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I would like to know what is the unconditional distribution of a variable that comes from a negative binomial

$ X\sim\operatorname{NB}(r,p) $

where

$ r\sim\operatorname{Pois}(\lambda) $

and

  • $r$ is the number of 'failures' until the experiment is stopped: that number is distributed as the above Poisson distribution with $k$ as the number of events in a certain experiment, and $\lambda$ as the average number of events in a given time interval
  • $p$ is the probability of obtaining an event classified as a 'success' in an experiment

Thanks in advance for any help!

EDIT

Hoping that this heps in any way... The trick I'm really looking forward is some sort of algebraic magic that allows me to express the unconditional distribution into some other known distribution -ideally, a Poisson-, and to understand how the parameters of $X$ and $r$ gets mapped into the ones of the unconditional distribution.

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    I answered but skimming again, I'm confused. What is this $k$ parameter in your Poisson distribution (last I checked Poisson had one parameter, $\lambda$)? In any event you plug whatever it is into the formula below. – 2017-01-28
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    You're obviously right, thanks for pointing that out and sorry for the mistake. I might have been a little too vague with my question (given that your answer is valuable indeed): what I'm trying to do here is to simplify the unconditional distribution into something that resembles another poisson distribution, and to understand how the parameters of $X$ and $r$ gets mapped into that last poisson. I'm not even sure this is possible, and if it is, I'm not even sure that the unconditional distribution would be of poisson form: I'm quite blindly trying... So, thanks again for any help – 2017-01-29

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We have a conditional negative binomial$$P(X = k| R=r) = {{k+r-1}\choose{k}}(1-p)^rp^k$$ where $R$ is Poisson $$ P(R=r) = \frac{\lambda^r}{r!}e^{-\lambda}$$

To get the unconditional we must average over $r$ giving $$ P(X=k) = \sum_{r=1}^\infty P(X=k|R=r)P(R=r),$$ so it's a matter of plugging the two expressions in and simplifying if possible.