Going from a rate to a probability

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The discussion centers on simulating a system where atoms pass a boundary at a specified rate, but the challenge lies in translating this rate into a probability for crossing over time. The user seeks guidance on how to model the stochastic nature of the system, given that the function provided is a rate rather than a direct probability. A suggestion is made that the Poisson distribution may be applicable for determining the expected time until the first atom crosses the boundary. The conversation highlights the need for a clearer understanding of how to convert the rate function into a probabilistic framework. Ultimately, the focus is on finding an effective method to simulate the crossing event based on the given rate.
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I am simulating a system where I have as a function of time the rate at which something happens, like for instance n atoms per second passes a boundary. Now what happens in the system is that as soon as one atom passes the boundary, everything goes back to initial values. So my problem is that I don't have a very good idea on how to simulate the stochastic nature of the system given that the function I have is not probability for crossing as a function of t but rather a rate. Does anyone have an idea on what to do?
Edit: So basically I want to know that given this rate function, when should I expect the first atom to cross the boundary on average.
 
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Not completely clear from you description... But it sounds as if you want the Poisson distribution.
 

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