Name of the distribution with density $P(x) e^{-x/\theta}$, where $P$ is a polynomial with positive coefficients.

Question

The Gamma distribution of shape $k$ and scale $\theta$ has density $\frac1{\Gamma(k)\theta(k)} x^{k-1} e^{-x/\theta}$. Consider the more general distribution with density (up to a normalizing constant) $P(x) e^{-x/\theta}$, where $P$ is a polynomial with positive coefficients. Does this distribution have a name?

Answer 1

This is a convex combination, or equivalently a finite mixture, of Gamma distributions with shape parameter $1,2,\dots,n$ and scale parameter $\theta$, where $n-1$ is the degree of $P$. The nonnegativity of the coefficients of $P$ is crucial to this statement, whereas it is not crucial to this density being well-defined. For example, there is such a distribution with $P(x)=x^2-x+1$, which is not a mixture of Gamma distributions.