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I've got a binomial distribution with unknown probability p (if it matters, it's somewhere around 0.01). If I estimate the value of p the obvious way by running a bunch of trials and dividing the number of successes by the number of trials, it takes a while: the quality of the estimate increases as the square of the number of trials. Is there a way to speed things up? --Carnildo (talk) 21:35, 1 February 2012 (UTC)
See Bayesian inference#Posterior distribution of the binomial parameter. The mean value of p is not the number of successes i divided by the number of trials n, but rather μ=(i+1)/(n+2), and the standard deviation is σ=√(μ(1−μ)/(n+3)). So if n=7 and i=0 you get p ≈ μ±σ = 0.111111±0.0947559. This may also be written p=0.11(9), see concise notation#Measurements. Bo Jacoby (talk) 11:49, 2 February 2012 (UTC).
Yes, but why not use the correct formula? The case n=i=0 gives p=0.5000(2887), which makes sense, while your formula, p=i/n=0/0, does not make sense. Solving the equations and gives and . Neither 0.0106(26) nor 0.0100(25) give integer values of n and i. There is no way to speed things up. Bo Jacoby (talk) 15:05, 3 February 2012 (UTC).