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(Apologies in advance for using wrong terms and probably asking the wrong question)

Given a standard distribution with some $\mu$ and $\sigma$, and a single measurement $m$. Can I calculate the likelihood that $m$ "belongs" to the given standard distribution?


For context: I'm working on a real-time monitoring solution. We measure a count at a regular interval. The measured counts follow a normal distribution, which give me my $\mu$ and $\sigma$. Now given an alerting threshold, I want to calculate the specificity (for false positives) and sensitivity (for false negatives). Calculating the specificity is straightforward using the CDF. I'm struggling with calculating the sensitivity though. If I can calculate the likelihood that a measurement "belongs" to the normal distribution, then I can calculate the sensitivity. But I don't know how to do this, hence this question.

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You just want to calculate how many standard deviations ($\sigma$) your observation ($m$) is away from the mean ($\mu$) and then apply z-score relations to find the probability of the observation.

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  • $\begingroup$ Could you elaborate a bit more on this? I'm using the z-scores for the specificity, as it tells me what % of samples I can expect to fall outside of the threshold given normal operations. But I don't know how to use z-scores to calculate the sensitivity. $\endgroup$
    – Tiddo
    Commented Sep 28, 2021 at 14:08
  • $\begingroup$ Can you explain more what you mean exactly by sensitivity? $\endgroup$
    – Adam Kells
    Commented Sep 28, 2021 at 14:27
  • $\begingroup$ Sensitivity is the "true negative" rate, i.e. the inverse of "false negatives". E.g. if it's 90%, then it tells me that 90% of the actual incidents will be detected, but 10% won't be. $\endgroup$
    – Tiddo
    Commented Sep 28, 2021 at 17:03

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