Name of distribution of the parameter of a Poissonian

Question

Consider a Poisson process $\hat{n}$ with with parameter $t$ and distribution

$$f_t(n) = e^{-t} \frac{t^n}{n!}$$

Now instead suppose to have a random variable $\hat{t} \in \mathbb{R}^+$ whose distribution, for some integer parameter $n$, is again

$$f_n(t) = f_t(n) = e^{-t} \frac{t^n}{n!}$$

How is this distribution called? Just "power law with exponential cutoff", or there exists some better name? Also, apart from nomenclature, do you know of applications where two related processes $\hat{n}$ and $\hat{t}$ appear together in a sensible manner?

Answer 1

This distribution is called Gamma distribution https://en.wikipedia.org/wiki/Gamma_distribution

Answer 2

This simply is the density of the $\Gamma$-distribution with parameter $\nu = n+1$ and $1$, often abbreviated by $\Gamma_{1,n+1}$. For a Poisson-process with parameter $\lambda = 1$ $\Gamma_{1,n+1}$ is the distribution of the $n$-th arrival.