Resampling Techniques, Bootstrap and Blocking

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Running many measurements

With the assumption that the average measurements $i$ are also defined as iid stochastic variables and have the same probability function $p$ , we defined the total average over $m$ experiments as

$\overline{X}=\frac{1}{m}\sum_{i} \overline{x}_{i}.$

and the total variance

$\sigma^2_{m}=\frac{1}{m}\sum_{i} \left( \overline{x}_{i}-\overline{X}\right)^2.$

These are the quantities we used in showing that if the individual mean values are iid stochastic variables, then in the limit $m\rightarrow \infty$ , the distribution for $\overline{X}$ is given by a Gaussian distribution with variance $\sigma^2_m$ .