# 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.

• Can you add a link to the paper? This looks strange indeed. Commented Sep 30, 2022 at 9:10
• Some fonts, especially in PostScript, occasionally do not show the vertical crossbar in the "+" symbol well. Most likely the numerator of the Binomial coefficient is $t+k,$ not $t=k.$ But that can't explain the appearance of "$t-k$" in the exponent of $1-p.$
– whuber
Commented Sep 30, 2022 at 15:59

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:
pp <- 0.2