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Plain Text, pasted on Jan 27:
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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   2.046
  duplicate       number       number 
  spacings       observed     expected
        0          63.       67.668
        1         131.      135.335
        2         142.      135.335
        3          93.       90.224
        4          41.       45.112
        5          19.       18.045
  6 to INF         11.        8.282
 Chisquare with  6 d.o.f. =     2.19 p-value=  .098778
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  2 to 25   1.968
  duplicate       number       number 
  spacings       observed     expected
        0          75.       67.668
        1         132.      135.335
        2         134.      135.335
        3          91.       90.224
        4          40.       45.112
        5          19.       18.045
  6 to INF          9.        8.282
 Chisquare with  6 d.o.f. =     1.59 p-value=  .046611
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  3 to 26   1.940
  duplicate       number       number 
  spacings       observed     expected
        0          76.       67.668
        1         142.      135.335
        2         121.      135.335
        3          91.       90.224
        4          46.       45.112
        5          17.       18.045
  6 to INF          7.        8.282
 Chisquare with  6 d.o.f. =     3.16 p-value=  .210937
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  4 to 27   2.058
  duplicate       number       number 
  spacings       observed     expected
        0          64.       67.668
        1         138.      135.335
        2         131.      135.335
        3          89.       90.224
        4          47.       45.112
        5          18.       18.045
  6 to INF         13.        8.282
 Chisquare with  6 d.o.f. =     3.17 p-value=  .213268
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  5 to 28   1.916
  duplicate       number       number 
  spacings       observed     expected
        0          68.       67.668
        1         154.      135.335
        2         118.      135.335
        3          96.       90.224
        4          47.       45.112
        5          13.       18.045
  6 to INF          4.        8.282
 Chisquare with  6 d.o.f. =     8.87 p-value=  .818933
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  6 to 29   1.958
  duplicate       number       number 
  spacings       observed     expected
        0          61.       67.668
        1         157.      135.335
        2         125.      135.335
        3          89.       90.224
        4          47.       45.112
        5          13.       18.045
  6 to INF          8.        8.282
 Chisquare with  6 d.o.f. =     6.43 p-value=  .623214
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  7 to 30   2.094
  duplicate       number       number 
  spacings       observed     expected
        0          56.       67.668
        1         123.      135.335
        2         149.      135.335
        3          93.       90.224
        4          54.       45.112
        5          21.       18.045
  6 to INF          4.        8.282
 Chisquare with  6 d.o.f. =     9.05 p-value=  .829229
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  8 to 31   2.030
  duplicate       number       number 
  spacings       observed     expected
        0          63.       67.668
        1         118.      135.335
        2         154.      135.335
        3          96.       90.224
        4          48.       45.112
        5          17.       18.045
  6 to INF          4.        8.282
 Chisquare with  6 d.o.f. =     7.95 p-value=  .757884
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           bbs.32          using bits  9 to 32   1.986
  duplicate       number       number 
  spacings       observed     expected
        0          61.       67.668
        1         137.      135.335
        2         142.      135.335
        3          94.       90.224
        4          48.       45.112
        5          12.       18.045
  6 to INF          6.        8.282
 Chisquare with  6 d.o.f. =     4.00 p-value=  .323627
  :::::::::::::::::::::::::::::::::::::::::
   The 9 p-values were
        .098778   .046611   .210937   .213268   .818933
        .623214   .829229   .757884   .323627
  A KSTEST for the 9 p-values yields  .327992

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

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     ::            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=147.002; p-value= .998737
           OPERM5 test for file bbs.32         
     For a sample of 1,000,000 consecutive 5-tuples,
 chisquare for 99 degrees of freedom=187.354; 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       202     211.4   .419543     .420
        29      5163    5134.0   .163694     .583
        30     23139   23103.0   .055951     .639
        31     11496   11551.5   .266888     .906
  chisquare=  .906 for 3 d. of f.; p-value= .346618
--------------------------------------------------------------
     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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       183     211.4  3.819843    3.820
        30      5130    5134.0   .003132    3.823
        31     23198   23103.0   .390256    4.213
        32     11489   11551.5   .338423    4.552
  chisquare= 4.552 for 3 d. of f.; p-value= .808715
--------------------------------------------------------------

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

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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          935       944.3        .092        .092
          r =5        21662     21743.9        .308        .400
          r =6        77403     77311.8        .108        .508
                        p=1-exp(-SUM/2)= .22418
        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          949       944.3        .023        .023
          r =5        21585     21743.9       1.161       1.185
          r =6        77466     77311.8        .308       1.492
                        p=1-exp(-SUM/2)= .52577
        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          939       944.3        .030        .030
          r =5        21377     21743.9       6.191       6.221
          r =6        77684     77311.8       1.792       8.013
                        p=1-exp(-SUM/2)= .98180
        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          907       944.3       1.473       1.473
          r =5        21674     21743.9        .225       1.698
          r =6        77419     77311.8        .149       1.847
                        p=1-exp(-SUM/2)= .60283
        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          979       944.3       1.275       1.275
          r =5        21642     21743.9        .478       1.753
          r =6        77379     77311.8        .058       1.811
                        p=1-exp(-SUM/2)= .59565
        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          983       944.3       1.586       1.586
          r =5        21618     21743.9        .729       2.315
          r =6        77399     77311.8        .098       2.413
                        p=1-exp(-SUM/2)= .70079
        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          922       944.3        .527        .527
          r =5        21705     21743.9        .070        .596
          r =6        77373     77311.8        .048        .645
                        p=1-exp(-SUM/2)= .27556
        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          977       944.3       1.132       1.132
          r =5        21588     21743.9       1.118       2.250
          r =6        77435     77311.8        .196       2.446
                        p=1-exp(-SUM/2)= .70571
        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          920       944.3        .625        .625
          r =5        21831     21743.9        .349        .974
          r =6        77249     77311.8        .051       1.025
                        p=1-exp(-SUM/2)= .40109
        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          932       944.3        .160        .160
          r =5        21629     21743.9        .607        .767
          r =6        77439     77311.8        .209        .977
                        p=1-exp(-SUM/2)= .38635
        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          916       944.3        .848        .848
          r =5        21551     21743.9       1.711       2.560
          r =6        77533     77311.8        .633       3.192
                        p=1-exp(-SUM/2)= .79733
        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          945       944.3        .001        .001
          r =5        21599     21743.9        .966        .966
          r =6        77456     77311.8        .269       1.235
                        p=1-exp(-SUM/2)= .46073
        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          947       944.3        .008        .008
          r =5        21622     21743.9        .683        .691
          r =6        77431     77311.8        .184        .875
                        p=1-exp(-SUM/2)= .35431
        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          962       944.3        .332        .332
          r =5        21542     21743.9       1.875       2.206
          r =6        77496     77311.8        .439       2.645
                        p=1-exp(-SUM/2)= .73357
        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          935       944.3        .092        .092
          r =5        21585     21743.9       1.161       1.253
          r =6        77480     77311.8        .366       1.619
                        p=1-exp(-SUM/2)= .55486
        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          925       944.3        .395        .395
          r =5        21541     21743.9       1.893       2.288
          r =6        77534     77311.8        .639       2.926
                        p=1-exp(-SUM/2)= .76851
        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          916       944.3        .848        .848
          r =5        21639     21743.9        .506       1.354
          r =6        77445     77311.8        .229       1.584
                        p=1-exp(-SUM/2)= .54701
        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          980       944.3       1.350       1.350
          r =5        21584     21743.9       1.176       2.525
          r =6        77436     77311.8        .200       2.725
                        p=1-exp(-SUM/2)= .74397
        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          920       944.3        .625        .625
          r =5        21666     21743.9        .279        .904
          r =6        77414     77311.8        .135       1.040
                        p=1-exp(-SUM/2)= .40535
        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          964       944.3        .411        .411
          r =5        21529     21743.9       2.124       2.535
          r =6        77507     77311.8        .493       3.028
                        p=1-exp(-SUM/2)= .77994
        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          976       944.3       1.064       1.064
          r =5        21359     21743.9       6.813       7.877
          r =6        77665     77311.8       1.614       9.491
                        p=1-exp(-SUM/2)= .99131
        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          976       944.3       1.064       1.064
          r =5        21665     21743.9        .286       1.350
          r =6        77359     77311.8        .029       1.379
                        p=1-exp(-SUM/2)= .49822
        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          945       944.3        .001        .001
          r =5        21659     21743.9        .331        .332
          r =6        77396     77311.8        .092        .424
                        p=1-exp(-SUM/2)= .19092
        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          929       944.3        .248        .248
          r =5        21739     21743.9        .001        .249
          r =6        77332     77311.8        .005        .254
                        p=1-exp(-SUM/2)= .11941
        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          930       944.3        .217        .217
          r =5        21860     21743.9        .620        .836
          r =6        77210     77311.8        .134        .971
                        p=1-exp(-SUM/2)= .38447
   TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices
 These should be 25 uniform [0,1] random variables:
     .224182     .525771     .981799     .602832     .595654
     .700795     .275561     .705708     .401094     .386354
     .797332     .460727     .354312     .733570     .554863
     .768511     .547007     .743974     .405349     .779935
     .991309     .498220     .190917     .119407     .384472
   brank test summary for bbs.32         
       The KS test for those 25 supposed UNI's yields
                    KS p-value= .745974

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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:  143655 missing words,    4.08 sigmas from mean, p-value= .99998
 tst no  2:  143492 missing words,    3.70 sigmas from mean, p-value= .99989
 tst no  3:  143648 missing words,    4.06 sigmas from mean, p-value= .99998
 tst no  4:  143430 missing words,    3.55 sigmas from mean, p-value= .99981
 tst no  5:  144164 missing words,    5.27 sigmas from mean, p-value=1.00000
 tst no  6:  143382 missing words,    3.44 sigmas from mean, p-value= .99971
 tst no  7:  143469 missing words,    3.64 sigmas from mean, p-value= .99987
 tst no  8:  143690 missing words,    4.16 sigmas from mean, p-value= .99998
 tst no  9:  143449 missing words,    3.60 sigmas from mean, p-value= .99984
 tst no 10:  144108 missing words,    5.14 sigmas from mean, p-value=1.00000
 tst no 11:  143443 missing words,    3.58 sigmas from mean, p-value= .99983
 tst no 12:  143521 missing words,    3.77 sigmas from mean, p-value= .99992
 tst no 13:  143785 missing words,    4.38 sigmas from mean, p-value= .99999
 tst no 14:  143482 missing words,    3.67 sigmas from mean, p-value= .99988
 tst no 15:  144113 missing words,    5.15 sigmas from mean, p-value=1.00000
 tst no 16:  143448 missing words,    3.60 sigmas from mean, p-value= .99984
 tst no 17:  143471 missing words,    3.65 sigmas from mean, p-value= .99987
 tst no 18:  143875 missing words,    4.59 sigmas from mean, p-value=1.00000
 tst no 19:  143541 missing words,    3.81 sigmas from mean, p-value= .99993
 tst no 20:  144047 missing words,    4.99 sigmas from mean, p-value=1.00000

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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        556044******* 1.0000
    OPSO for bbs.32          using bits 22 to 31        556021******* 1.0000
    OPSO for bbs.32          using bits 21 to 30        556191******* 1.0000
    OPSO for bbs.32          using bits 20 to 29        556130******* 1.0000
    OPSO for bbs.32          using bits 19 to 28        556033******* 1.0000
    OPSO for bbs.32          using bits 18 to 27        556680******* 1.0000
    OPSO for bbs.32          using bits 17 to 26        556939******* 1.0000
    OPSO for bbs.32          using bits 16 to 25        555897******* 1.0000
    OPSO for bbs.32          using bits 15 to 24        556034******* 1.0000
    OPSO for bbs.32          using bits 14 to 23        556063******* 1.0000
    OPSO for bbs.32          using bits 13 to 22        556181******* 1.0000
    OPSO for bbs.32          using bits 12 to 21        556088******* 1.0000
    OPSO for bbs.32          using bits 11 to 20        555978******* 1.0000
    OPSO for bbs.32          using bits 10 to 19        556623******* 1.0000
    OPSO for bbs.32          using bits  9 to 18        557106******* 1.0000
    OPSO for bbs.32          using bits  8 to 17        556027******* 1.0000
    OPSO for bbs.32          using bits  7 to 16        556000******* 1.0000
    OPSO for bbs.32          using bits  6 to 15        556117******* 1.0000
    OPSO for bbs.32          using bits  5 to 14        556231******* 1.0000
    OPSO for bbs.32          using bits  4 to 13        556157******* 1.0000
    OPSO for bbs.32          using bits  3 to 12        555983******* 1.0000
    OPSO for bbs.32          using bits  2 to 11        556656******* 1.0000
    OPSO for bbs.32          using bits  1 to 10        557002******* 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        554451******* 1.0000
    OQSO for bbs.32          using bits 27 to 31        554657******* 1.0000
    OQSO for bbs.32          using bits 26 to 30        554543******* 1.0000
    OQSO for bbs.32          using bits 25 to 29        554300******* 1.0000
    OQSO for bbs.32          using bits 24 to 28        554419******* 1.0000
    OQSO for bbs.32          using bits 23 to 27        554326******* 1.0000
    OQSO for bbs.32          using bits 22 to 26        554390******* 1.0000
    OQSO for bbs.32          using bits 21 to 25        554629******* 1.0000
    OQSO for bbs.32          using bits 20 to 24        554508******* 1.0000
    OQSO for bbs.32          using bits 19 to 23        554570******* 1.0000
    OQSO for bbs.32          using bits 18 to 22        554574******* 1.0000
    OQSO for bbs.32          using bits 17 to 21        554179******* 1.0000
    OQSO for bbs.32          using bits 16 to 20        554394******* 1.0000
    OQSO for bbs.32          using bits 15 to 19        554367******* 1.0000
    OQSO for bbs.32          using bits 14 to 18        554465******* 1.0000
    OQSO for bbs.32          using bits 13 to 17        554580******* 1.0000
    OQSO for bbs.32          using bits 12 to 16        554406******* 1.0000
    OQSO for bbs.32          using bits 11 to 15        554556******* 1.0000
    OQSO for bbs.32          using bits 10 to 14        554551******* 1.0000
    OQSO for bbs.32          using bits  9 to 13        554235******* 1.0000
    OQSO for bbs.32          using bits  8 to 12        554477******* 1.0000
    OQSO for bbs.32          using bits  7 to 11        554273******* 1.0000
    OQSO for bbs.32          using bits  6 to 10        554595******* 1.0000
    OQSO for bbs.32          using bits  5 to  9        554624******* 1.0000
    OQSO for bbs.32          using bits  4 to  8        554468******* 1.0000
    OQSO for bbs.32          using bits  3 to  7        554684******* 1.0000
    OQSO for bbs.32          using bits  2 to  6        554602******* 1.0000
    OQSO for bbs.32          using bits  1 to  5        554299******* 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        553983******* 1.0000
     DNA for bbs.32          using bits 30 to 31        554454******* 1.0000
     DNA for bbs.32          using bits 29 to 30        554507******* 1.0000
     DNA for bbs.32          using bits 28 to 29        554595******* 1.0000
     DNA for bbs.32          using bits 27 to 28        553789******* 1.0000
     DNA for bbs.32          using bits 26 to 27        554648******* 1.0000
     DNA for bbs.32          using bits 25 to 26        554889******* 1.0000
     DNA for bbs.32          using bits 24 to 25        554417******* 1.0000
     DNA for bbs.32          using bits 23 to 24        553857******* 1.0000
     DNA for bbs.32          using bits 22 to 23        554470******* 1.0000
     DNA for bbs.32          using bits 21 to 22        554379******* 1.0000
     DNA for bbs.32          using bits 20 to 21        554478******* 1.0000
     DNA for bbs.32          using bits 19 to 20        553947******* 1.0000
     DNA for bbs.32          using bits 18 to 19        554751******* 1.0000
     DNA for bbs.32          using bits 17 to 18        554720******* 1.0000
     DNA for bbs.32          using bits 16 to 17        554601******* 1.0000
     DNA for bbs.32          using bits 15 to 16        553971******* 1.0000
     DNA for bbs.32          using bits 14 to 15        554489******* 1.0000
     DNA for bbs.32          using bits 13 to 14        554487******* 1.0000
     DNA for bbs.32          using bits 12 to 13        554490******* 1.0000
     DNA for bbs.32          using bits 11 to 12        553876******* 1.0000
     DNA for bbs.32          using bits 10 to 11        554751******* 1.0000
     DNA for bbs.32          using bits  9 to 10        554617******* 1.0000
     DNA for bbs.32          using bits  8 to  9        554448******* 1.0000
     DNA for bbs.32          using bits  7 to  8        554003******* 1.0000
     DNA for bbs.32          using bits  6 to  7        554534******* 1.0000
     DNA for bbs.32          using bits  5 to  6        554456******* 1.0000
     DNA for bbs.32          using bits  4 to  5        554431******* 1.0000
     DNA for bbs.32          using bits  3 to  4        553940******* 1.0000
     DNA for bbs.32          using bits  2 to  3        554672******* 1.0000
     DNA for bbs.32          using bits  1 to  2        554748******* 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          10844.84    118.014     1.000000
 byte stream for bbs.32          11197.53    123.002     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  2502.96       .042      .516674
           bits  2 to  9  2431.26      -.972      .165510
           bits  3 to 10  2638.48      1.958      .974911
           bits  4 to 11  2446.33      -.759      .223936
           bits  5 to 12  2619.88      1.695      .954994
           bits  6 to 13  2487.37      -.179      .429143
           bits  7 to 14  2495.13      -.069      .472543
           bits  8 to 15  2493.78      -.088      .464977
           bits  9 to 16  2477.33      -.321      .374258
           bits 10 to 17  2584.70      1.198      .884515
           bits 11 to 18  2638.29      1.956      .974750
           bits 12 to 19  2493.99      -.085      .466143
           bits 13 to 20  2539.12       .553      .709964
           bits 14 to 21  2433.14      -.946      .172180
           bits 15 to 22  2488.89      -.157      .437574
           bits 16 to 23  2612.95      1.597      .944906
           bits 17 to 24  2509.74       .138      .554775
           bits 18 to 25  2529.68       .420      .662638
           bits 19 to 26  2452.17      -.676      .249398
           bits 20 to 27  2404.22     -1.355      .087774
           bits 21 to 28  2411.41     -1.253      .105123
           bits 22 to 29  2480.67      -.273      .392258
           bits 23 to 30  2506.54       .093      .536866
           bits 24 to 31  2519.82       .280      .610355
           bits 25 to 32  2507.45       .105      .541979

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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: 3506    z-score:  -.776 p-value: .218799
            Successes: 3541    z-score:   .822 p-value: .794438
            Successes: 3535    z-score:   .548 p-value: .708135
            Successes: 3480    z-score: -1.963 p-value: .024796
            Successes: 3547    z-score:  1.096 p-value: .863437
            Successes: 3476    z-score: -2.146 p-value: .015932
            Successes: 3548    z-score:  1.142 p-value: .873180
            Successes: 3540    z-score:   .776 p-value: .781201
            Successes: 3551    z-score:  1.279 p-value: .899470
            Successes: 3524    z-score:   .046 p-value: .518210
 
           square size   avg. no.  parked   sample sigma
             100.            3524.800       26.551
            KSTEST for the above 10: p=  .781492

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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    1.5626    .8398     .792047
          10    1.5134   1.1995     .781510
          15     .7072   1.0606     .508735
          20     .0317    .8967     .031319
          25     .9343    .9578     .608993
          30     .2187    .8499     .197353
          35    1.1373    .8655     .681157
          40     .2205    .8226     .198787
          45    1.4047    .8169     .756292
          50     .3830    .7987     .319490
          55     .1387    .7652     .130142
          60    1.1169    .7551     .674540
          65     .3054    .7424     .264329
          70    2.8579    .7551     .943431
          75    2.7509    .7713     .937006
          80    2.8622    .8273     .943674
          85     .4990    .8110     .394369
          90    3.6190    .8614     .973672
          95     .1502    .8747     .140152
         100     .9247    .8628     .605176
     MINIMUM DISTANCE TEST for bbs.32         
          Result of KS test on 20 transformed mindist^2's:
                                  p-value= .495479

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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=   6.376     p-value= .19147
 sample no:  2     r^3=  16.483     p-value= .42272
 sample no:  3     r^3=  57.796     p-value= .85435
 sample no:  4     r^3=  15.579     p-value= .40506
 sample no:  5     r^3= 214.059     p-value= .99920
 sample no:  6     r^3=  23.327     p-value= .54047
 sample no:  7     r^3=   2.732     p-value= .08703
 sample no:  8     r^3=  34.392     p-value= .68222
 sample no:  9     r^3=  19.009     p-value= .46935
 sample no: 10     r^3=  71.366     p-value= .90735
 sample no: 11     r^3=  20.297     p-value= .49165
 sample no: 12     r^3=  22.199     p-value= .52287
 sample no: 13     r^3=  83.751     p-value= .93868
 sample no: 14     r^3=  22.177     p-value= .52252
 sample no: 15     r^3=  51.553     p-value= .82065
 sample no: 16     r^3=  45.434     p-value= .78008
 sample no: 17     r^3=  11.800     p-value= .32520
 sample no: 18     r^3=  61.038     p-value= .86927
 sample no: 19     r^3=  28.374     p-value= .61163
 sample no: 20     r^3=  14.798     p-value= .38936
  A KS test is applied to those 20 p-values.
---------------------------------------------------------
       3DSPHERES test for file bbs.32               p-value= .801257
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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      .5    -2.8    -2.4    -4.3      .1
      .5     -.5     1.1    -2.1     3.0    -1.1
    -3.2    -1.5    -1.1    -1.4    -2.1     -.2
     1.4     1.6    -1.8     4.8     3.0     1.7
     1.8    -1.9      .2     -.2      .5      .0
    -1.6      .4     2.1     1.4    -2.6      .1
     2.1     5.5     3.0    -1.8      .9    -1.0
    -1.1
           Chi-square with 42 degrees of freedom:186.535
              z-score= 15.770  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  .208261
                Test no.  2      p-value  .389122
                Test no.  3      p-value  .823191
                Test no.  4      p-value  .427107
                Test no.  5      p-value  .164340
                Test no.  6      p-value  .077852
                Test no.  7      p-value  .715234
                Test no.  8      p-value  .820594
                Test no.  9      p-value  .269488
                Test no. 10      p-value  .856772
   Results of the OSUM test for bbs.32         
        KSTEST on the above 10 p-values:  .108177

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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: .894457
     runs down; ks test for 10 p's: .668952
                 Run test for bbs.32         :
       runs up; ks test for 10 p's: .394742
     runs down; ks test for 10 p's: .231693

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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
                                98508    98585.86
                  98508= No. of wins, z-score= -.348 pvalue= .36383
   Analysis of Throws-per-Game:
 Chisq=  13.37 for 20 degrees of freedom, p=  .13905
               Throws Observed Expected  Chisq     Sum
                  1    66442    66666.7    .757     .757
                  2    37808    37654.3    .627    1.384
                  3    26863    26954.7    .312    1.697
                  4    19321    19313.5    .003    1.699
                  5    13927    13851.4    .412    2.112
                  6    10140     9943.5   3.881    5.993
                  7     7050     7145.0   1.264    7.257
                  8     5109     5139.1    .176    7.433
                  9     3768     3699.9   1.255    8.688
                 10     2630     2666.3    .494    9.182
                 11     1905     1923.3    .175    9.357
                 12     1399     1388.7    .076    9.432
                 13     1012     1003.7    .068    9.501
                 14      743      726.1    .391    9.892
                 15      512      525.8    .364   10.256
                 16      352      381.2   2.229   12.486
                 17      283      276.5    .151   12.637
                 18      205      200.8    .087   12.723
                 19      147      146.0    .007   12.730
                 20       98      106.2    .635   13.366
                 21      286      287.1    .004   13.370
            SUMMARY  FOR bbs.32         
                p-value for no. of wins: .363833
                p-value for throws/game: .139054

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 Results of DIEHARD battery of tests sent to file bbs.out2       



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