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.
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 $a<b$, null recurrent chain if $a=b$ and a positive recurrent chain if $a>b$ with invariant distribution $f(x) = (1-a/b) (a/b)^x$.