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I have next statistical problem: I have a box which is empty initially. A ball is introduced in the box randomly with a given probability, $P_{in}$. After some time we count the number of balls. The probability of finding $N$ balls in the box is given by the Poisson distribution $$P_N=\frac{e^{-\lambda}\lambda^{N}}{N!},$$ being $\lambda$ the average number of balls found in the box after a time.

In the problem I have, there is a probability of introducing a ball to the box, $P_{in}$, but also to extract it, $P_{out}$. I would like to know if this probability distribution is known. Thanks for any help.

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    What does “being $N$ the average number of balls found in the box” mean? Is it an exercise from some textbook. Would you give the reference and provide context? – 2017-01-05
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    Thank you for the comment. I have corrected this sentence which was wrong. This is a problem I found during my PhD in statistical physics (I have no reference), where I get two processes similarly to adding or extracting a ball. – 2017-01-05

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You can think of the number of balls in the box being governed by a discrete-time birth-death process (which is a Markov chain), with birth rate $b=P_{in}$ and death rate $a=P_{out}$.

Lets assume $a \neq b$. Then, by (8) in the link above, we have a transient chain if $ab$ with invariant distribution $f(x) = (1-a/b) (a/b)^x$.

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    I think the positive recurrent distribution should be f(x) = (1-b/a)(b/a)^x – 2017-01-05
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    While I'm on the subject: I think the positive recurrent distribution is good for predicting changes in the number of balls, if starting with a large number, but not for "empty initially." – 2017-01-05
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    Yeah, thanks. This problem isn't particularly well specified, but oh well. – 2017-01-05