Trying to understand how to find an outlier in a Poisson process. Using example below to help me understand.

A machine creates widgets at a rate of 10 per hour ($\lambda =10$) under normal circumstances (as a Poisson process).

Normally, you would expect the time between widgets to be about 6 minutes. However, it has now been 10 hours since a widget was created.

If I understand this correctly, the probability of this occurring is equal to $e^{-10*10}$. What is the probability that this time between two events (widget-creations) is "abnormal" or an outlier? Is that the same as the probability of the occurrence?

My terminology may not be correct & this question might not even be valid, would appreciate any guidance on this problem.


Whenever you talk about outliers, essentially what you're saying is that the model is a mixture of your real model and an outlier model. So, as you know, the waiting time has a real distribution $X$ that is exponential. If you want to talk about outliers, then you've essentially got a mixture model $Y$ that is a mixture of $X$ and some other distribution $Z$. Decide what $Z$ is, and then it's easy to find the likelihood that an event comes from $X$ or $Z$.


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