Going from a rate to a probability

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SUMMARY

The discussion centers on simulating a system where a rate function, specifically the rate of atoms crossing a boundary, is used to determine the expected time for the first atom to cross. The key insight provided is the application of the Poisson distribution to model the stochastic nature of this process. Participants emphasize the need to convert the rate of occurrence into a probability framework to accurately predict the timing of events. This approach allows for a clearer understanding of the average time until the first atom crosses the boundary.

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  • Understanding of stochastic processes
  • Familiarity with the Poisson distribution
  • Knowledge of rate functions in probability theory
  • Basic concepts of simulation modeling
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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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