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Suppose we have a system of N balls, each of which can be in one of two boxes. A ball in box I stays there for a random amount of time with exponential(lambda) distribution and then moves instantaneously to box II. A ball in box II stays there for a random amount of time with exponential(lambda) distribution and then moves instantaneously to box I. All balls act independently of each other. Let Xt be the number of balls in box I at time t.

  • a) I'm trying to show that X is a birth and death process and specify the birth and death rates.

  • b) How can we find the stationary distribution of the process.

For part (a), if we can show it satisfies a Yule process, this is essentially what we're trying to do. And for part (b), I also want some clarification on the detailed balanced equations for this problem.

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    this is the ehrenfest urn model in continuous time. At the event times of a poisson $ \lambda$ process $X_t$ goes up or down by 1.2012-04-13
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    Like @mike said, except the intensity of the Poisson process is $N\lambda$. (Unrelated: *Yule processes* are pure birth hence they cannot model bounded populations like here.)2012-04-14

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