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Talk:Monte Carlo methods: Difference between revisions
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: If you want to use the stddev formula, then each <math>x_i</math> takes the value of either 1 (landing in circle) or 0 (not). The average is <math>\mu = p</math> as mentioned above; now <math>\sigma^2 = {1\over N} \sum (x_i - p)^2</math>. Note that there are going to be about <math>Np</math> of those <math>x_i</math>s with value <math>1</math>, and <math>N(1-p)</math> with value <math>0</math>, so <math>\sum(x_i - p)^2 \approx Np(1-p)^2 + N(1-p)(0-p)^2 = Np(1-p)</math>. See how it comes back to the same formula? --[[User:Ledrug|Ledrug]] ([[User talk:Ledrug|talk]]) 06:17, 5 May 2014 (UTC)
:: Thank you very much that explains the origin of the formula. Even though following your reasoning the formula should be:
:: <math> \sum(x_i - p)^2 \approx Np(1-p)</math>,
:: But because we have a factor of <math> {1\over N} </math> we must take into account, then the resulting stddev should be:
:: <math>\sigma^2 = p(1-p)</math>,
:: Meaning that the formula implemented has an extra factor of <math> {1\over N} </math> inside the square root and a factor of <math> p </math> outside of the square root, meaning:
:: error = val * sqrt(val * (1 - val) / sampled) * 4, when it should be:
:: error = sqrt(val * (1 - val)) * 4;
:: Am I missing something else here? Sorry for the intrigue I'm no expert in probability, but I'm curious as to the implementation.-[[User:Chibby0ne|Chibby0ne]] ([[User talk:Chibby0ne|talk]]) 17:18, 17 May 2014 (UTC)
==Formulae hidden to most browsers by under-tested cosmetic edits at 18:00, 11 September 2016 ==
Under-tested cosmetic edits made to the task page at 18:00, 11 September 2016, including the injection of spaces around expressions in <math> tags, have left some or all of the task description formulae completely invisible to all browsers which display the graphic file version of formulae rather than processing the MathML (this is, in fact, the majority of browsers). The MediaWiki processor does not currently expect such spaces, and generates syntactically ill-formed HTML if they are introduced. Other aspects of these cosmetic edits may further compound the problem. [[User:Hout|Hout]] ([[User talk:Hout|talk]]) 21:20, 22 September 2016 (UTC)
: Now repaired [[User:Hout|Hout]] ([[User talk:Hout|talk]]) 00:20, 16 November 2016 (UTC)
==other REXX (output) examples==
Here are some other REXX examples (runs):
<pre>
scale: 1
234567890123456789012345678901234567890123456789012345678901234567890123
true pi= 3.141592653589793238462643383279502884197169399375105820974944592307816406+
times= 5000000000000
100,000 repetitions: Monte Carlo pi is accurate to 3 places.
200,000 repetitions: Monte Carlo pi is accurate to 4 places.
2,700,000 repetitions: Monte Carlo pi is accurate to 5 places.
2,800,000 repetitions: Monte Carlo pi is accurate to 6 places.
17,700,000 repetitions: Monte Carlo pi is accurate to 8 places.
682,600,000 repetitions: Monte Carlo pi is accurate to 10 places.
</pre>
<pre>
scale: 1
234567890123456789012345678901234567890123456789012345678901234567890123
true pi= 3.141592653589793238462643383279502884197169399375105820974944592307816406+
times= 5000000000000
100,000 repetitions: Monte Carlo pi is accurate to 4 places.
700,000 repetitions: Monte Carlo pi is accurate to 5 places.
6,000,000 repetitions: Monte Carlo pi is accurate to 6 places.
6,100,000 repetitions: Monte Carlo pi is accurate to 7 places.
133,700,000 repetitions: Monte Carlo pi is accurate to 8 places.
362,300,000 repetitions: Monte Carlo pi is accurate to 9 places.
623,300,000 repetitions: Monte Carlo pi is accurate to 10 places.
</pre>
<pre>
1 2 3 4 5 6 7
scale: 1
234567890123456789012345678901234567890123456789012345678901234567890123
true pi= 3.141592653589793238462643383279502884197169399375105820974944592307816406+
times= 5000000000000
100,000 repetitions: Monte Carlo pi is accurate to 4 places.
12,900,000 repetitions: Monte Carlo pi is accurate to 5 places.
36,200,000 repetitions: Monte Carlo pi is accurate to 6 places.
38,400,000 repetitions: Monte Carlo pi is accurate to 7 places.
69,500,000 repetitions: Monte Carlo pi is accurate to 8 places.
476,200,000 repetitions: Monte Carlo pi is accurate to 10 places.
1,959,100,000 repetitions: Monte Carlo pi is accurate to 11 places.
4,006,300,000 repetitions: Monte Carlo pi is accurate to 12 places.
</pre>
<pre>
scale: 1
234567890123456789012345678901234567890123456789012345678901234567890123
true pi= 3.141592653589793238462643383279502884197169399375105820974944592307816406+
times= 5000000000000
100,000 repetitions: Monte Carlo pi is accurate to 3 places.
900,000 repetitions: Monte Carlo pi is accurate to 4 places.
2,100,000 repetitions: Monte Carlo pi is accurate to 5 places.
4,900,000 repetitions: Monte Carlo pi is accurate to 6 places.
471,400,000 repetitions: Monte Carlo pi is accurate to 7 places.
471,600,000 repetitions: Monte Carlo pi is accurate to 8 places.
491,100,000 repetitions: Monte Carlo pi is accurate to 9 places.
</pre>
== Error in all languages ==
Hello!
Regarding Monte Carlo methods the results are not very good in most languages. You can even get worse result with more samples.
Please if anyone with insight could explain this!
Have you tested RDRAND and Monte Carlo methods? I have found some results on MASM32.com.
Two links:
http://masm32.com/board/index.php?topic=2441.0
http://masm32.com/board/index.php?topic=2432.0
The last is for 64 bit and the mc1.zip is OK.
It looks good for Intel Ivy Bridge but not so good for AMD (slower and higher error).
Best regards
Oldboy 1948
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