 ### fix typos

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 ... ... @@ -122,8 +122,8 @@ which simply is the estimated standard deviation of the set of $\langle r^2(L)\r #### Approximate analytical formulas Apart from the numerical bootstrap procedure , there are also approximate analytical formulas. When computing the error of the estimate in Eq. (2) we must remember that both$w_k$and$r_k^2\$ are random variables. Following [this stackexchange post](https://stats.stackexchange.com/questions/25895/computing-standard-error-in-weighted-mean-estimation/33959#33959) the best approximation comes from theory that deals with approximating the error of the expectation value of a ratio of random variables. In a compact form (equivalent to the formula in stackexchange post), the error can be written as $$s(\langle r^2(L)\rangle) = \sqrt{\frac{N}{(N-1)} \frac{\sum_{k=1}^N \left(w_k^{(L)})^2 \left(r^2_k(L) - \langle r^2(L)\rangle\right)^2}{\left(\sum_{k=1}^N w_k^{(L)}\right)^2} }\tag{5}$$ Note that this is an approximation (derived for example in [W.G. Cochrane, Sampling Techniques](https://books.google.com/books/about/Sampling_Techniques.html?id=8Y4QAQAAIAAJ)). [Gatz and Smith](https://doi.org/10.1016/1352-2310(94)00210-C) compared various approximate formulas to the result of bootstrapping for some weather data and found that Eq. (5) gave the best agreement. We confirmed this also for the polymer project. $$s(\langle r^2(L)\rangle) = \sqrt{\frac{N}{(N-1)} \frac{\sum_{k=1}^N \left(w_k^{(L)}\right)^2 \left(r^2_k(L) - \langle r^2(L)\rangle\right)^2}{\left(\sum_{k=1}^N w_k^{(L)}\right)^2} }\tag{5}$$ Note that this is an approximation (derived for example in [W.G. Cochran, Sampling Techniques](https://books.google.com/books/about/Sampling_Techniques.html?id=8Y4QAQAAIAAJ)). [Gatz and Smith](https://doi.org/10.1016/1352-2310(94)00210-C) compared various approximate formulas to the result of bootstrapping for some weather data and found that Eq. (5) gave the best agreement. We confirmed this also for the polymer project. ### The problem with Rosenbluth ... ...
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