
:: by a designated string of 5 consecutive bits from the test ::
:: file, elements of which are assumed 32-bit random integers. ::
:: The mean number of missing words in a sequence of 2^21 four- ::
:: letter words, (2^21+3 "keystrokes"), is again 141909, with ::
:: sigma = 295. The mean is based on theory; sigma comes from ::
:: extensive simulation. ::
:: ::
:: The DNA test considers an alphabet of 4 letters:: C,G,A,T,::
:: determined by two designated bits in the sequence of random ::
:: integers being tested. It considers 10-letter words, so that ::
:: as in OPSO and OQSO, there are 2^20 possible words, and the ::
:: mean number of missing words from a string of 2^21 (over- ::
:: lapping) 10-letter words (2^21+9 "keystrokes") is 141909. ::
:: The standard deviation sigma=339 was determined as for OQSO ::
:: by simulation. (Sigma for OPSO, 290, is the true value (to ::
:: three places), not determined by simulation. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
OPSO test for generator xr232utest.dat
Output: No. missing words (mw), equiv normal variate (z), p-value (p)
mw z p
OPSO for xr232utest.dat using bits 23 to 32 141927 .061 .5243
OPSO for xr232utest.dat using bits 22 to 31 141605 -1.049 .1470
OPSO for xr232utest.dat using bits 21 to 30 142052 .492 .6886
OPSO for xr232utest.dat using bits 20 to 29 141679 -.794 .2135
OPSO for xr232utest.dat using bits 19 to 28 141645 -.911 .1810
OPSO for xr232utest.dat using bits 18 to 27 142489 1.999 .9772
OPSO for xr232utest.dat using bits 17 to 26 142040 .451 .6739
OPSO for xr232utest.dat using bits 16 to 25 141838 -.246 .4029
OPSO for xr232utest.dat using bits 15 to 24 141459 -1.553 .0602
OPSO for xr232utest.dat using bits 14 to 23 141990 .278 .6096
OPSO for xr232utest.dat using bits 13 to 22 142391 1.661 .9516
OPSO for xr232utest.dat using bits 12 to 21 142772 2.975 .9985
OPSO for xr232utest.dat using bits 11 to 20 141771 -.477 .3167
OPSO for xr232utest.dat using bits 10 to 19 141557 -1.215 .1122
OPSO for xr232utest.dat using bits 9 to 18 142261 1.213 .8874
OPSO for xr232utest.dat using bits 8 to 17 141865 -.153 .4393
OPSO for xr232utest.dat using bits 7 to 16 141624 -.984 .1626
OPSO for xr232utest.dat using bits 6 to 15 141611 -1.029 .1518
OPSO for xr232utest.dat using bits 5 to 14 141952 .147 .5585
OPSO for xr232utest.dat using bits 4 to 13 141465 -1.532 .0627
OPSO for xr232utest.dat using bits 3 to 12 141749 -.553 .2902
OPSO for xr232utest.dat using bits 2 to 11 141813 -.332 .3699
OPSO for xr232utest.dat using bits 1 to 10 142330 1.451 .9266
OQSO test for generator xr232utest.dat
Output: No. missing words (mw), equiv normal variate (z), p-value (p)
mw z p
OQSO for xr232utest.dat using bits 28 to 32 142551 2.175 .9852
OQSO for xr232utest.dat using bits 27 to 31 141859 -.171 .4323
OQSO for xr232utest.dat using bits 26 to 30 141967 .195 .5775
OQSO for xr232utest.dat using bits 25 to 29 142229 1.084 .8607
OQSO for xr232utest.dat using bits 24 to 28 141526 -1.299 .0969
OQSO for xr232utest.dat using bits 23 to 27 142159 .846 .8013
OQSO for xr232utest.dat using bits 22 to 26 142497 1.992 .9768
OQSO for xr232utest.dat using bits 21 to 25 142129 .745 .7718
OQSO for xr232utest.dat using bits 20 to 24 142148 .809 .7908
OQSO for xr232utest.dat using bits 19 to 23 142193 .962 .8319
OQSO for xr232utest.dat using bits 18 to 22 142312 1.365 .9139
OQSO for xr232utest.dat using bits 17 to 21 141975 .223 .5881
OQSO for xr232utest.dat using bits 16 to 20 141920 .036 .5144
OQSO for xr232utest.dat using bits 15 to 19 142101 .650 .7421
OQSO for xr232utest.dat using bits 14 to 18 142509 2.033 .9790
OQSO for xr232utest.dat using bits 13 to 17 142084 .592 .7231
OQSO for xr232utest.dat using bits 12 to 16 141928 .063 .5252
OQSO for xr232utest.dat using bits 11 to 15 141784 -.425 .3355
OQSO for xr232utest.dat using bits 10 to 14 142306 1.345 .9106
OQSO for xr232utest.dat using bits 9 to 13 141395 -1.743 .0406
OQSO for xr232utest.dat using bits 8 to 12 141706 -.689 .2453
OQSO for xr232utest.dat using bits 7 to 11 141772 -.466 .3208
OQSO for xr232utest.dat using bits 6 to 10 142108 .673 .7497
OQSO for xr232utest.dat using bits 5 to 9 141742 -.567 .2853
OQSO for xr232utest.dat using bits 4 to 8 142031 .412 .6600
OQSO for xr232utest.dat using bits 3 to 7 141880 -.099 .4604
OQSO for xr232utest.dat using bits 2 to 6 141892 -.059 .4766
OQSO for xr232utest.dat using bits 1 to 5 141807 -.347 .3643
DNA test for generator xr232utest.dat
Output: No. missing words (mw), equiv normal variate (z), p-value (p)
mw z p
DNA for xr232utest.dat using bits 31 to 32 141802 -.317 .3758
DNA for xr232utest.dat using bits 30 to 31 142393 1.427 .9232
DNA for xr232utest.dat using bits 29 to 30 141436 -1.396 .0813
DNA for xr232utest.dat using bits 28 to 29 141808 -.299 .3825
DNA for xr232utest.dat using bits 27 to 28 142313 1.191 .8831
DNA for xr232utest.dat using bits 26 to 27 141785 -.367 .3569
DNA for xr232utest.dat using bits 25 to 26 142385 1.403 .9197
DNA for xr232utest.dat using bits 24 to 25 142504 1.754 .9603
DNA for xr232utest.dat using bits 23 to 24 141777 -.390 .3481
DNA for xr232utest.dat using bits 22 to 23 142056 .433 .6674
DNA for xr232utest.dat using bits 21 to 22 142188 .822 .7945
DNA for xr232utest.dat using bits 20 to 21 142236 .964 .8324
DNA for xr232utest.dat using bits 19 to 20 142271 1.067 .8570
DNA for xr232utest.dat using bits 18 to 19 141440 -1.384 .0831
DNA for xr232utest.dat using bits 17 to 18 141619 -.856 .1959
DNA for xr232utest.dat using bits 16 to 17 142164 .751 .7737
DNA for xr232utest.dat using bits 15 to 16 142698 2.326 .9900
DNA for xr232utest.dat using bits 14 to 15 141837 -.213 .4155
DNA for xr232utest.dat using bits 13 to 14 141541 -1.087 .1386
DNA for xr232utest.dat using bits 12 to 13 142163 .748 .7729
DNA for xr232utest.dat using bits 11 to 12 141847 -.184 .4271
DNA for xr232utest.dat using bits 10 to 11 141557 -1.039 .1493
DNA for xr232utest.dat using bits 9 to 10 142292 1.129 .8705
DNA for xr232utest.dat using bits 8 to 9 142067 .465 .6791
DNA for xr232utest.dat using bits 7 to 8 141198 -2.098 .0179
DNA for xr232utest.dat using bits 6 to 7 141965 .164 .5652
DNA for xr232utest.dat using bits 5 to 6 141564 -1.019 .1542
DNA for xr232utest.dat using bits 4 to 5 141947 .111 .5442
DNA for xr232utest.dat using bits 3 to 4 141694 -.635 .2627
DNA for xr232utest.dat using bits 2 to 3 142010 .297 .6168
DNA for xr232utest.dat using bits 1 to 2 141991 .241 .5952
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: This is the COUNT-THE-1's TEST on a stream of bytes. ::
:: Consider the file under test as a stream of bytes (four per ::
:: 32 bit integer). Each byte can contain from 0 to 8 1's, ::
:: with probabilities 1,8,28,56,70,56,28,8,1 over 256. Now let ::
:: the stream of bytes provide a string of overlapping 5-letter ::
:: words, each "letter" taking values A,B,C,D,E. The letters are ::
:: determined by the number of 1's in a byte:: 0,1,or 2 yield A,::
:: 3 yields B, 4 yields C, 5 yields D and 6,7 or 8 yield E. Thus ::
:: we have a monkey at a typewriter hitting five keys with vari- ::
:: ous probabilities (37,56,70,56,37 over 256). There are 5^5 ::
:: possible 5-letter words, and from a string of 256,000 (over- ::
:: lapping) 5-letter words, counts are made on the frequencies ::
:: for each word. The quadratic form in the weak inverse of ::
:: the covariance matrix of the cell counts provides a chisquare ::
:: test:: Q5-Q4, the difference of the naive Pearson sums of ::
:: (OBS-EXP)^2/EXP on counts for 5- and 4-letter cell counts. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Test results for xr232utest.dat
Chi-square with 5^5-5^4=2500 d.of f. for sample size:2560000
chisquare equiv normal p-value
Results fo COUNT-THE-1's in successive bytes:
byte stream for xr232utest.dat 2564.81 .916 .820295
byte stream for xr232utest.dat 2551.57 .729 .767097
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: This is the COUNT-THE-1's TEST for specific bytes. ::
:: Consider the file under test as a stream of 32-bit integers. ::
:: From each integer, a specific byte is chosen , say the left- ::
:: most:: bits 1 to 8. Each byte can contain from 0 to 8 1's, ::
:: with probabilitie 1,8,28,56,70,56,28,8,1 over 256. Now let ::
:: the specified bytes from successive integers provide a string ::
:: of (overlapping) 5-letter words, each "letter" taking values ::
:: A,B,C,D,E. The letters are determined by the number of 1's, ::
:: in that byte:: 0,1,or 2 ---> A, 3 ---> B, 4 ---> C, 5 ---> D,::
:: and 6,7 or 8 ---> E. Thus we have a monkey at a typewriter ::
:: hitting five keys with with various probabilities:: 37,56,70,::
:: 56,37 over 256. There are 5^5 possible 5-letter words, and ::
:: from a string of 256,000 (overlapping) 5-letter words, counts ::
:: are made on the frequencies for each word. The quadratic form ::
:: in the weak inverse of the covariance matrix of the cell ::
:: counts provides a chisquare test:: Q5-Q4, the difference of ::
:: the naive Pearson sums of (OBS-EXP)^2/EXP on counts for 5- ::
:: and 4-letter cell counts. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Chi-square with 5^5-5^4=2500 d.of f. for sample size: 256000
chisquare equiv normal p value
Results for COUNT-THE-1's in specified bytes:
bits 1 to 8 2415.77 -1.191 .116779
bits 2 to 9 2626.16 1.784 .962803
bits 3 to 10 2426.79 -1.035 .150238
bits 4 to 11 2513.20 .187 .574060
bits 5 to 12 2586.91 1.229 .890482
bits 6 to 13 2554.41 .770 .779211
bits 7 to 14 2509.58 .136 .553913
bits 8 to 15 2633.35 1.886 .970346
bits 9 to 16 2522.82 .323 .626567
bits 10 to 17 2428.48 -1.011 .155917
bits 11 to 18 2522.76 .322 .626233
bits 12 to 19 2443.60 -.798 .212529
bits 13 to 20 2619.81 1.694 .954902
bits 14 to 21 2497.76 -.032 .487373
bits 15 to 22 2545.08 .638 .738110
bits 16 to 23 2561.51 .870 .807815
bits 17 to 24 2592.04 1.302 .903484
bits 18 to 25 2544.43 .628 .735087
bits 19 to 26 2434.24 -.930 .176203
bits 20 to 27 2382.40 -1.663 .048148
bits 21 to 28 2560.52 .856 .803986
bits 22 to 29 2461.32 -.547 .292179
bits 23 to 30 2463.13 -.521 .301018
bits 24 to 31 2469.44 -.432 .332807
bits 25 to 32 2526.70 .378 .647144
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: THIS IS A PARKING LOT TEST ::
:: In a square of side 100, randomly "park" a car---a circle of ::
:: radius 1. Then try to park a 2nd, a 3rd, and so on, each ::
:: time parking "by ear". That is, if an attempt to park a car ::
:: causes a crash with one already parked, try again at a new ::
:: random location. (To avoid path problems, consider parking ::
:: helicopters rather than cars.) Each attempt leads to either ::
:: a crash or a success, the latter followed by an increment to ::
:: the list of cars already parked. If we plot n: the number of ::
:: attempts, versus k:: the number successfully parked, we get a::
:: curve that should be similar to those provided by a perfect ::
:: random number generator. Theory for the behavior of such a ::
:: random curve seems beyond reach, and as graphics displays are ::
:: not available for this battery of tests, a simple characteriz ::
:: ation of the random experiment is used: k, the number of cars ::
:: successfully parked after n=12,000 attempts. Simulation shows ::
:: that k should average 3523 with sigma 21.9 and is very close ::
:: to normally distributed. Thus (k-3523)/21.9 should be a st- ::
:: andard normal variable, which, converted to a uniform varia- ::
:: ble, provides input to a KSTEST based on a sample of 10. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
CDPARK: result of ten tests on file xr232utest.dat
Of 12,000 tries, the average no. of successes
should be 3523 with sigma=21.9
Successes: 3552 z-score: 1.324 p-value: .907282
Successes: 3476 z-score: -2.146 p-value: .015932
Successes: 3522 z-score: -.046 p-value: .481790
Successes: 3515 z-score: -.365 p-value: .357445
Successes: 3500 z-score: -1.050 p-value: .146807
Successes: 3574 z-score: 2.329 p-value: .990064
Successes: 3566 z-score: 1.963 p-value: .975204
Successes: 3565 z-score: 1.918 p-value: .972432
Successes: 3545 z-score: 1.005 p-value: .842447
Successes: 3486 z-score: -1.689 p-value: .045562
square size avg. no. parked sample sigma
100. 3530.100 33.472
KSTEST for the above 10: p= .946442
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: THE MINIMUM DISTANCE TEST ::
:: It does this 100 times:: choose n=8000 random points in a ::
:: square of side 10000. Find d, the minimum distance between ::
:: the (n^2-n)/2 pairs of points. If the points are truly inde- ::
:: pendent uniform, then d^2, the square of the minimum distance ::
:: should be (very close to) exponentially distributed with mean ::
:: .995 . Thus 1-exp(-d^2/.995) should be uniform on [0,1) and ::
:: a KSTEST on the resulting 100 values serves as a test of uni- ::
:: formity for random points in the square. Test numbers=0 mod 5 ::
:: are printed but the KSTEST is based on the full set of 100 ::
:: random choices of 8000 points in the 10000x10000 square. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
This is the MINIMUM DISTANCE test
for random integers in the file xr232utest.dat
Sample no. d^2 avg equiv uni
5 .2954 1.1600 .256888
10 .5895 .7988 .447043
15 .0728 .8092 .070522
20 .6374 .8280 .473021
25 .7586 .8089 .533449
30 1.2466 .8598 .714308
35 .5947 .8664 .449922
40 3.2424 .9794 .961562
45 .7559 1.0153 .532194
50 .0296 .9858 .029334
55 .2579 .9335 .228337
60 .0547 .9774 .053494
65 .7562 .9855 .532311
70 .9705 .9743 .622949
75 .2959 .9302 .257257
80 .7805 .9427 .543615
85 .7496 .9083 .529215
90 3.2456 .9242 .961686
95 .1177 .9122 .111581
100 1.4335 .9507 .763237
MINIMUM DISTANCE TEST for xr232utest.dat
Result of KS test on 20 transformed mindist^2's:
p-value= .153708
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: THE 3DSPHERES TEST ::
:: Choose 4000 random points in a cube of edge 1000. At each ::
:: point, center a sphere large enough to reach the next closest ::
:: point. Then the volume of the smallest such sphere is (very ::
:: close to) exponentially distributed with mean 120pi/3. Thus ::
:: the radius cubed is exponential with mean 30. (The mean is ::
:: obtained by extensive simulation). The 3DSPHERES test gener- ::
:: ates 4000 such spheres 20 times. Each min radius cubed leads ::
:: to a uniform variable by means of 1-exp(-r^3/30.), then a ::
:: KSTEST is done on the 20 p-values. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
The 3DSPHERES test for file xr232utest.dat
sample no: 1 r^3= 2.755 p-value= .08774
sample no: 2 r^3= 11.907 p-value= .32759
sample no: 3 r^3= 7.062 p-value= .20975
sample no: 4 r^3= 8.434 p-value= .24508
sample no: 5 r^3= 19.784 p-value= .48288
sample no: 6 r^3= 14.727 p-value= .38792
sample no: 7 r^3= 7.071 p-value= .20999
sample no: 8 r^3= 20.417 p-value= .49366
sample no: 9 r^3= 5.766 p-value= .17485
sample no: 10 r^3= 28.923 p-value= .61867
sample no: 11 r^3= 8.065 p-value= .23572
sample no: 12 r^3= 23.795 p-value= .54759
sample no: 13 r^3= 2.627 p-value= .08385
sample no: 14 r^3= 17.576 p-value= .44337
sample no: 15 r^3= 38.318 p-value= .72120
sample no: 16 r^3= 27.848 p-value= .60476
sample no: 17 r^3= 41.159 p-value= .74640
sample no: 18 r^3= 41.454 p-value= .74888
sample no: 19 r^3= 64.838 p-value= .88482
sample no: 20 r^3= 16.370 p-value= .42055
A KS test is applied to those 20 p-values.
---------------------------------------------------------
3DSPHERES test for file xr232utest.dat p-value= .672302
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: This is the SQEEZE test ::
:: Random integers are floated to get uniforms on [0,1). Start- ::
:: ing with k=2^31=2147483647, the test finds j, the number of ::
:: iterations necessary to reduce k to 1, using the reduction ::
:: k=ceiling(k*U), with U provided by floating integers from ::
:: the file being tested. Such j's are found 100,000 times, ::
:: then counts for the number of times j was <=6,7,...,47,>=48 ::
:: are used to provide a chi-square test for cell frequencies. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
RESULTS OF SQUEEZE TEST FOR xr232utest.dat
Table of standardized frequency counts
( (obs-exp)/sqrt(exp) )^2
for j taking values <=6,7,8,...,47,>=48:
1.3 -.3 .6 .3 -1.2 -.7
-.9 -1.0 .3 .1 -.6 2.1
.5 -1.1 -.7 .2 .7 -1.1
.8 .5 .4 -.5 -.7 .5
.6 -.5 .0 -.9 .0 -.3
-.3 1.3 -.4 -.6 1.0 -.5
.0 -.4 1.3 -1.3 .1 -1.0
.8
Chi-square with 42 degrees of freedom: 27.105
z-score= -1.625 p-value= .036355
______________________________________________________________
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: The OVERLAPPING SUMS test ::
:: Integers are floated to get a sequence U(1),U(2),... of uni- ::
:: form [0,1) variables. Then overlapping sums, ::
:: S(1)=U(1)+...+U(100), S2=U(2)+...+U(101),... are formed. ::
:: The S's are virtually normal with a certain covariance mat- ::
:: rix. A linear transformation of the S's converts them to a ::
:: sequence of independent standard normals, which are converted ::
:: to uniform variables for a KSTEST. The p-values from ten ::
:: KSTESTs are given still another KSTEST. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Test no. 1 p-value .162178
Test no. 2 p-value .214245
Test no. 3 p-value .297837
Test no. 4 p-value .861613
Test no. 5 p-value .016242
Test no. 6 p-value .608891
Test no. 7 p-value .963947
Test no. 8 p-value .599817
Test no. 9 p-value .988628
Test no. 10 p-value .761837
Results of the OSUM test for xr232utest.dat
KSTEST on the above 10 p-values: .429538
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: This is the RUNS test. It counts runs up, and runs down, ::
:: in a sequence of uniform [0,1) variables, obtained by float- ::
:: ing the 32-bit integers in the specified file. This example ::
:: shows how runs are counted: .123,.357,.789,.425,.224,.416,.95::
:: contains an up-run of length 3, a down-run of length 2 and an ::
:: up-run of (at least) 2, depending on the next values. The ::
:: covariance matrices for the runs-up and runs-down are well ::
:: known, leading to chisquare tests for quadratic forms in the ::
:: weak inverses of the covariance matrices. Runs are counted ::
:: for sequences of length 10,000. This is done ten times. Then ::
:: repeated. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
The RUNS test for file xr232utest.dat
Up and down runs in a sample of 10000
_________________________________________________
Run test for xr232utest.dat :
runs up; ks test for 10 p's: .335159
runs down; ks test for 10 p's: .115278
Run test for xr232utest.dat :
runs up; ks test for 10 p's: .399468
runs down; ks test for 10 p's: .987244
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: This is the CRAPS TEST. It plays 200,000 games of craps, finds::
:: the number of wins and the number of throws necessary to end ::
:: each game. The number of wins should be (very close to) a ::
:: normal with mean 200000p and variance 200000p(1-p), with ::
:: p=244/495. Throws necessary to complete the game can vary ::
:: from 1 to infinity, but counts for all>21 are lumped with 21. ::
:: A chi-square test is made on the no.-of-throws cell counts. ::
:: Each 32-bit integer from the test file provides the value for ::
:: the throw of a die, by floating to [0,1), multiplying by 6 ::
:: and taking 1 plus the integer part of the result. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Results of craps test for xr232utest.dat
No. of wins: Observed Expected
98705 98585.86
98705= No. of wins, z-score= .533 pvalue= .70294
Analysis of Throws-per-Game:
Chisq= 12.52 for 20 degrees of freedom, p= .10291
Throws Observed Expected Chisq Sum
1 66827 66666.7 .386 .386
2 37760 37654.3 .297 .682
3 27020 26954.7 .158 .840
4 19168 19313.5 1.096 1.936
5 13734 13851.4 .995 2.931
6 9987 9943.5 .190 3.121
7 7191 7145.0 .296 3.417
8 5107 5139.1 .200 3.617
9 3610 3699.9 2.183 5.800
10 2671 2666.3 .008 5.808
11 1965 1923.3 .903 6.711
12 1373 1388.7 .178 6.889
13 952 1003.7 2.665 9.554
14 710 726.1 .359 9.913
15 513 525.8 .313 10.226
16 383 381.2 .009 10.235
17 268 276.5 .264 10.499
18 203 200.8 .023 10.522
19 163 146.0 1.983 12.505
20 106 106.2 .000 12.506
21 289 287.1 .012 12.518
SUMMARY FOR xr232utest.dat
p-value for no. of wins: .702937
p-value for throws/game: .102910
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Results of DIEHARD battery of tests sent to file xr232utest.txt