What is the name of the random variable $X$ with distribution $\mathbb P(X = i) = {n-i \choose k-1}/{n \choose k}$?

Question

Fix integers $1 \le k \le n$ and let ${n \choose k}$ be the binomial coefficient.

Question. What is the name of the random variable $X$ supported on $\{1,2,\ldots,n\}$ such that $\mathbb P(X = i) = {n-i \choose k-1}/{n \choose k}$ ?

Answer 1

This is a location-transformed special case of the negative hypergeometric distribution $$\Pr[X = k] = \frac{\binom{k+r-1}{k}\binom{N-r-k}{K-k}}{\binom{N}{k}}, \quad k \in \{0, 1, 2, \ldots, K\}.$$

To see this, let $k = i-1$, $r = 1$, $N = n$, and $K = n-k$: we get

$$\Pr[X = i] = \frac{\binom{1}{1} \binom{n-1-(i-1)}{n-k-(i-1)}}{\binom{n}{n-k}} = \frac{\binom{n-i}{(n-i)-(k-1)}}{\binom{n}{k}} = \frac{\binom{n-i}{k-1}}{\binom{n}{k}}, \quad i \in \{1, 2, \ldots, n-k+1\},$$ where we have used the reflection identity $\binom{n}{n-k} = \binom{n}{k}$. The location transformation applies in the sense that the support is shifted by $1$. Note that your support is too large; for $i > n-k+1$, the numerator is zero.

I will leave it to the reader to browse the Wikipedia article to develop an interpretation of the meaning of such a distribution for this special case.