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 Jan 7 comment Confidence interval for sum of random subsequence generated by coin tossing Maybe, my first formulation of this question was ambiguous. As already said, I am sorry for that. However, it was also not clear the way you interpreted it. It would be useful to ask for clarification, before answering immediately and complaining afterwards. Jan 7 comment Confidence interval for sum of random subsequence generated by coin tossing I am sorry for your inconvenience, this is my first question here. I think the formulation of the question was somewhat misleading due to a small typo. See edit. Jan 7 revised Confidence interval for sum of random subsequence generated by coin tossing added 186 characters in body; edited title Jan 7 comment Confidence interval for sum of random subsequence generated by coin tossing Do not understand me wrong, I appreciate your answer below (although my reputation is still too low to rate it as useful). It is the best I have so far. I agree that a general solution might not exist. However, there may be a better solution for the special case $\pi_i = (1-\rho) \rho^{i-1}$. At least for another special case ($\pi_i=\frac{1}{n}$ if $i\leq n$ and $\pi_i=0$ else) there exists a better one: Since $n X$ is binomial, an optimal confidence interval can be constructed easily and calculated using numerical means. Jan 7 revised Confidence interval for sum of random subsequence generated by coin tossing added 439 characters in body Jan 7 revised Confidence interval for sum of random subsequence generated by coin tossing added 439 characters in body Jan 7 suggested rejected edit on Sum of random subsequence generated by coin tossing Jan 7 comment Confidence interval for sum of random subsequence generated by coin tossing Assume $\pi_i = \frac{1}{N}$ for $i\leq N$ and $\pi_i = 0$ else. Then $X$ corresponds to a binomial distribution, for which the standard normal approximation is justified for $\max(Np,N(1-p))\gg 1$. Therefore, due to similar reasons, I doubt that your approximation works well for the geometric sequence, if for example $\rho=0.5$ and $p=0.9$. I would like to know a method, which allows an accurate and efficient numerical evaluation of such confidence intervals for any $\rho$ and $p$. Jan 7 comment Confidence interval for sum of random subsequence generated by coin tossing Thanks for your reply. However, I miss some condition, for which this standard normal approximation works well. Jan 7 awarded Editor Jan 7 awarded Student Jan 7 revised Confidence interval for sum of random subsequence generated by coin tossing deleted 2 characters in body Jan 7 asked Confidence interval for sum of random subsequence generated by coin tossing