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I’m trying to figure out the best split of time between measuring either background or signal+background in a counting experiment in the case where we have no prior knowledge about the mean signal count rate.

Assuming we have only a limited time T to perform measurements and there is no timing uncertainty.

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    $\begingroup$ this depends very much from the background to signal strength, so you should make a short measurement of the two, to determine it. if the signal strength is high, you take longer for the background , otherwise more for the combined measurement. $\endgroup$
    – trula
    Commented Oct 20, 2023 at 12:13
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    $\begingroup$ (+1) Adding a bit to what @trula commented, the relative undesirability of false positives and false negatives may be a factor. If no answer is posted here, your question might get a good answer over at the stats stack exchange (there are many Poisson distribution questions and answers there) or at the signal processing stack exchange. $\endgroup$
    – Ed V
    Commented Oct 20, 2023 at 12:28

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Assuming you're trying to measure a flux, and not just trying to falsify a null hypothesis, I'd suggest equal time on background and signal+background. It's not optimum generally, but it's not far off optimum for any particular case.

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