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Topic: PI on duty to check a PRNG
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Luis A. Afonso

Posts: 4,617
From: LIsbon (Portugal)
Registered: 2/16/05
PI on duty to check a PRNG
Posted: Mar 1, 2013 9:31 AM
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PI on duty to check a PRNG

Suppose 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__________ 0.00

Therefore 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 , 3.1417

The stared value is anomalous. . . The PRNG´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 TIMER
LOCATE 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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