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
- Does this distribution have a name on its own ?
- 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.
- More generally, how to find more info in cases like that, when one knows the probability distribution, but no name ?
Cheers,
