deletions | additions       diff --git a/Burst_Weights_Theory.tex b/Burst_Weights_Theory.tex  index e78f5f6..68bb230 100644  --- a/Burst_Weights_Theory.tex  +++ b/Burst_Weights_Theory.tex 
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\operatorname{Var} (E_i) = \frac{E_p\,(1 - E_p)}{n_{ti}}  \end{equation}  Bursts with higher counts carries a counts, provide  more accurate estimations of the population PR PR,  since their PR  variance will be smaller. is smaller (eq.~\ref{eq:E_var}).  Therefore, in estimating the population PR we need to "focus"   on bigger bursts.  Traditionally, this is accomplished by merely discarding bursts 
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When the statistics $\hat{p}$ is an unbiased estimator of a distribution   parameter and the equality holds in eq.~\ref{eq:cramer_rao},  the estimator is a minimum-variance unbiasedestimator  (MVUB)and the  estimator and it  is said to be efficient (meaning that it does an optimal use the information contained in the sample to estimate the  parameter).  A population of $N$ bursts can be modeled by a set of $N$ binomial  variables with same success probability $E_p$ and varying number of successes  equal to the burst size. Now, notice An estimator for $E_p$ can be constructed  noticing  that the sum of binomial RV with same success probability is still a binomial (with number of trial equal to   the sum of the number of trials).  Therefore we can build an estimator for $E_p$ by computing the proportion of  success from Taking  the sumover all bursts  of acceptor counts o over all bursts  divided by the total number of photons as in eq.~\ref{eq:E_estim}. eq.~\ref{eq:E_estim}, we obtain   the proportion of successes $\hat{E}$  \begin{equation}  \label{eq:E_estim}  \hat{E} = \frac{\sum_i n_{ai}}{\sum_i n_{ti}}  \end{equation}  The variance of$\hat{E}$  (eq.~\ref{eq:E_variance}) is equal to the inverse of the Fisher information $\mathcal{I}(\hat{E})$ and therefore $\hat{E}$ is a MVUB   estimator for $E_p$.