# reference on some hitting time

I am looking for references about some probability distribution.

Here are some elements of definition.

### Associated probabilities

• Support : $T(\Omega) = \mathbb{N} \backslash \{0,1\}$

• Probability mass function : $\forall k \in \mathbb{N}^*, P(T = k) = \frac{k-1}{k!}$ equivalent to :

• Cumulated probability function : $\forall k \in \mathbb{N}, P(T > k) = \frac{1}{k!}$
• Expected value : $E[T] = \mathrm{e}$

### One way to realize the distribution

Consider a "Polyá-like" urn, starting with only one green ball.

Each time a green ball is picked, it is replaced back in, together with a red ball.

$T$ is then realized as the number of turns so as to pick a red ball for the first time.

### What I would like to know

1. Does this distribution have a name on its own ?
2. Is there some family of distributions that this belongs to ? A natural candidate seems to be that obtained by changing every parameter of the urn model above.
3. More generally, how to find more info in cases like that, when one knows the probability distribution, but no name ?

Cheers,