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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 outdiv.bin     
                   For a sample of size 500:     mean   
           outdiv.bin      using bits  1 to 24   1.982
  duplicate       number       number 
  spacings       observed     expected
        0          73.       67.668
        1         135.      135.335
        2         138.      135.335
        3          81.       90.224
        4          41.       45.112
        5          22.       18.045
  6 to INF         10.        8.282
 Chisquare with  6 d.o.f. =     3.01 p-value=  .192992
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           outdiv.bin      using bits  2 to 25   1.880
  duplicate       number       number 
  spacings       observed     expected
        0          68.       67.668
        1         156.      135.335
        2         127.      135.335
        3          87.       90.224
        4          43.       45.112
        5          18.       18.045
  6 to INF          1.        8.282
 Chisquare with  6 d.o.f. =    10.29 p-value=  .886927
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           outdiv.bin      using bits  3 to 26   1.946
  duplicate       number       number 
  spacings       observed     expected
        0          64.       67.668
        1         136.      135.335
        2         149.      135.335
        3          90.       90.224
        4          42.       45.112
        5          15.       18.045
  6 to INF          4.        8.282
 Chisquare with  6 d.o.f. =     4.52 p-value=  .393922
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           outdiv.bin      using bits  4 to 27   2.084
  duplicate       number       number 
  spacings       observed     expected
        0          56.       67.668
        1         128.      135.335
        2         150.      135.335
        3          90.       90.224
        4          48.       45.112
        5          18.       18.045
  6 to INF         10.        8.282
 Chisquare with  6 d.o.f. =     4.54 p-value=  .396053
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           outdiv.bin      using bits  5 to 28   1.972
  duplicate       number       number 
  spacings       observed     expected
        0          69.       67.668
        1         133.      135.335
        2         142.      135.335
        3          88.       90.224
        4          43.       45.112
        5          20.       18.045
  6 to INF          5.        8.282
 Chisquare with  6 d.o.f. =     2.06 p-value=  .085972
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           outdiv.bin      using bits  6 to 29   1.918
  duplicate       number       number 
  spacings       observed     expected
        0          81.       67.668
        1         131.      135.335
        2         141.      135.335
        3          80.       90.224
        4          42.       45.112
        5          16.       18.045
  6 to INF          9.        8.282
 Chisquare with  6 d.o.f. =     4.67 p-value=  .413207
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           outdiv.bin      using bits  7 to 30   1.958
  duplicate       number       number 
  spacings       observed     expected
        0          68.       67.668
        1         134.      135.335
        2         132.      135.335
        3         105.       90.224
        4          43.       45.112
        5          15.       18.045
  6 to INF          3.        8.282
 Chisquare with  6 d.o.f. =     6.50 p-value=  .630245
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           outdiv.bin      using bits  8 to 31   2.028
  duplicate       number       number 
  spacings       observed     expected
        0          64.       67.668
        1         135.      135.335
        2         132.      135.335
        3          93.       90.224
        4          51.       45.112
        5          18.       18.045
  6 to INF          7.        8.282
 Chisquare with  6 d.o.f. =     1.33 p-value=  .030269
  :::::::::::::::::::::::::::::::::::::::::
                   For a sample of size 500:     mean   
           outdiv.bin      using bits  9 to 32   2.100
  duplicate       number       number 
  spacings       observed     expected
        0          52.       67.668
        1         142.      135.335
        2         135.      135.335
        3          86.       90.224
        4          57.       45.112
        5          18.       18.045
  6 to INF         10.        8.282
 Chisquare with  6 d.o.f. =     7.64 p-value=  .734628
  :::::::::::::::::::::::::::::::::::::::::
   The 9 p-values were
        .192992   .886927   .393922   .396053   .085972
        .413207   .630245   .030269   .734628
  A KSTEST for the 9 p-values yields  .298221

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

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     ::            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 outdiv.bin     
     For a sample of 1,000,000 consecutive 5-tuples,
 chisquare for 99 degrees of freedom= 97.813; p-value= .485120
           OPERM5 test for file outdiv.bin     
     For a sample of 1,000,000 consecutive 5-tuples,
 chisquare for 99 degrees of freedom=103.107; p-value= .631297
     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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 outdiv.bin     
         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       226     211.4  1.005753    1.006
        29      5217    5134.0  1.341505    2.347
        30     23106   23103.0   .000377    2.348
        31     11451   11551.5   .874790    3.222
  chisquare= 3.222 for 3 d. of f.; p-value= .677732
--------------------------------------------------------------
     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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 outdiv.bin     
         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       242     211.4  4.423738    4.424
        30      5057    5134.0  1.155155    5.579
        31     23146   23103.0   .079858    5.659
        32     11555   11551.5   .001046    5.660
  chisquare= 5.660 for 3 d. of f.; p-value= .879292
--------------------------------------------------------------

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

     :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::        
     :: 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 outdiv.bin     
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits  1 to  8
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          926       944.3        .355        .355
          r =5        21976     21743.9       2.477       2.832
          r =6        77098     77311.8        .591       3.423
                        p=1-exp(-SUM/2)= .81945
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits  2 to  9
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          920       944.3        .625        .625
          r =5        21862     21743.9        .641       1.267
          r =6        77218     77311.8        .114       1.381
                        p=1-exp(-SUM/2)= .49859
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits  3 to 10
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          933       944.3        .135        .135
          r =5        21741     21743.9        .000        .136
          r =6        77326     77311.8        .003        .138
                        p=1-exp(-SUM/2)= .06679
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits  4 to 11
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          873       944.3       5.384       5.384
          r =5        21686     21743.9        .154       5.538
          r =6        77441     77311.8        .216       5.754
                        p=1-exp(-SUM/2)= .94369
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits  5 to 12
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          886       944.3       3.600       3.600
          r =5        21750     21743.9        .002       3.601
          r =6        77364     77311.8        .035       3.636
                        p=1-exp(-SUM/2)= .83769
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits  6 to 13
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          942       944.3        .006        .006
          r =5        21882     21743.9        .877        .883
          r =6        77176     77311.8        .239       1.121
                        p=1-exp(-SUM/2)= .42915
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits  7 to 14
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          953       944.3        .080        .080
          r =5        21885     21743.9        .916        .996
          r =6        77162     77311.8        .290       1.286
                        p=1-exp(-SUM/2)= .47429
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits  8 to 15
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          907       944.3       1.473       1.473
          r =5        21819     21743.9        .259       1.733
          r =6        77274     77311.8        .018       1.751
                        p=1-exp(-SUM/2)= .58341
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits  9 to 16
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          951       944.3        .048        .048
          r =5        21761     21743.9        .013        .061
          r =6        77288     77311.8        .007        .068
                        p=1-exp(-SUM/2)= .03357
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 10 to 17
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          940       944.3        .020        .020
          r =5        21819     21743.9        .259        .279
          r =6        77241     77311.8        .065        .344
                        p=1-exp(-SUM/2)= .15794
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 11 to 18
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          935       944.3        .092        .092
          r =5        21911     21743.9       1.284       1.376
          r =6        77154     77311.8        .322       1.698
                        p=1-exp(-SUM/2)= .57213
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 12 to 19
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4         1012       944.3       4.853       4.853
          r =5        21774     21743.9        .042       4.895
          r =6        77214     77311.8        .124       5.019
                        p=1-exp(-SUM/2)= .91868
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 13 to 20
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          910       944.3       1.246       1.246
          r =5        21823     21743.9        .288       1.534
          r =6        77267     77311.8        .026       1.560
                        p=1-exp(-SUM/2)= .54152
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 14 to 21
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          960       944.3        .261        .261
          r =5        21799     21743.9        .140        .401
          r =6        77241     77311.8        .065        .465
                        p=1-exp(-SUM/2)= .20763
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 15 to 22
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          934       944.3        .112        .112
          r =5        21943     21743.9       1.823       1.935
          r =6        77123     77311.8        .461       2.397
                        p=1-exp(-SUM/2)= .69828
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 16 to 23
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          926       944.3        .355        .355
          r =5        21976     21743.9       2.477       2.832
          r =6        77098     77311.8        .591       3.423
                        p=1-exp(-SUM/2)= .81945
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 17 to 24
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          910       944.3       1.246       1.246
          r =5        21704     21743.9        .073       1.319
          r =6        77386     77311.8        .071       1.390
                        p=1-exp(-SUM/2)= .50103
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 18 to 25
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          985       944.3       1.754       1.754
          r =5        21456     21743.9       3.812       5.566
          r =6        77559     77311.8        .790       6.356
                        p=1-exp(-SUM/2)= .95834
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 19 to 26
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          961       944.3        .295        .295
          r =5        21772     21743.9        .036        .332
          r =6        77267     77311.8        .026        .358
                        p=1-exp(-SUM/2)= .16372
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 20 to 27
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          960       944.3        .261        .261
          r =5        21561     21743.9       1.538       1.799
          r =6        77479     77311.8        .362       2.161
                        p=1-exp(-SUM/2)= .66058
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 21 to 28
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          903       944.3       1.806       1.806
          r =5        21813     21743.9        .220       2.026
          r =6        77284     77311.8        .010       2.036
                        p=1-exp(-SUM/2)= .63868
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 22 to 29
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          926       944.3        .355        .355
          r =5        21800     21743.9        .145        .499
          r =6        77274     77311.8        .018        .518
                        p=1-exp(-SUM/2)= .22814
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 23 to 30
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          916       944.3        .848        .848
          r =5        21883     21743.9        .890       1.738
          r =6        77201     77311.8        .159       1.897
                        p=1-exp(-SUM/2)= .61265
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 24 to 31
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          937       944.3        .056        .056
          r =5        21831     21743.9        .349        .405
          r =6        77232     77311.8        .082        .488
                        p=1-exp(-SUM/2)= .21641
        Rank of a 6x8 binary matrix,
     rows formed from eight bits of the RNG outdiv.bin     
     b-rank test for bits 25 to 32
                     OBSERVED   EXPECTED     (O-E)^2/E      SUM
          r<=4          977       944.3       1.132       1.132
          r =5        21711     21743.9        .050       1.182
          r =6        77312     77311.8        .000       1.182
                        p=1-exp(-SUM/2)= .44624
   TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices
 These should be 25 uniform [0,1] random variables:
     .819446     .498586     .066788     .943691     .837689
     .429150     .474293     .583413     .033572     .157944
     .572128     .918685     .541523     .207631     .698283
     .819446     .501025     .958340     .163716     .660582
     .638683     .228145     .612651     .216405     .446242
   brank test summary for outdiv.bin     
       The KS test for those 25 supposed UNI's yields
                    KS p-value= .089596

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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:  142924 missing words,    2.37 sigmas from mean, p-value= .99112
 tst no  2:  141989 missing words,     .19 sigmas from mean, p-value= .57384
 tst no  3:  142731 missing words,    1.92 sigmas from mean, p-value= .97256
 tst no  4:  142488 missing words,    1.35 sigmas from mean, p-value= .91182
 tst no  5:  142170 missing words,     .61 sigmas from mean, p-value= .72875
 tst no  6:  142396 missing words,    1.14 sigmas from mean, p-value= .87225
 tst no  7:  143016 missing words,    2.59 sigmas from mean, p-value= .99514
 tst no  8:  142557 missing words,    1.51 sigmas from mean, p-value= .93489
 tst no  9:  142727 missing words,    1.91 sigmas from mean, p-value= .97196
 tst no 10:  141846 missing words,    -.15 sigmas from mean, p-value= .44119
 tst no 11:  141474 missing words,   -1.02 sigmas from mean, p-value= .15455
 tst no 12:  142135 missing words,     .53 sigmas from mean, p-value= .70100
 tst no 13:  142674 missing words,    1.79 sigmas from mean, p-value= .96300
 tst no 14:  142430 missing words,    1.22 sigmas from mean, p-value= .88811
 tst no 15:  142711 missing words,    1.87 sigmas from mean, p-value= .96947
 tst no 16:  142042 missing words,     .31 sigmas from mean, p-value= .62171
 tst no 17:  142416 missing words,    1.18 sigmas from mean, p-value= .88176
 tst no 18:  142192 missing words,     .66 sigmas from mean, p-value= .74552
 tst no 19:  142216 missing words,     .72 sigmas from mean, p-value= .76317
 tst no 20:  143005 missing words,    2.56 sigmas from mean, p-value= .99477

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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 outdiv.bin     
  Output: No. missing words (mw), equiv normal variate (z), p-value (p)
                                                           mw     z     p
    OPSO for outdiv.bin      using bits 23 to 32        142704  2.740  .9969
    OPSO for outdiv.bin      using bits 22 to 31        142383  1.633  .9488
    OPSO for outdiv.bin      using bits 21 to 30        142316  1.402  .9196
    OPSO for outdiv.bin      using bits 20 to 29        142815  3.123  .9991
    OPSO for outdiv.bin      using bits 19 to 28        143173  4.357 1.0000
    OPSO for outdiv.bin      using bits 18 to 27        143576  5.747 1.0000
    OPSO for outdiv.bin      using bits 17 to 26        143571  5.730 1.0000
    OPSO for outdiv.bin      using bits 16 to 25        142885  3.364  .9996
    OPSO for outdiv.bin      using bits 15 to 24        142460  1.899  .9712
    OPSO for outdiv.bin      using bits 14 to 23        142740  2.864  .9979
    OPSO for outdiv.bin      using bits 13 to 22        142792  3.044  .9988
    OPSO for outdiv.bin      using bits 12 to 21        143024  3.844  .9999
    OPSO for outdiv.bin      using bits 11 to 20        143122  4.182 1.0000
    OPSO for outdiv.bin      using bits 10 to 19        143315  4.847 1.0000
    OPSO for outdiv.bin      using bits  9 to 18        143682  6.113 1.0000
    OPSO for outdiv.bin      using bits  8 to 17        142667  2.613  .9955
    OPSO for outdiv.bin      using bits  7 to 16        142343  1.495  .9326
    OPSO for outdiv.bin      using bits  6 to 15        142429  1.792  .9634
    OPSO for outdiv.bin      using bits  5 to 14        142446  1.851  .9679
    OPSO for outdiv.bin      using bits  4 to 13        143361  5.006 1.0000
    OPSO for outdiv.bin      using bits  3 to 12        143010  3.795  .9999
    OPSO for outdiv.bin      using bits  2 to 11        143285  4.744 1.0000
    OPSO for outdiv.bin      using bits  1 to 10        143770  6.416 1.0000
 OQSO test for generator outdiv.bin     
  Output: No. missing words (mw), equiv normal variate (z), p-value (p)
                                                           mw     z     p
    OQSO for outdiv.bin      using bits 28 to 32        142043   .453  .6748
    OQSO for outdiv.bin      using bits 27 to 31        141732  -.601  .2739
    OQSO for outdiv.bin      using bits 26 to 30        141957   .162  .5642
    OQSO for outdiv.bin      using bits 25 to 29        141842  -.228  .4097
    OQSO for outdiv.bin      using bits 24 to 28        141246 -2.249  .0123
    OQSO for outdiv.bin      using bits 23 to 27        142033   .419  .6625
    OQSO for outdiv.bin      using bits 22 to 26        142156   .836  .7985
    OQSO for outdiv.bin      using bits 21 to 25        141844  -.221  .4124
    OQSO for outdiv.bin      using bits 20 to 24        141733  -.598  .2750
    OQSO for outdiv.bin      using bits 19 to 23        142105   .663  .7464
    OQSO for outdiv.bin      using bits 18 to 22        141837  -.245  .4032
    OQSO for outdiv.bin      using bits 17 to 21        142478  1.928  .9731
    OQSO for outdiv.bin      using bits 16 to 20        141804  -.357  .3605
    OQSO for outdiv.bin      using bits 15 to 19        142255  1.172  .8794
    OQSO for outdiv.bin      using bits 14 to 18        142183   .928  .8232
    OQSO for outdiv.bin      using bits 13 to 17        141923   .046  .5185
    OQSO for outdiv.bin      using bits 12 to 16        142131   .751  .7738
    OQSO for outdiv.bin      using bits 11 to 15        142149   .812  .7917
    OQSO for outdiv.bin      using bits 10 to 14        142397  1.653  .9508
    OQSO for outdiv.bin      using bits  9 to 13        143045  3.850  .9999
    OQSO for outdiv.bin      using bits  8 to 12        141949   .134  .5535
    OQSO for outdiv.bin      using bits  7 to 11        142753  2.860  .9979
    OQSO for outdiv.bin      using bits  6 to 10        141683  -.767  .2215
    OQSO for outdiv.bin      using bits  5 to  9        142283  1.267  .8974
    OQSO for outdiv.bin      using bits  4 to  8        142426  1.751  .9601
    OQSO for outdiv.bin      using bits  3 to  7        141164 -2.527  .0058
    OQSO for outdiv.bin      using bits  2 to  6        142406  1.684  .9539
    OQSO for outdiv.bin      using bits  1 to  5        141149 -2.577  .0050
  DNA test for generator outdiv.bin     
  Output: No. missing words (mw), equiv normal variate (z), p-value (p)
                                                           mw     z     p
     DNA for outdiv.bin      using bits 31 to 32        142042   .391  .6522
     DNA for outdiv.bin      using bits 30 to 31        141771  -.408  .3416
     DNA for outdiv.bin      using bits 29 to 30        141932   .067  .5267
     DNA for outdiv.bin      using bits 28 to 29        142160   .739  .7702
     DNA for outdiv.bin      using bits 27 to 28        141623  -.845  .1992
     DNA for outdiv.bin      using bits 26 to 27        142383  1.397  .9188
     DNA for outdiv.bin      using bits 25 to 26        141899  -.030  .4878
     DNA for outdiv.bin      using bits 24 to 25        141514 -1.166  .1218
     DNA for outdiv.bin      using bits 23 to 24        141760  -.440  .3298
     DNA for outdiv.bin      using bits 22 to 23        142505  1.757  .9606
     DNA for outdiv.bin      using bits 21 to 22        141620  -.853  .1967
     DNA for outdiv.bin      using bits 20 to 21        141689  -.650  .2579
     DNA for outdiv.bin      using bits 19 to 20        142033   .365  .6424
     DNA for outdiv.bin      using bits 18 to 19        142209   .884  .8116
     DNA for outdiv.bin      using bits 17 to 18        141852  -.169  .4329
     DNA for outdiv.bin      using bits 16 to 17        142076   .492  .6885
     DNA for outdiv.bin      using bits 15 to 16        141715  -.573  .2832
     DNA for outdiv.bin      using bits 14 to 15        142093   .542  .7060
     DNA for outdiv.bin      using bits 13 to 14        141431 -1.411  .0791
     DNA for outdiv.bin      using bits 12 to 13        141915   .017  .5067
     DNA for outdiv.bin      using bits 11 to 12        141988   .232  .5918
     DNA for outdiv.bin      using bits 10 to 11        141988   .232  .5918
     DNA for outdiv.bin      using bits  9 to 10        141534 -1.107  .1341
     DNA for outdiv.bin      using bits  8 to  9        142158   .734  .7684
     DNA for outdiv.bin      using bits  7 to  8        142136   .669  .7481
     DNA for outdiv.bin      using bits  6 to  7        142460  1.624  .9479
     DNA for outdiv.bin      using bits  5 to  6        142439  1.562  .9409
     DNA for outdiv.bin      using bits  4 to  5        142045   .400  .6555
     DNA for outdiv.bin      using bits  3 to  4        142727  2.412  .9921
     DNA for outdiv.bin      using bits  2 to  3        141859  -.148  .4410
     DNA for outdiv.bin      using bits  1 to  2        141981   .211  .5837

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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 outdiv.bin     
 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 outdiv.bin       2442.82      -.809      .209373
 byte stream for outdiv.bin       2302.93     -2.787      .002660

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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  2517.01       .241      .595057
           bits  2 to  9  2462.34      -.533      .297164
           bits  3 to 10  2556.57       .800      .788156
           bits  4 to 11  2520.45       .289      .613813
           bits  5 to 12  2485.64      -.203      .419511
           bits  6 to 13  2518.82       .266      .604944
           bits  7 to 14  2553.19       .752      .774033
           bits  8 to 15  2489.32      -.151      .439963
           bits  9 to 16  2625.58      1.776      .962130
           bits 10 to 17  2567.38       .953      .829667
           bits 11 to 18  2538.34       .542      .706171
           bits 12 to 19  2539.76       .562      .713064
           bits 13 to 20  2422.39     -1.098      .136200
           bits 14 to 21  2535.41       .501      .691757
           bits 15 to 22  2426.46     -1.040      .149166
           bits 16 to 23  2486.57      -.190      .424655
           bits 17 to 24  2574.05      1.047      .852488
           bits 18 to 25  2549.63       .702      .758614
           bits 19 to 26  2488.81      -.158      .437119
           bits 20 to 27  2519.02       .269      .606022
           bits 21 to 28  2575.60      1.069      .857484
           bits 22 to 29  2510.96       .155      .561566
           bits 23 to 30  2643.00      2.022      .978428
           bits 24 to 31  2555.32       .782      .783011
           bits 25 to 32  2538.02       .538      .704608

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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 outdiv.bin     
            Of 12,000 tries, the average no. of successes
                 should be 3523 with sigma=21.9
            Successes: 3520    z-score:  -.137 p-value: .445521
            Successes: 3538    z-score:   .685 p-value: .753306
            Successes: 3468    z-score: -2.511 p-value: .006012
            Successes: 3530    z-score:   .320 p-value: .625377
            Successes: 3515    z-score:  -.365 p-value: .357445
            Successes: 3551    z-score:  1.279 p-value: .899470
            Successes: 3558    z-score:  1.598 p-value: .944998
            Successes: 3519    z-score:  -.183 p-value: .427537
            Successes: 3533    z-score:   .457 p-value: .676028
            Successes: 3548    z-score:  1.142 p-value: .873180
 
           square size   avg. no.  parked   sample sigma
             100.            3528.000       24.216
            KSTEST for the above 10: p=  .629880

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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 outdiv.bin     
     Sample no.    d^2     avg     equiv uni            
           5     .3569    .7062     .301378
          10    1.3184    .5711     .734194
          15     .6724    .6205     .491227
          20     .0105    .7785     .010479
          25     .9794    .7639     .626323
          30     .7994    .7013     .552225
          35    1.2185    .6763     .706145
          40    2.2612    .7247     .896950
          45     .7887    .8014     .547375
          50    1.2189    .8426     .706261
          55    1.1795    .8935     .694390
          60     .1192    .8470     .112878
          65    1.1714    .8521     .691883
          70     .6646    .8741     .487252
          75    2.0129    .8823     .867747
          80     .1384    .9024     .129883
          85    2.4513    .8885     .914873
          90     .3418    .9185     .290759
          95     .9452    .9123     .613235
         100     .9808    .9112     .626818
     MINIMUM DISTANCE TEST for outdiv.bin     
          Result of KS test on 20 transformed mindist^2's:
                                  p-value= .830309

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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 outdiv.bin     
 sample no:  1     r^3=  25.128     p-value= .56725
 sample no:  2     r^3=  57.412     p-value= .85247
 sample no:  3     r^3=   9.269     p-value= .26580
 sample no:  4     r^3=  31.345     p-value= .64824
 sample no:  5     r^3=   8.695     p-value= .25161
 sample no:  6     r^3=  13.252     p-value= .35707
 sample no:  7     r^3=  68.874     p-value= .89932
 sample no:  8     r^3=  49.897     p-value= .81047
 sample no:  9     r^3=  48.028     p-value= .79829
 sample no: 10     r^3=  11.551     p-value= .31957
 sample no: 11     r^3=   5.379     p-value= .16414
 sample no: 12     r^3=  42.369     p-value= .75642
 sample no: 13     r^3= 102.956     p-value= .96767
 sample no: 14     r^3=  42.012     p-value= .75350
 sample no: 15     r^3=  28.034     p-value= .60721
 sample no: 16     r^3=  33.675     p-value= .67453
 sample no: 17     r^3=  12.939     p-value= .35034
 sample no: 18     r^3=  14.394     p-value= .38110
 sample no: 19     r^3=  15.754     p-value= .40852
 sample no: 20     r^3=   3.757     p-value= .11771
  A KS test is applied to those 20 p-values.
---------------------------------------------------------
       3DSPHERES test for file outdiv.bin           p-value= .354591
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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 outdiv.bin     
         Table of standardized frequency counts
     ( (obs-exp)/sqrt(exp) )^2
        for j taking values <=6,7,8,...,47,>=48:
     -.8      .1     -.8      .2    -2.1    -1.5
     1.3     1.2    -1.8    -1.2    -1.1     -.3
      .1      .7      .1     -.1     -.6      .2
      .9      .3     1.0    -1.1     -.4      .9
     -.8     -.4     1.4      .4      .3     -.5
     1.3    -1.5      .6    -1.4      .3      .6
      .7      .5      .1     1.0     1.6      .0
      .8
           Chi-square with 42 degrees of freedom: 37.212
              z-score=  -.522  p-value= .318882
______________________________________________________________

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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  .501186
                Test no.  2      p-value  .411127
                Test no.  3      p-value  .128923
                Test no.  4      p-value  .939062
                Test no.  5      p-value  .421814
                Test no.  6      p-value  .321254
                Test no.  7      p-value  .005718
                Test no.  8      p-value  .650870
                Test no.  9      p-value  .980092
                Test no. 10      p-value  .446905
   Results of the OSUM test for outdiv.bin     
        KSTEST on the above 10 p-values:  .319289

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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 outdiv.bin     
     Up and down runs in a sample of 10000
_________________________________________________ 
                 Run test for outdiv.bin     :
       runs up; ks test for 10 p's: .336458
     runs down; ks test for 10 p's: .385198
                 Run test for outdiv.bin     :
       runs up; ks test for 10 p's: .673492
     runs down; ks test for 10 p's: .750297

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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 outdiv.bin     
  No. of wins:  Observed Expected
                                98797    98585.86
                  98797= No. of wins, z-score=  .944 pvalue= .82750
   Analysis of Throws-per-Game:
 Chisq=  19.09 for 20 degrees of freedom, p=  .48370
               Throws Observed Expected  Chisq     Sum
                  1    67057    66666.7   2.285    2.285
                  2    37712    37654.3    .088    2.374
                  3    26765    26954.7   1.335    3.709
                  4    19305    19313.5    .004    3.713
                  5    13692    13851.4   1.835    5.548
                  6     9971     9943.5    .076    5.623
                  7     7194     7145.0    .336    5.959
                  8     5085     5139.1    .569    6.528
                  9     3558     3699.9   5.440   11.968
                 10     2658     2666.3    .026   11.993
                 11     1907     1923.3    .139   12.132
                 12     1419     1388.7    .659   12.791
                 13      998     1003.7    .033   12.824
                 14      749      726.1    .720   13.544
                 15      537      525.8    .237   13.781
                 16      398      381.2    .745   14.525
                 17      275      276.5    .009   14.534
                 18      212      200.8    .621   15.155
                 19      130      146.0   1.750   16.906
                 20       91      106.2   2.180   19.085
                 21      287      287.1    .000   19.085
            SUMMARY  FOR outdiv.bin     
                p-value for no. of wins: .827503
                p-value for throws/game: .483702

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 Results of DIEHARD battery of tests sent to file outdiv.txt     



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