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Plain Text, pasted on Feb 1:
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       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 bbs.32         
                   For a sample of size 500:     mean   
           bbs.32          using bits  1 to 24   1.974
  duplicate       number       number 
  spacings       observed     expected
        0          74.       67.668
        1         145.      135.335
        2         116.      135.335
        3          90.       90.224
        4          49.       45.112
        5          19.       18.045
  6 to INF          7.        8.282
 Chisquare with  6 d.o.f. =     4.63 p-value=  .407914
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  2 to 25   2.006
  duplicate       number       number 
  spacings       observed     expected
        0          62.       67.668
        1         125.      135.335
        2         144.      135.335
        3         115.       90.224
        4          34.       45.112
        5          14.       18.045
  6 to INF          6.        8.282
 Chisquare with  6 d.o.f. =    12.89 p-value=  .955266
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  3 to 26   1.952
  duplicate       number       number 
  spacings       observed     expected
        0          74.       67.668
        1         139.      135.335
        2         130.      135.335
        3          91.       90.224
        4          43.       45.112
        5          13.       18.045
  6 to INF         10.        8.282
 Chisquare with  6 d.o.f. =     2.77 p-value=  .163428
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  4 to 27   2.018
  duplicate       number       number 
  spacings       observed     expected
        0          60.       67.668
        1         154.      135.335
        2         126.      135.335
        3          82.       90.224
        4          50.       45.112
        5          17.       18.045
  6 to INF         11.        8.282
 Chisquare with  6 d.o.f. =     6.32 p-value=  .611556
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  5 to 28   1.894
  duplicate       number       number 
  spacings       observed     expected
        0          78.       67.668
        1         146.      135.335
        2         129.      135.335
        3          81.       90.224
        4          40.       45.112
        5          19.       18.045
  6 to INF          7.        8.282
 Chisquare with  6 d.o.f. =     4.49 p-value=  .388760
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  6 to 29   2.098
  duplicate       number       number 
  spacings       observed     expected
        0          52.       67.668
        1         137.      135.335
        2         132.      135.335
        3         101.       90.224
        4          51.       45.112
        5          22.       18.045
  6 to INF          5.        8.282
 Chisquare with  6 d.o.f. =     7.95 p-value=  .758471
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  7 to 30   1.902
  duplicate       number       number 
  spacings       observed     expected
        0          76.       67.668
        1         139.      135.335
        2         130.      135.335
        3          95.       90.224
        4          40.       45.112
        5          13.       18.045
  6 to INF          7.        8.282
 Chisquare with  6 d.o.f. =     3.78 p-value=  .293096
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  8 to 31   2.038
  duplicate       number       number 
  spacings       observed     expected
        0          62.       67.668
        1         127.      135.335
        2         139.      135.335
        3          98.       90.224
        4          54.       45.112
        5          16.       18.045
  6 to INF          4.        8.282
 Chisquare with  6 d.o.f. =     5.95 p-value=  .571665
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  9 to 32   1.896
  duplicate       number       number 
  spacings       observed     expected
        0          80.       67.668
        1         142.      135.335
        2         128.      135.335
        3          75.       90.224
        4          52.       45.112
        5          21.       18.045
  6 to INF          2.        8.282
 Chisquare with  6 d.o.f. =    11.84 p-value=  .934427
  :::::::::::::::::::::::::::::::::::::::::
   The 9 p-values were
        .407914   .955266   .163428   .611556   .388760
        .758471   .293096   .571665   .934427
  A KSTEST for the 9 p-values yields  .237399

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     ::            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 bbs.32         
     For a sample of 1,000,000 consecutive 5-tuples,
 chisquare for 99 degrees of freedom=109.577; p-value= .780459
           OPERM5 test for file bbs.32         
     For a sample of 1,000,000 consecutive 5-tuples,
 chisquare for 99 degrees of freedom= 88.425; p-value= .231897
     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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 bbs.32         
         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       202     211.4   .419543     .420
        29      5040    5134.0  1.721447    2.141
        30     23212   23103.0   .513819    2.655
        31     11546   11551.5   .002642    2.657
  chisquare= 2.657 for 3 d. of f.; p-value= .604027
--------------------------------------------------------------
     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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 bbs.32         
         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       208     211.4   .055259     .055
        30      5111    5134.0   .103130     .158
        31     23110   23103.0   .002093     .160
        32     11571   11551.5   .032835     .193
  chisquare=  .193 for 3 d. of f.; p-value= .351424
--------------------------------------------------------------

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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 bbs.32         
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits  1 to  8
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          926       944.3        .355        .355
          r =5        21903     21743.9       1.164       1.519
          r =6        77171     77311.8        .256       1.775
                        p=1-exp(-SUM/2)= .58837
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits  2 to  9
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          992       944.3       2.409       2.409
          r =5        21812     21743.9        .213       2.623
          r =6        77196     77311.8        .173       2.796
                        p=1-exp(-SUM/2)= .75292
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits  3 to 10
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          905       944.3       1.636       1.636
          r =5        21838     21743.9        .407       2.043
          r =6        77257     77311.8        .039       2.082
                        p=1-exp(-SUM/2)= .64686
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits  4 to 11
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          961       944.3        .295        .295
          r =5        21595     21743.9       1.020       1.315
          r =6        77444     77311.8        .226       1.541
                        p=1-exp(-SUM/2)= .53722
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits  5 to 12
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          947       944.3        .008        .008
          r =5        21564     21743.9       1.488       1.496
          r =6        77489     77311.8        .406       1.902
                        p=1-exp(-SUM/2)= .61370
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits  6 to 13
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          928       944.3        .281        .281
          r =5        22065     21743.9       4.742       5.023
          r =6        77007     77311.8       1.202       6.225
                        p=1-exp(-SUM/2)= .95551
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits  7 to 14
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4         1002       944.3       3.526       3.526
          r =5        21898     21743.9       1.092       4.618
          r =6        77100     77311.8        .580       5.198
                        p=1-exp(-SUM/2)= .92565
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits  8 to 15
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          919       944.3        .678        .678
          r =5        21896     21743.9       1.064       1.742
          r =6        77185     77311.8        .208       1.950
                        p=1-exp(-SUM/2)= .62278
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits  9 to 16
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          950       944.3        .034        .034
          r =5        21874     21743.9        .778        .813
          r =6        77176     77311.8        .239       1.051
                        p=1-exp(-SUM/2)= .40885
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 10 to 17
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          943       944.3        .002        .002
          r =5        21654     21743.9        .372        .373
          r =6        77403     77311.8        .108        .481
                        p=1-exp(-SUM/2)= .21379
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 11 to 18
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          958       944.3        .199        .199
          r =5        21628     21743.9        .618        .817
          r =6        77414     77311.8        .135        .952
                        p=1-exp(-SUM/2)= .37861
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 12 to 19
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          936       944.3        .073        .073
          r =5        21664     21743.9        .294        .367
          r =6        77400     77311.8        .101        .467
                        p=1-exp(-SUM/2)= .20832
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 13 to 20
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          924       944.3        .436        .436
          r =5        21650     21743.9        .406        .842
          r =6        77426     77311.8        .169       1.011
                        p=1-exp(-SUM/2)= .39669
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 14 to 21
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          952       944.3        .063        .063
          r =5        21623     21743.9        .672        .735
          r =6        77425     77311.8        .166        .901
                        p=1-exp(-SUM/2)= .36260
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 15 to 22
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          917       944.3        .789        .789
          r =5        21920     21743.9       1.426       2.216
          r =6        77163     77311.8        .286       2.502
                        p=1-exp(-SUM/2)= .71377
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 16 to 23
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          935       944.3        .092        .092
          r =5        22038     21743.9       3.978       4.069
          r =6        77027     77311.8       1.049       5.119
                        p=1-exp(-SUM/2)= .92264
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 17 to 24
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          946       944.3        .003        .003
          r =5        22047     21743.9       4.225       4.228
          r =6        77007     77311.8       1.202       5.430
                        p=1-exp(-SUM/2)= .93379
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 18 to 25
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4         1000       944.3       3.285       3.285
          r =5        21618     21743.9        .729       4.014
          r =6        77382     77311.8        .064       4.078
                        p=1-exp(-SUM/2)= .86984
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 19 to 26
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          984       944.3       1.669       1.669
          r =5        21540     21743.9       1.912       3.581
          r =6        77476     77311.8        .349       3.930
                        p=1-exp(-SUM/2)= .85982
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 20 to 27
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          950       944.3        .034        .034
          r =5        21864     21743.9        .663        .698
          r =6        77186     77311.8        .205        .902
                        p=1-exp(-SUM/2)= .36316
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 21 to 28
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          949       944.3        .023        .023
          r =5        21781     21743.9        .063        .087
          r =6        77270     77311.8        .023        .109
                        p=1-exp(-SUM/2)= .05318
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 22 to 29
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          962       944.3        .332        .332
          r =5        21859     21743.9        .609        .941
          r =6        77179     77311.8        .228       1.169
                        p=1-exp(-SUM/2)= .44265
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 23 to 30
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4         1031       944.3       7.960       7.960
          r =5        21756     21743.9        .007       7.967
          r =6        77213     77311.8        .126       8.093
                        p=1-exp(-SUM/2)= .98252
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 24 to 31
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          949       944.3        .023        .023
          r =5        21858     21743.9        .599        .622
          r =6        77193     77311.8        .183        .805
                        p=1-exp(-SUM/2)= .33125
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG bbs.32         
     b-rank test for bits 25 to 32
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          954       944.3        .100        .100
          r =5        21757     21743.9        .008        .108
          r =6        77289     77311.8        .007        .114
                        p=1-exp(-SUM/2)= .05552
   TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices
 These should be 25 uniform [0,1] random variables:
     .588370     .752923     .646859     .537217     .613696
     .955508     .925648     .622777     .408848     .213789
     .378609     .208317     .396685     .362605     .713771
     .922644     .933789     .869845     .859824     .363156
     .053176     .442649     .982517     .331246     .055516
   brank test summary for bbs.32         
       The KS test for those 25 supposed UNI's yields
                    KS p-value= .615427

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     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     ::                   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:  141553 missing words,    -.83 sigmas from mean, p-value= .20255
 tst no  2:  141650 missing words,    -.61 sigmas from mean, p-value= .27229
 tst no  3:  141811 missing words,    -.23 sigmas from mean, p-value= .40915
 tst no  4:  141774 missing words,    -.32 sigmas from mean, p-value= .37593
 tst no  5:  141923 missing words,     .03 sigmas from mean, p-value= .51274
 tst no  6:  141900 missing words,    -.02 sigmas from mean, p-value= .49131
 tst no  7:  142267 missing words,     .84 sigmas from mean, p-value= .79833
 tst no  8:  141612 missing words,    -.69 sigmas from mean, p-value= .24362
 tst no  9:  142282 missing words,     .87 sigmas from mean, p-value= .80805
 tst no 10:  141521 missing words,    -.91 sigmas from mean, p-value= .18212
 tst no 11:  141486 missing words,    -.99 sigmas from mean, p-value= .16131
 tst no 12:  141330 missing words,   -1.35 sigmas from mean, p-value= .08794
 tst no 13:  141517 missing words,    -.92 sigmas from mean, p-value= .17966
 tst no 14:  141255 missing words,   -1.53 sigmas from mean, p-value= .06316
 tst no 15:  141580 missing words,    -.77 sigmas from mean, p-value= .22081
 tst no 16:  141655 missing words,    -.59 sigmas from mean, p-value= .27618
 tst no 17:  142142 missing words,     .54 sigmas from mean, p-value= .70665
 tst no 18:  141852 missing words,    -.13 sigmas from mean, p-value= .44672
 tst no 19:  142621 missing words,    1.66 sigmas from mean, p-value= .95182
 tst no 20:  141397 missing words,   -1.20 sigmas from mean, p-value= .11565

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     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     ::             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 bbs.32         
  Output: No. missing words (mw), equiv normal variate (z), p-value (p)
                                                           mw     z     p
    OPSO for bbs.32          using bits 23 to 32        786526******* 1.0000
    OPSO for bbs.32          using bits 22 to 31        786515******* 1.0000
    OPSO for bbs.32          using bits 21 to 30        786497******* 1.0000
    OPSO for bbs.32          using bits 20 to 29        786510******* 1.0000
    OPSO for bbs.32          using bits 19 to 28        786502******* 1.0000
    OPSO for bbs.32          using bits 18 to 27        786517******* 1.0000
    OPSO for bbs.32          using bits 17 to 26        786514******* 1.0000
    OPSO for bbs.32          using bits 16 to 25        696403******* 1.0000
    OPSO for bbs.32          using bits 15 to 24        670688******* 1.0000
    OPSO for bbs.32          using bits 14 to 23        786516******* 1.0000
    OPSO for bbs.32          using bits 13 to 22        786525******* 1.0000
    OPSO for bbs.32          using bits 12 to 21        786511******* 1.0000
    OPSO for bbs.32          using bits 11 to 20        786520******* 1.0000
    OPSO for bbs.32          using bits 10 to 19        786530******* 1.0000
    OPSO for bbs.32          using bits  9 to 18        786525******* 1.0000
    OPSO for bbs.32          using bits  8 to 17        533939******* 1.0000
    OPSO for bbs.32          using bits  7 to 16        142025   .399  .6550
    OPSO for bbs.32          using bits  6 to 15        141952   .147  .5585
    OPSO for bbs.32          using bits  5 to 14        142731  2.833  .9977
    OPSO for bbs.32          using bits  4 to 13        144030  7.313 1.0000
    OPSO for bbs.32          using bits  3 to 12        146989 17.516 1.0000
    OPSO for bbs.32          using bits  2 to 11        147959 20.861 1.0000
    OPSO for bbs.32          using bits  1 to 10        146192 14.768 1.0000
 OQSO test for generator bbs.32         
  Output: No. missing words (mw), equiv normal variate (z), p-value (p)
                                                           mw     z     p
    OQSO for bbs.32          using bits 28 to 32        141415 -1.676  .0469
    OQSO for bbs.32          using bits 27 to 31        141394 -1.747  .0403
    OQSO for bbs.32          using bits 26 to 30        141699  -.713  .2379
    OQSO for bbs.32          using bits 25 to 29        142150   .816  .7927
    OQSO for bbs.32          using bits 24 to 28        152494 35.880 1.0000
    OQSO for bbs.32          using bits 23 to 27        157435 52.629 1.0000
    OQSO for bbs.32          using bits 22 to 26        154070 41.223 1.0000
    OQSO for bbs.32          using bits 21 to 25        149920 27.155 1.0000
    OQSO for bbs.32          using bits 20 to 24        146468 15.453 1.0000
    OQSO for bbs.32          using bits 19 to 23        148650 22.850 1.0000
    OQSO for bbs.32          using bits 18 to 22        151480 32.443 1.0000
    OQSO for bbs.32          using bits 17 to 21        151068 31.046 1.0000
    OQSO for bbs.32          using bits 16 to 20        149082 24.314 1.0000
    OQSO for bbs.32          using bits 15 to 19        148051 20.819 1.0000
    OQSO for bbs.32          using bits 14 to 18        146428 15.318 1.0000
    OQSO for bbs.32          using bits 13 to 17        145575 12.426 1.0000
    OQSO for bbs.32          using bits 12 to 16        143695  6.053 1.0000
    OQSO for bbs.32          using bits 11 to 15        144124  7.507 1.0000
    OQSO for bbs.32          using bits 10 to 14        146627 15.992 1.0000
    OQSO for bbs.32          using bits  9 to 13        144878 10.063 1.0000
    OQSO for bbs.32          using bits  8 to 12        143323  4.792 1.0000
    OQSO for bbs.32          using bits  7 to 11        143096  4.023 1.0000
    OQSO for bbs.32          using bits  6 to 10        142512  2.043  .9795
    OQSO for bbs.32          using bits  5 to  9        142174   .897  .8152
    OQSO for bbs.32          using bits  4 to  8        142061   .514  .6964
    OQSO for bbs.32          using bits  3 to  7        141386 -1.774  .0380
    OQSO for bbs.32          using bits  2 to  6        142256  1.175  .8800
    OQSO for bbs.32          using bits  1 to  5        142334  1.440  .9250
  DNA test for generator bbs.32         
  Output: No. missing words (mw), equiv normal variate (z), p-value (p)
                                                           mw     z     p
     DNA for bbs.32          using bits 31 to 32        144116  6.509 1.0000
     DNA for bbs.32          using bits 30 to 31        145360 10.179 1.0000
     DNA for bbs.32          using bits 29 to 30        150735 26.034 1.0000
     DNA for bbs.32          using bits 28 to 29        143966  6.067 1.0000
     DNA for bbs.32          using bits 27 to 28        144761  8.412 1.0000
     DNA for bbs.32          using bits 26 to 27        147121 15.374 1.0000
     DNA for bbs.32          using bits 25 to 26        152718 31.884 1.0000
     DNA for bbs.32          using bits 24 to 25        148152 18.415 1.0000
     DNA for bbs.32          using bits 23 to 24        143681  5.226 1.0000
     DNA for bbs.32          using bits 22 to 23        143434  4.498 1.0000
     DNA for bbs.32          using bits 21 to 22        147139 15.427 1.0000
     DNA for bbs.32          using bits 20 to 21        143406  4.415 1.0000
     DNA for bbs.32          using bits 19 to 20        142923  2.990  .9986
     DNA for bbs.32          using bits 18 to 19        143632  5.082 1.0000
     DNA for bbs.32          using bits 17 to 18        145788 11.442 1.0000
     DNA for bbs.32          using bits 16 to 17        146503 13.551 1.0000
     DNA for bbs.32          using bits 15 to 16        142548  1.884  .9702
     DNA for bbs.32          using bits 14 to 15        142787  2.589  .9952
     DNA for bbs.32          using bits 13 to 14        145713 11.220 1.0000
     DNA for bbs.32          using bits 12 to 13        143727  5.362 1.0000
     DNA for bbs.32          using bits 11 to 12        143096  3.501  .9998
     DNA for bbs.32          using bits 10 to 11        143962  6.055 1.0000
     DNA for bbs.32          using bits  9 to 10        144058  6.338 1.0000
     DNA for bbs.32          using bits  8 to  9        142138   .675  .7500
     DNA for bbs.32          using bits  7 to  8        141548 -1.066  .1432
     DNA for bbs.32          using bits  6 to  7        142377  1.380  .9161
     DNA for bbs.32          using bits  5 to  6        141811  -.290  .3859
     DNA for bbs.32          using bits  4 to  5        142150   .710  .7611
     DNA for bbs.32          using bits  3 to  4        141585  -.957  .1694
     DNA for bbs.32          using bits  2 to  3        142149   .707  .7602
     DNA for bbs.32          using bits  1 to  2        141678  -.682  .2475

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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 bbs.32         
 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 bbs.32         368585.00   5177.224     1.000000
 byte stream for bbs.32         367294.70   5158.977     1.000000

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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  2594.83      1.341      .910057
           bits  2 to  9  2494.66      -.076      .469895
           bits  3 to 10  2440.62      -.840      .200517
           bits  4 to 11  2457.26      -.604      .272785
           bits  5 to 12  2457.20      -.605      .272492
           bits  6 to 13  2384.35     -1.636      .050966
           bits  7 to 14  2522.20       .314      .623210
           bits  8 to 15  2539.22       .555      .710440
           bits  9 to 16  2492.49      -.106      .457716
           bits 10 to 17  2417.71     -1.164      .122258
           bits 11 to 18  2639.77      1.977      .975962
           bits 12 to 19  2529.52       .418      .661846
           bits 13 to 20  2520.49       .290      .613990
           bits 14 to 21  2388.58     -1.576      .057544
           bits 15 to 22  2473.63      -.373      .354580
           bits 16 to 23  2426.23     -1.043      .148409
           bits 17 to 24  2581.95      1.159      .876759
           bits 18 to 25  2578.96      1.117      .867918
           bits 19 to 26  2554.61       .772      .780034
           bits 20 to 27  2590.96      1.286      .900843
           bits 21 to 28  2596.06      1.359      .912856
           bits 22 to 29  2488.09      -.168      .433113
           bits 23 to 30  2484.48      -.219      .413134
           bits 24 to 31  2555.28       .782      .782834
           bits 25 to 32  2545.30       .641      .739104

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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 bbs.32         
            Of 12,000 tries, the average no. of successes
                 should be 3523 with sigma=21.9
            Successes: 3481    z-score: -1.918 p-value: .027568
            Successes: 3526    z-score:   .137 p-value: .554479
            Successes: 3545    z-score:  1.005 p-value: .842447
            Successes: 3542    z-score:   .868 p-value: .807188
            Successes: 3489    z-score: -1.553 p-value: .060270
            Successes: 3513    z-score:  -.457 p-value: .323972
            Successes: 3498    z-score: -1.142 p-value: .126820
            Successes: 3523    z-score:   .000 p-value: .500000
            Successes: 3507    z-score:  -.731 p-value: .232514
            Successes: 3546    z-score:  1.050 p-value: .853193
 
           square size   avg. no.  parked   sample sigma
             100.            3517.000       22.177
            KSTEST for the above 10: p=  .360847

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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 bbs.32         
     Sample no.    d^2     avg     equiv uni            
           5     .9378    .5297     .610370
          10     .0629    .5144     .061273
          15     .3572    .7165     .301612
          20     .7039    .6736     .507099
          25     .9253    .7817     .605439
          30    3.2093    .9323     .960261
          35     .1252    .9610     .118231
          40     .5581    .9381     .429317
          45     .7992    .9365     .552116
          50     .4488    .8936     .363062
          55     .1062   1.0378     .101271
          60    1.0445   1.0319     .649974
          65     .4247    .9740     .347416
          70     .1390    .9863     .130365
          75     .3401    .9629     .289520
          80     .2111    .9212     .191144
          85     .1768    .9454     .162821
          90     .8316    .9932     .566460
          95     .2094    .9922     .189748
         100     .1097    .9738     .104389
     MINIMUM DISTANCE TEST for bbs.32         
          Result of KS test on 20 transformed mindist^2's:
                                  p-value= .484088

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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 bbs.32         
 sample no:  1     r^3=  57.425     p-value= .85254
 sample no:  2     r^3=  78.185     p-value= .92618
 sample no:  3     r^3=  34.952     p-value= .68810
 sample no:  4     r^3=  36.078     p-value= .69959
 sample no:  5     r^3=   1.748     p-value= .05661
 sample no:  6     r^3=  27.541     p-value= .60070
 sample no:  7     r^3=  27.772     p-value= .60375
 sample no:  8     r^3=   5.883     p-value= .17806
 sample no:  9     r^3=  45.855     p-value= .78314
 sample no: 10     r^3=   6.554     p-value= .19625
 sample no: 11     r^3=  27.579     p-value= .60120
 sample no: 12     r^3=   5.811     p-value= .17611
 sample no: 13     r^3=  10.421     p-value= .29345
 sample no: 14     r^3=  15.396     p-value= .40141
 sample no: 15     r^3=   2.275     p-value= .07301
 sample no: 16     r^3=  47.912     p-value= .79751
 sample no: 17     r^3=  27.933     p-value= .60588
 sample no: 18     r^3=  46.603     p-value= .78848
 sample no: 19     r^3=  39.484     p-value= .73183
 sample no: 20     r^3=  14.490     p-value= .38308
  A KS test is applied to those 20 p-values.
---------------------------------------------------------
       3DSPHERES test for file bbs.32               p-value= .251159
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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 bbs.32         
         Table of standardized frequency counts
     ( (obs-exp)/sqrt(exp) )^2
        for j taking values <=6,7,8,...,47,>=48:
     -.1      .5     -.4     -.5     -.1     1.9
     1.2    -1.2      .0      .7     -.6     -.1
     -.3     -.8    -1.3      .2     -.7      .1
      .4     -.5     2.4      .5     1.1     -.9
      .1     -.4     -.5     1.5     -.7      .6
    -1.4      .9    -1.9      .8      .1     -.7
     -.5      .5      .1     -.1     -.6      .0
      .8
           Chi-square with 42 degrees of freedom: 32.204
              z-score= -1.069  p-value= .137417
______________________________________________________________

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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  .856594
                Test no.  2      p-value  .181666
                Test no.  3      p-value  .294062
                Test no.  4      p-value  .883345
                Test no.  5      p-value  .686900
                Test no.  6      p-value  .955357
                Test no.  7      p-value  .301129
                Test no.  8      p-value  .666933
                Test no.  9      p-value  .681837
                Test no. 10      p-value  .976020
   Results of the OSUM test for bbs.32         
        KSTEST on the above 10 p-values:  .860299

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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 bbs.32         
     Up and down runs in a sample of 10000
_________________________________________________ 
                 Run test for bbs.32         :
       runs up; ks test for 10 p's: .168184
     runs down; ks test for 10 p's: .480434
                 Run test for bbs.32         :
       runs up; ks test for 10 p's: .801154
     runs down; ks test for 10 p's: .705166

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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 bbs.32         
  No. of wins:  Observed Expected
                                98663    98585.86
                  98663= No. of wins, z-score=  .345 pvalue= .63496
   Analysis of Throws-per-Game:
 Chisq=  21.71 for 20 degrees of freedom, p=  .64328
               Throws Observed Expected  Chisq     Sum
                  1    66613    66666.7    .043     .043
                  2    37548    37654.3    .300     .343
                  3    26994    26954.7    .057     .401
                  4    19149    19313.5   1.400    1.801
                  5    13995    13851.4   1.488    3.289
                  6     9964     9943.5    .042    3.332
                  7     7180     7145.0    .171    3.503
                  8     5214     5139.1   1.092    4.595
                  9     3744     3699.9    .526    5.122
                 10     2615     2666.3    .987    6.109
                 11     2025     1923.3   5.375   11.483
                 12     1368     1388.7    .310   11.793
                 13      932     1003.7   5.124   16.917
                 14      715      726.1    .171   17.088
                 15      554      525.8   1.509   18.596
                 16      382      381.2    .002   18.598
                 17      253      276.5   2.004   20.602
                 18      196      200.8    .116   20.718
                 19      146      146.0    .000   20.718
                 20      111      106.2    .216   20.934
                 21      302      287.1    .772   21.705
            SUMMARY  FOR bbs.32         
                p-value for no. of wins: .634960
                p-value for throws/game: .643281

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 Results of DIEHARD battery of tests sent to file bbs.out3       



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