Unconditional distribution of a negative binomial with poisson mean

Question

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.

Answer 1

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.