Given $X \sim \text{NBin}(n,p)$, I've seen two different calculations for $\mathbb{E} (X)$:
\begin{align*} &1. \mathbb{E} (X) = \frac{n}{p}, \quad \text{or}\\ &2. \mathbb{E} (Y) = \frac{n(1-p)}{p} \end{align*}
Proof for 1.: Proof for the calculation of mean in negative binomial distribution
Proof for 2: Although I can't find a concrete proof on stackexchange, this is the expected value used in the wikipedia article for negative binomials, and I have also seen this value used in some questions here.
I've heard someone say that both are valid depending on the way you define the negative binomial, but I still don't quite understand the difference between the set-ups for the two different $\mathbb{E} (X)$.
Could someone explain their differences? Thank you!