```Date: Mar 1, 2013 9:31 AM
Author: Luis A. Afonso
Subject: PI on duty to check a PRNG

PI on duty to check a PRNGSuppose we are estimate by a random simulation / Monte Carlo procedure pi=3.14159? . The Probability to have at most a difference relative to the true pi value  is given by the DKW formula:en.wikipedia.org/wiki/Dvoretzky-Kiefer-Wolfowitz inequality  Table of values x and their probabilities, N=16E06 trials.___x<3.1412_____p=0.00___x=3.1412_____p=0.01_____3.1413_______0.13_____3.1414_______0.60_____3.1415_______(1.51)_____3.1416_______(2.00)_____3.1417_______(1.39)____  3.1418_______0.51_____3.1419_______0.10_____3.1420_______0.01>3.1420__________ 0.00Therefore we must observe our empirical results inside the interval [3.1412, 3.1420] if the simulations are working properly. Experience setting:In the Cartesian plane (orthogonal axis, x, y) let be the square with vertices on (0, 0), (1, 0), (1, 1), (0, 1) and C= (0.5, 0.5) the centre of the circumference (radius= 0.5) tangent to the square sides. When a point W= (randomX, randomY) falls inside the circumference with area pi*0.5*0.5 one has:_____(x - .5)^2 + (y - .5)^2 <= 0.25,condition to which a success is counted. Outputs:_3.1417  , 3.1416  , 3.1416  , 3.1416  , 3.1417  , 3.1416  , _3.1422*, 3.1415  , 3.1416  , 3.1417  , 3.1415  , 3.1414  ,_3.1417  , 3.1417  , 3.1417The stared value is anomalous.  .  . The PRNGĀ“s performance is not as good as it was desired. It should be used only for illustrative or pedagogical, not research, purposes.Luis A. Afonso        REM "minus"        CLS        PRINT " (x-.5)^2+(y-.5)^2 <= .25   "        PRINT        all = 1.6E+07        FOR rpt = 1 TO 10        inn = 0        COLOR 13        FOR j = 1 TO all        RANDOMIZE TIMERLOCATE 4, 50: PRINT USING "##########"; all - j        u = (RND - .5) ^ 2 + (RND - .5) ^ 2        IF u <= .5 ^ 2 THEN inn = inn + 1        LOCATE 10 + rpt, 40        PRINT USING "#.####   "; 4 * inn / j        NEXT j        NEXT rpt        END
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