Assume we have k success in a sequences of N independent experiments. The probability mass function of $N$, given $k$, is proportional to $$P(N=x) \propto \dfrac{x!}{(x-k)!} (1-p)^{x-k}$$ where $p$ is the probability of sucess. My question is this: does this law have a name as a known standard distribution?
With a litte modification, we could say that $N_k = N-k \sim$ Negative Binomial distribution, but that is not correct. In fact, the Negative Binomial assumes that the last experiment is a success, but in my case there is no conditionning.