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a canonical normal distribution F(x) where |
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and - x0 is the value of x when y = 0. (I.e., F(y), which describes the distribution of  is transformed to F(x) which describes the canonical normal distribution with mean 0 and variance 1.) The tail of a normal distribution is well approximated by |
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Using the same line of reasoning (but now with (N - n)observations of x1, etc.) we have |
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From this we see that both q' and q" are functions of the variances and means as well as the total number of trials, N, and the number of trials, n,given x'. More importantly, both q' and 1 - q" decrease exponentially with n,yielding |
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with the approximation being quite good even for relatively small n. For p =½ this reduces to |
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where the error is less than min {(q')2, (1 - q")2}. (If one random variable is a priori more likely than the other to be best, i.e., if p ¹ ½,then we can see from |
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