Approaching the Solution

With the scilab function solutionSetApproach(l,n,N,K) one can take a set $\cal{G}$ $^{\emptyset}$ with $n$ many members of length $l$ with $N$ many events. This can be repeated $K$ times. For every repetition the number of events $N$ will be multiplied by $K$ thereby increasing the number of events.Some empirical results can be seen below.

A first table (and diagram 3.16) shows 20 successive selections of n=32 individuals using N=64 until N=1280 events. While the mean value is mostly around 100% shows the half-distance between the maximum and the minimum of the mean values a decrease from about 19-17% to about 11-12% while increasing the number of events. Similarly shows the standard deviation in percentage a decrease from about 9.5 - 2.8. The 'real' mean values MEAN1 are centered around 0.031 with a standard deviation STD1 from 0.029 running down to 0.0008.

Figure 3.16: Population with n=32 and increasing events from 64 to 1280 stepsize=64

-->l=5, n=32, N=32, K=20, [MDX]=solutionSetApproach(l,n,N,K)


 MDX  =
 
    N         MEAN [%]	   DIST2 [%]     STD          MEAN1      STD1
    --------------------------------------------------------------------
    64.      101.30208    17.1875      9.5991352    0.0316569  0.0029997
 
    128.     99.699519    19.53125     8.8566371    0.0311561  0.0027677
    192.     100.         15.364583    8.0806361    0.03125    0.0025252
 
    256.     100.0625     13.476563    6.2994203    0.0312695  0.0019686
 
    320.     99.951923    8.59375      5.5318223    0.0312350  0.0017287
 
    384.     100.20461    7.9427083    4.1850025    0.0313139  0.0013078
 
    448.     99.848214    9.8214286    4.419781     0.0312026  0.0013812
 
    512.     100.49716    8.7890625    4.7191576    0.0314054  0.0014747
 
    576.     99.807099    11.892361    5.4225295    0.0311897  0.0016945
 
    640.     100.07813    6.71875      3.1863929    0.0312744  0.0009957
 
    704.     99.784091    7.2443182    3.228669     0.0311825  0.0010090
 
    768.     100.02325    8.1380208    3.4860207    0.0312573  0.0010894
 
    832.     99.961367    8.8341346    3.4679514    0.0312379  0.0010837
 
    896.     99.946121    5.859375     3.1022874    0.0312332  0.0009695
 
    960.     99.934414    6.6666667    3.04064      0.0312295  0.0009502
 
    1024.    100.10102    4.9804688    2.470903     0.0312816  0.0007722
 
    1088.    100.17398    7.0772059    3.6741721    0.0313044  0.0011482
 
    1152.    99.855324    5.3819444    2.952377     0.0312048  0.0009226
 
    1216.    100.         5.7976974    2.8538176    0.03125    0.0008918
 
    1280.    99.933036    5.0390625    2.8314077    0.0312291 0.0008848

In a last example the number of events has been increased until 12800. One can observe that the one-half difference becomes even more smaller approaching 1.7-2.8% deviance from the mean and the standard deviation runs from 3.3 to about 0.7 - 1.1 (cf. diagram 3.17).

 
-->l=5, n=32, N=1280, K=10,
[MDX]=solutionSetApproach(l,n,N,K)

 MDX  =

    N         MEAN [%]	   DIST2 [%]    STD
    ---------------------------------------------
    1280.     100.18415    6.8359375    3.3885234  
    2560.     100.08594    3.2421875    1.7525262  
    3840.     99.927263    2.9817708    1.4970031  
    5120.     99.945313    3.4960938    1.7487093  
    6400.     99.972917    3.0703125    1.2020543  
    7680.     100.         2.0898438    1.0388803  
    8960.     100.01368    2.4497768    1.021788   
    10240.    100.         1.4501953    0.8440676  
    11520.    99.988799    1.7447917    0.7871516  
    12800.    100.01347    2.8632813    1.1348433

Figure 3.17: Population with n=32 and increasing events from N=1280 to N=12800 stepsize=1280