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1. Coin toss

I'm not sure this approach is correct. You cannot say that Y = X^86,381 because toss successions are correlated. If we consider Enuma example and take the particular result HTH then knowing that HT is not 2 heads entails that TH is not two heads. We should introduce some kind of correlation...
2. Prediction Intervals (Critical Levels)

Hello, you want to predict X_{n+1} from X_{1}...X_{n} without knowing \mu and \sigma. So, as you said, 1. the estimated variance of \bar{X}_{n} is S_{n}^{2}/n 2. the estimated variance of X_{i} is S_{n} 3. as a result the estimated variance of X_{n + 1} - \bar{X}_{n} is S_{n}^2 (1 + 1/n)...
3. The mean and variance

If you know exactly the pdf (probability density function) f(x), the formula for the mean is \mu = E[x] = \int x f(x) dx and for the variance \sigma^{2} = E[(x - E[x])^{2}] = \int (x - E[x])^{2} f(x) dx If you only have experimental data, you can estimate the mean and variance of the...
4. Chi square problem

But you do not know the exact frequency of the old technique. So I would suggest to tests : H0 : there is no difference between the two techniques As you observe a large number of events, you can assume gaussian distribution and use a Pearson chi2 test like : chi2 = (250 - 175)^2/(250 +...