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Plain Text, pasted on Jan 25:
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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   2.068
  duplicate       number       number 
  spacings       observed     expected
        0          62.       67.668
        1         120.      135.335
        2         140.      135.335
        3         112.       90.224
        4          40.       45.112
        5          22.       18.045
  6 to INF          4.        8.282
 Chisquare with  6 d.o.f. =    11.29 p-value=  .920159
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  2 to 25   1.886
  duplicate       number       number 
  spacings       observed     expected
        0          70.       67.668
        1         146.      135.335
        2         145.      135.335
        3          77.       90.224
        4          42.       45.112
        5          14.       18.045
  6 to INF          6.        8.282
 Chisquare with  6 d.o.f. =     5.30 p-value=  .493928
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  3 to 26   1.908
  duplicate       number       number 
  spacings       observed     expected
        0          72.       67.668
        1         133.      135.335
        2         151.      135.335
        3          85.       90.224
        4          40.       45.112
        5          12.       18.045
  6 to INF          7.        8.282
 Chisquare with  6 d.o.f. =     5.24 p-value=  .486048
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  4 to 27   2.022
  duplicate       number       number 
  spacings       observed     expected
        0          60.       67.668
        1         137.      135.335
        2         139.      135.335
        3          91.       90.224
        4          47.       45.112
        5          22.       18.045
  6 to INF          4.        8.282
 Chisquare with  6 d.o.f. =     4.16 p-value=  .344292
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  5 to 28   1.892
  duplicate       number       number 
  spacings       observed     expected
        0          73.       67.668
        1         149.      135.335
        2         133.      135.335
        3          78.       90.224
        4          46.       45.112
        5          15.       18.045
  6 to INF          6.        8.282
 Chisquare with  6 d.o.f. =     4.66 p-value=  .411396
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  6 to 29   1.866
  duplicate       number       number 
  spacings       observed     expected
        0          74.       67.668
        1         139.      135.335
        2         148.      135.335
        3          83.       90.224
        4          39.       45.112
        5          10.       18.045
  6 to INF          7.        8.282
 Chisquare with  6 d.o.f. =     7.07 p-value=  .685417
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  7 to 30   1.908
  duplicate       number       number 
  spacings       observed     expected
        0          76.       67.668
        1         144.      135.335
        2         135.      135.335
        3          75.       90.224
        4          46.       45.112
        5          18.       18.045
  6 to INF          6.        8.282
 Chisquare with  6 d.o.f. =     4.80 p-value=  .429843
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  8 to 31   1.974
  duplicate       number       number 
  spacings       observed     expected
        0          84.       67.668
        1         129.      135.335
        2         121.      135.335
        3          88.       90.224
        4          53.       45.112
        5          14.       18.045
  6 to INF         11.        8.282
 Chisquare with  6 d.o.f. =     8.99 p-value=  .825854
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  9 to 32   2.050
  duplicate       number       number 
  spacings       observed     expected
        0          71.       67.668
        1         118.      135.335
        2         137.      135.335
        3          99.       90.224
        4          50.       45.112
        5          18.       18.045
  6 to INF          7.        8.282
 Chisquare with  6 d.o.f. =     3.99 p-value=  .321564
  :::::::::::::::::::::::::::::::::::::::::
   The 9 p-values were
        .920159   .493928   .486048   .344292   .411396
        .685417   .429843   .825854   .321564
  A KSTEST for the 9 p-values yields  .537902

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

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     ::            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=218.733; p-value=1.000000
           OPERM5 test for file bbs.32         
     For a sample of 1,000,000 consecutive 5-tuples,
 chisquare for 99 degrees of freedom=245.401; p-value=1.000000
     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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       197     211.4   .983261     .983
        29      5159    5134.0   .121637    1.105
        30     23059   23103.0   .083977    1.189
        31     11585   11551.5   .097010    1.286
  chisquare= 1.286 for 3 d. of f.; p-value= .395917
--------------------------------------------------------------
     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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       221     211.4   .434279     .434
        30      5108    5134.0   .131775     .566
        31     22998   23103.0   .477636    1.044
        32     11673   11551.5  1.277435    2.321
  chisquare= 2.321 for 3 d. of f.; p-value= .555328
--------------------------------------------------------------

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

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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          957       944.3        .171        .171
          r =5        21535     21743.9       2.007       2.178
          r =6        77508     77311.8        .498       2.676
                        p=1-exp(-SUM/2)= .73758
        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          932       944.3        .160        .160
          r =5        21844     21743.9        .461        .621
          r =6        77224     77311.8        .100        .721
                        p=1-exp(-SUM/2)= .30260
        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          894       944.3       2.679       2.679
          r =5        21989     21743.9       2.763       5.442
          r =6        77117     77311.8        .491       5.933
                        p=1-exp(-SUM/2)= .94852
        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          963       944.3        .370        .370
          r =5        21879     21743.9        .839       1.210
          r =6        77158     77311.8        .306       1.516
                        p=1-exp(-SUM/2)= .53132
        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          922       944.3        .527        .527
          r =5        21930     21743.9       1.593       2.119
          r =6        77148     77311.8        .347       2.467
                        p=1-exp(-SUM/2)= .70866
        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          936       944.3        .073        .073
          r =5        21871     21743.9        .743        .816
          r =6        77193     77311.8        .183        .998
                        p=1-exp(-SUM/2)= .39301
        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          935       944.3        .092        .092
          r =5        21693     21743.9        .119        .211
          r =6        77372     77311.8        .047        .258
                        p=1-exp(-SUM/2)= .12087
        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          972       944.3        .812        .812
          r =5        21633     21743.9        .566       1.378
          r =6        77395     77311.8        .090       1.468
                        p=1-exp(-SUM/2)= .51992
        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          953       944.3        .080        .080
          r =5        21790     21743.9        .098        .178
          r =6        77257     77311.8        .039        .217
                        p=1-exp(-SUM/2)= .10269
        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          940       944.3        .020        .020
          r =5        21832     21743.9        .357        .377
          r =6        77228     77311.8        .091        .467
                        p=1-exp(-SUM/2)= .20840
        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          920       944.3        .625        .625
          r =5        21806     21743.9        .177        .803
          r =6        77274     77311.8        .018        .821
                        p=1-exp(-SUM/2)= .33676
        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          946       944.3        .003        .003
          r =5        22051     21743.9       4.337       4.340
          r =6        77003     77311.8       1.233       5.574
                        p=1-exp(-SUM/2)= .93839
        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          936       944.3        .073        .073
          r =5        21994     21743.9       2.877       2.950
          r =6        77070     77311.8        .756       3.706
                        p=1-exp(-SUM/2)= .84323
        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          931       944.3        .187        .187
          r =5        21994     21743.9       2.877       3.064
          r =6        77075     77311.8        .725       3.789
                        p=1-exp(-SUM/2)= .84963
        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          977       944.3       1.132       1.132
          r =5        21871     21743.9        .743       1.875
          r =6        77152     77311.8        .330       2.206
                        p=1-exp(-SUM/2)= .66805
        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          871       944.3       5.690       5.690
          r =5        21668     21743.9        .265       5.955
          r =6        77461     77311.8        .288       6.243
                        p=1-exp(-SUM/2)= .95591
        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          909       944.3       1.320       1.320
          r =5        21557     21743.9       1.607       2.926
          r =6        77534     77311.8        .639       3.565
                        p=1-exp(-SUM/2)= .83176
        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          908       944.3       1.396       1.396
          r =5        21907     21743.9       1.223       2.619
          r =6        77185     77311.8        .208       2.827
                        p=1-exp(-SUM/2)= .75670
        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          878       944.3       4.655       4.655
          r =5        21960     21743.9       2.148       6.803
          r =6        77162     77311.8        .290       7.093
                        p=1-exp(-SUM/2)= .97118
        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          915       944.3        .909        .909
          r =5        21726     21743.9        .015        .924
          r =6        77359     77311.8        .029        .953
                        p=1-exp(-SUM/2)= .37897
        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          946       944.3        .003        .003
          r =5        21760     21743.9        .012        .015
          r =6        77294     77311.8        .004        .019
                        p=1-exp(-SUM/2)= .00949
        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          947       944.3        .008        .008
          r =5        21897     21743.9       1.078       1.086
          r =6        77156     77311.8        .314       1.400
                        p=1-exp(-SUM/2)= .50334
        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          952       944.3        .063        .063
          r =5        21789     21743.9        .094        .156
          r =6        77259     77311.8        .036        .192
                        p=1-exp(-SUM/2)= .09171
        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          920       944.3        .625        .625
          r =5        21900     21743.9       1.121       1.746
          r =6        77180     77311.8        .225       1.971
                        p=1-exp(-SUM/2)= .62670
        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          944       944.3        .000        .000
          r =5        21714     21743.9        .041        .041
          r =6        77342     77311.8        .012        .053
                        p=1-exp(-SUM/2)= .02615
   TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices
 These should be 25 uniform [0,1] random variables:
     .737581     .302597     .948520     .531315     .708658
     .393007     .120867     .519925     .102694     .208396
     .336756     .938389     .843227     .849633     .668048
     .955906     .831765     .756697     .971176     .378970
     .009493     .503336     .091706     .626697     .026155
   brank test summary for bbs.32         
       The KS test for those 25 supposed UNI's yields
                    KS p-value= .306570

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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:  151580 missing words,   22.60 sigmas from mean, p-value=1.00000
 tst no  2:  143620 missing words,    4.00 sigmas from mean, p-value= .99997
 tst no  3:  143365 missing words,    3.40 sigmas from mean, p-value= .99966
 tst no  4:  143361 missing words,    3.39 sigmas from mean, p-value= .99965
 tst no  5:  143372 missing words,    3.42 sigmas from mean, p-value= .99968
 tst no  6:  143372 missing words,    3.42 sigmas from mean, p-value= .99968
 tst no  7:  143710 missing words,    4.21 sigmas from mean, p-value= .99999
 tst no  8:  143383 missing words,    3.44 sigmas from mean, p-value= .99971
 tst no  9:  143724 missing words,    4.24 sigmas from mean, p-value= .99999
 tst no 10:  143749 missing words,    4.30 sigmas from mean, p-value= .99999
 tst no 11:  143909 missing words,    4.67 sigmas from mean, p-value=1.00000
 tst no 12:  143847 missing words,    4.53 sigmas from mean, p-value=1.00000
 tst no 13:  143836 missing words,    4.50 sigmas from mean, p-value=1.00000
 tst no 14:  143767 missing words,    4.34 sigmas from mean, p-value= .99999
 tst no 15:  143370 missing words,    3.41 sigmas from mean, p-value= .99968
 tst no 16:  143350 missing words,    3.37 sigmas from mean, p-value= .99962
 tst no 17:  143388 missing words,    3.45 sigmas from mean, p-value= .99972
 tst no 18:  143489 missing words,    3.69 sigmas from mean, p-value= .99989
 tst no 19:  143726 missing words,    4.24 sigmas from mean, p-value= .99999
 tst no 20:  143393 missing words,    3.47 sigmas from mean, p-value= .99974

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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        786437******* 1.0000
    OPSO for bbs.32          using bits 22 to 31        786277******* 1.0000
    OPSO for bbs.32          using bits 21 to 30        786241******* 1.0000
    OPSO for bbs.32          using bits 20 to 29        786367******* 1.0000
    OPSO for bbs.32          using bits 19 to 28        786343******* 1.0000
    OPSO for bbs.32          using bits 18 to 27        786616******* 1.0000
    OPSO for bbs.32          using bits 17 to 26        786477******* 1.0000
    OPSO for bbs.32          using bits 16 to 25        786234******* 1.0000
    OPSO for bbs.32          using bits 15 to 24        786468******* 1.0000
    OPSO for bbs.32          using bits 14 to 23        786330******* 1.0000
    OPSO for bbs.32          using bits 13 to 22        786291******* 1.0000
    OPSO for bbs.32          using bits 12 to 21        786436******* 1.0000
    OPSO for bbs.32          using bits 11 to 20        786147******* 1.0000
    OPSO for bbs.32          using bits 10 to 19        786550******* 1.0000
    OPSO for bbs.32          using bits  9 to 18        786307******* 1.0000
    OPSO for bbs.32          using bits  8 to 17        786296******* 1.0000
    OPSO for bbs.32          using bits  7 to 16        786435******* 1.0000
    OPSO for bbs.32          using bits  6 to 15        786336******* 1.0000
    OPSO for bbs.32          using bits  5 to 14        786279******* 1.0000
    OPSO for bbs.32          using bits  4 to 13        786373******* 1.0000
    OPSO for bbs.32          using bits  3 to 12        786330******* 1.0000
    OPSO for bbs.32          using bits  2 to 11        786571******* 1.0000
    OPSO for bbs.32          using bits  1 to 10        786348******* 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        785994******* 1.0000
    OQSO for bbs.32          using bits 27 to 31        785612******* 1.0000
    OQSO for bbs.32          using bits 26 to 30        785732******* 1.0000
    OQSO for bbs.32          using bits 25 to 29        785797******* 1.0000
    OQSO for bbs.32          using bits 24 to 28        786024******* 1.0000
    OQSO for bbs.32          using bits 23 to 27        785661******* 1.0000
    OQSO for bbs.32          using bits 22 to 26        785777******* 1.0000
    OQSO for bbs.32          using bits 21 to 25        785763******* 1.0000
    OQSO for bbs.32          using bits 20 to 24        785831******* 1.0000
    OQSO for bbs.32          using bits 19 to 23        785682******* 1.0000
    OQSO for bbs.32          using bits 18 to 22        785746******* 1.0000
    OQSO for bbs.32          using bits 17 to 21        785883******* 1.0000
    OQSO for bbs.32          using bits 16 to 20        786026******* 1.0000
    OQSO for bbs.32          using bits 15 to 19        785707******* 1.0000
    OQSO for bbs.32          using bits 14 to 18        785770******* 1.0000
    OQSO for bbs.32          using bits 13 to 17        785813******* 1.0000
    OQSO for bbs.32          using bits 12 to 16        785893******* 1.0000
    OQSO for bbs.32          using bits 11 to 15        785595******* 1.0000
    OQSO for bbs.32          using bits 10 to 14        785819******* 1.0000
    OQSO for bbs.32          using bits  9 to 13        785752******* 1.0000
    OQSO for bbs.32          using bits  8 to 12        786004******* 1.0000
    OQSO for bbs.32          using bits  7 to 11        785654******* 1.0000
    OQSO for bbs.32          using bits  6 to 10        785791******* 1.0000
    OQSO for bbs.32          using bits  5 to  9        785703******* 1.0000
    OQSO for bbs.32          using bits  4 to  8        785815******* 1.0000
    OQSO for bbs.32          using bits  3 to  7        785595******* 1.0000
    OQSO for bbs.32          using bits  2 to  6        785779******* 1.0000
    OQSO for bbs.32          using bits  1 to  5        785840******* 1.0000
  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        785939******* 1.0000
     DNA for bbs.32          using bits 30 to 31        785646******* 1.0000
     DNA for bbs.32          using bits 29 to 30        785653******* 1.0000
     DNA for bbs.32          using bits 28 to 29        785595******* 1.0000
     DNA for bbs.32          using bits 27 to 28        785770******* 1.0000
     DNA for bbs.32          using bits 26 to 27        785640******* 1.0000
     DNA for bbs.32          using bits 25 to 26        785992******* 1.0000
     DNA for bbs.32          using bits 24 to 25        785451******* 1.0000
     DNA for bbs.32          using bits 23 to 24        785975******* 1.0000
     DNA for bbs.32          using bits 22 to 23        785780******* 1.0000
     DNA for bbs.32          using bits 21 to 22        785676******* 1.0000
     DNA for bbs.32          using bits 20 to 21        785657******* 1.0000
     DNA for bbs.32          using bits 19 to 20        785679******* 1.0000
     DNA for bbs.32          using bits 18 to 19        785575******* 1.0000
     DNA for bbs.32          using bits 17 to 18        785929******* 1.0000
     DNA for bbs.32          using bits 16 to 17        785502******* 1.0000
     DNA for bbs.32          using bits 15 to 16        785836******* 1.0000
     DNA for bbs.32          using bits 14 to 15        785630******* 1.0000
     DNA for bbs.32          using bits 13 to 14        785597******* 1.0000
     DNA for bbs.32          using bits 12 to 13        785600******* 1.0000
     DNA for bbs.32          using bits 11 to 12        785892******* 1.0000
     DNA for bbs.32          using bits 10 to 11        785658******* 1.0000
     DNA for bbs.32          using bits  9 to 10        785894******* 1.0000
     DNA for bbs.32          using bits  8 to  9        785477******* 1.0000
     DNA for bbs.32          using bits  7 to  8        785974******* 1.0000
     DNA for bbs.32          using bits  6 to  7        785653******* 1.0000
     DNA for bbs.32          using bits  5 to  6        785593******* 1.0000
     DNA for bbs.32          using bits  4 to  5        785588******* 1.0000
     DNA for bbs.32          using bits  3 to  4        785870******* 1.0000
     DNA for bbs.32          using bits  2 to  3        785617******* 1.0000
     DNA for bbs.32          using bits  1 to  2        785912******* 1.0000

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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          21617.59    270.364     1.000000
 byte stream for bbs.32          22759.46    286.512     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  2437.07      -.890      .186756
           bits  2 to  9  2395.31     -1.481      .069365
           bits  3 to 10  2439.90      -.850      .197680
           bits  4 to 11  2506.35       .090      .535791
           bits  5 to 12  2556.84       .804      .789239
           bits  6 to 13  2527.62       .391      .651974
           bits  7 to 14  2454.02      -.650      .257782
           bits  8 to 15  2529.21       .413      .660219
           bits  9 to 16  2495.93      -.058      .477027
           bits 10 to 17  2464.19      -.506      .306267
           bits 11 to 18  2433.76      -.937      .174453
           bits 12 to 19  2538.73       .548      .708073
           bits 13 to 20  2596.19      1.360      .913127
           bits 14 to 21  2474.42      -.362      .358762
           bits 15 to 22  2558.40       .826      .795558
           bits 16 to 23  2535.67       .504      .693021
           bits 17 to 24  2516.94       .240      .594658
           bits 18 to 25  2431.04      -.975      .164707
           bits 19 to 26  2423.80     -1.078      .140603
           bits 20 to 27  2522.63       .320      .625542
           bits 21 to 28  2488.43      -.164      .435011
           bits 22 to 29  2572.55      1.026      .847565
           bits 23 to 30  2533.52       .474      .682258
           bits 24 to 31  2546.63       .659      .745177
           bits 25 to 32  2536.06       .510      .694976

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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: 3508    z-score:  -.685 p-value: .246694
            Successes: 3528    z-score:   .228 p-value: .590298
            Successes: 3545    z-score:  1.005 p-value: .842447
            Successes: 3528    z-score:   .228 p-value: .590298
            Successes: 3517    z-score:  -.274 p-value: .392053
            Successes: 3551    z-score:  1.279 p-value: .899470
            Successes: 3514    z-score:  -.411 p-value: .340551
            Successes: 3547    z-score:  1.096 p-value: .863437
            Successes: 3550    z-score:  1.233 p-value: .891189
            Successes: 3534    z-score:   .502 p-value: .692266
 
           square size   avg. no.  parked   sample sigma
             100.            3532.200       14.965
            KSTEST for the above 10: p=  .797436

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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     .4675    .2114     .374928
          10     .2583    .2928     .228663
          15     .3605    .2960     .303946
          20     .2556    .3076     .226528
          25     .0693    .3395     .067278
          30    1.3575    .5066     .744436
          35     .4369    .5011     .355405
          40     .0138    .4597     .013770
          45     .7937    .5813     .549613
          50     .6344    .5587     .471426
          55     .0241    .5315     .023901
          60    3.6627    .6113     .974804
          65    1.3575    .6199     .744436
          70     .3694    .6873     .310146
          75     .6123    .7046     .459585
          80     .7353    .7094     .522383
          85     .7950    .6967     .550237
          90     .1611    .6795     .149480
          95     .5772    .6819     .440182
         100    2.3185    .7235     .902724
     MINIMUM DISTANCE TEST for bbs.32         
          Result of KS test on 20 transformed mindist^2's:
                                  p-value= .999899

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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=  13.782     p-value= .36833
 sample no:  2     r^3=   7.376     p-value= .21796
 sample no:  3     r^3=   1.727     p-value= .05595
 sample no:  4     r^3=  14.314     p-value= .37944
 sample no:  5     r^3=   5.073     p-value= .15559
 sample no:  6     r^3=  16.754     p-value= .42792
 sample no:  7     r^3=  12.961     p-value= .35082
 sample no:  8     r^3=   6.564     p-value= .19653
 sample no:  9     r^3=   5.559     p-value= .16914
 sample no: 10     r^3=  50.501     p-value= .81425
 sample no: 11     r^3=   9.747     p-value= .27740
 sample no: 12     r^3=  33.213     p-value= .66949
 sample no: 13     r^3=  26.516     p-value= .58682
 sample no: 14     r^3=  72.139     p-value= .90970
 sample no: 15     r^3=   8.232     p-value= .23997
 sample no: 16     r^3=  33.603     p-value= .67375
 sample no: 17     r^3=  60.640     p-value= .86752
 sample no: 18     r^3=  15.863     p-value= .41067
 sample no: 19     r^3=  34.433     p-value= .68265
 sample no: 20     r^3=  44.712     p-value= .77472
  A KS test is applied to those 20 p-values.
---------------------------------------------------------
       3DSPHERES test for file bbs.32               p-value= .316078
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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    -2.4    -2.0    -3.2     3.4    -1.0
     1.1      .8     1.6     1.0      .0    -1.4
      .8     1.6    -2.7    -2.3    -2.7    -1.8
    -3.2     3.0      .5    -1.4     2.6     1.8
     1.7     8.1    -7.1     4.1     1.1     2.9
     3.1     -.9    -2.1    -2.2    -2.7     3.1
     3.6     -.7    -2.4     2.6    -1.3    -1.0
    -1.1
           Chi-square with 42 degrees of freedom:312.079
              z-score= 29.468  p-value=1.000000
______________________________________________________________

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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  .606056
                Test no.  2      p-value  .142778
                Test no.  3      p-value  .846881
                Test no.  4      p-value  .620060
                Test no.  5      p-value  .527181
                Test no.  6      p-value  .113921
                Test no.  7      p-value  .607919
                Test no.  8      p-value  .220080
                Test no.  9      p-value  .583735
                Test no. 10      p-value  .647270
   Results of the OSUM test for bbs.32         
        KSTEST on the above 10 p-values:  .474557

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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: .932132
     runs down; ks test for 10 p's: .159730
                 Run test for bbs.32         :
       runs up; ks test for 10 p's: .741187
     runs down; ks test for 10 p's: .771585

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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
                                98366    98585.86
                  98366= No. of wins, z-score= -.983 pvalue= .16272
   Analysis of Throws-per-Game:
 Chisq=  27.94 for 20 degrees of freedom, p=  .88911
               Throws Observed Expected  Chisq     Sum
                  1    66374    66666.7   1.285    1.285
                  2    37435    37654.3   1.277    2.562
                  3    27034    26954.7    .233    2.795
                  4    19440    19313.5    .829    3.625
                  5    13925    13851.4    .391    4.015
                  6    10103     9943.5   2.557    6.573
                  7     7047     7145.0   1.345    7.917
                  8     5107     5139.1    .200    8.118
                  9     3854     3699.9   6.421   14.539
                 10     2650     2666.3    .100   14.638
                 11     1858     1923.3   2.219   16.857
                 12     1475     1388.7   5.358   22.215
                 13     1017     1003.7    .176   22.391
                 14      761      726.1   1.673   24.065
                 15      519      525.8    .089   24.153
                 16      371      381.2    .270   24.424
                 17      282      276.5    .108   24.532
                 18      225      200.8   2.909   27.441
                 19      141      146.0    .170   27.611
                 20      103      106.2    .097   27.708
                 21      279      287.1    .229   27.937
            SUMMARY  FOR bbs.32         
                p-value for no. of wins: .162720
                p-value for throws/game: .889112

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



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