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User talk:Ledrug: Difference between revisions
→Finding pi using MonteCarlo method C Implementation
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:: 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)
:: --00:17, 18 May 2014 (UTC)▼
::: What I wrote above was only to demonstrate how <math>p(1-p)</math> comes about; it's not actually how variance is derived. For that you need to take the binomial distribution, and calculate the variance of observed <math>n</math> (see [[wp:Variance#Binomial_distribution]]). When you throw <math>N</math> events, and receive <math>n</math> positives, the variance of <math>n</math> is <math>\mathrm{Var}(n) = Np(1-p)</math>. Without getting into too much details, let's say your result really means <math>N{\pi\over 4} = n\pm \sqrt{Np(1-p)}</math>, which leads to <math>\pi = {4n\over N} \pm 4 \sqrt{p(1-p)\over N}</math>, i.e., <code>error = sqrt(val*(1 - val)/sampled)*4</code>. I must admit that the first <tt>val</tt> in the original code is spurious; I don't know what I was thinking. But that shouldn't give an error that's off by orders of magnitude.
▲--00:17, 18 May 2014 (UTC)
::: As a rule of thumb, sum of random samples deviates from "true" value by <math>\sqrt{N}</math>, while mean of random samples deviates by <math>1/\sqrt{N}</math>, as long as the distribution is something reasonable. This is why repeating an experiment many times reduces statistical uncertainty, but repeating too many times might not be worth the effort (because of the square root). Now I should go fix the code. --[[User:Ledrug|Ledrug]] ([[User talk:Ledrug|talk]]) 08:14, 18 May 2014 (UTC)
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