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Got Variance?
Posted By: Timothy Chow
Date: Tuesday, 25 May 2010, at 9:35 p.m.
In Response To: Got Variance? (Tom Keith)
There are a couple of different distributions floating around here.
One is the distribution of the value V of the position. V is an integervalued random variable. When you win or lose, you win or lose an integer number of points. Obviously this is not a Gaussian distribution because it's discrete.
If we take n independent random samples of V and average the results, then we get another random variable E_{n}. What the central limit theorem says is that E_{n}, suitably normalized, converges in distribution to a Gaussian random variable (provided V satisfies some mild hypotheses, which might not actually be satisfied if the cube is unbounded, but let's ignore that).
If V itself were actually Gaussian then we wouldn't need to pass to the limit; E_{n} would be exactly Gaussian for any n. But because V is not Gaussian, there's a question as to how fast the convergence is. The conventional wisdom, as you say, is that "several hundred trials ought to be enough." But according to several people, it doesn't seem to be enough. So is this a psychological misperception on their part or is this a real phenomenon? This is what we're discussing. To go about it, the first step would seem to be to reconstruct the distribution of V itself as accurately as possible. Once we have that then we will have a handle on E_{n} and will be able to study its rate of convergence.

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