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Suppose some event $X$ occurs on average $10$ times per minute. The events are independent of each other. Now, if I have understood correctly, this can be modeled as a Poisson process, and I can ask questions such as:

What is the probability that $X$ occurs more than 15 times per minute?

Now, I have a certain classifier for detecting the events $X$ automatically. Knowing that the process is Poisson, could I help the classifier? I am thinking of something like when the time since the last event increases, the probability of the next event occurring also increases. This probability could be used as an additional parameter for the classifier. Does that make sense?

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I am thinking of something like that when the time since the last event increases, the probability of the next event occurring also increases.

One of the properties of a Poisson process is that its inter-event time is exponentially distributed -- which is memoryless. So what you're thinking --- that "when the time since the last event increases, the probability of the next event occurring also increases" isn't the case. The elapsed time makes no difference to the expected time remaining until the next event.

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As pointed out in the previous answer by Glen_b, if it's a Poisson process, the wait time distribution is memoryless, i.e. the wait time until the next event does not depend on how much time has elapsed already.

However, other point processes models may be appropriate where the probability of the next event does depend on the elapsed wait time and process history more generally, see e.g. self-correcting point processes or Hawkes processes. A good introductory text which discusses both are the lecture notes on temporal point processes by Rassmussen.

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