Weird definition of negative binomial distribution In a paper I am reading, they define the negative binomial as the following: random variable $X$ has a negative binomial distribution with parameters $p \in (0,1),k \in \mathbb{N}$ if
$$\mathbb{P}[X=t]=\begin{pmatrix}t-k \\ k-1\end{pmatrix}p^{k}(1-p)^{t-k},$$
for all $t \in \mathbb{N}$.
I haven't seen this form for the negative binomial before, and I am a little confused. My questions are

*

*What is the interpretation of this form for the negative binomial?

*It's possible that we can have $k>t$ in which case the binomial coefficient isn't well-defined, surely?

I feel like I am missing something. Thanks.

This formula makes no sense, there must be an error somewhere. For instance, if we have parameters $p=0.2$ and $k=3$ (so the negbin, in the standard interpretation, would give the probability of $t$ failures before we see $k=3$ successes in a sequence of Bernoulli trials with success probability $p=0.2$ - but the interpretation here may be different), we get the following PMF:

This is the wrong way around, the probabilities should (potentially) first rise, then fall.
R code:
pp <- 0.2
kk <- 3
tt <- seq(0,3*kk)
(probs <- choose(tt-kk,kk-1)*pp^kk*(1-pp)^(tt-kk))
plot(tt,probs,pch=19)

