Special case of parameters in the Gamma distribution function?

Question

I am currently working with images using Voronoi tessellations and evaluating the distribution of the corresponding cell areas. These areas are said to be gamma-distributed, derived from the 1D case.

The version of the Gamma distribution (pdf) on Wikipedia has a shape parameter (alpha) and a rate parameter (beta).

See here --> Wikipedia version of the Gamma distribution

I am confused by a paper (and some other papers referencing this one). Here, a different version is said to be "the Gamma function".

See here--> Gamma distribution in the paper, page 436

My question now is: Can someone explain to me in relatively simple terms if this is a special case of a Gamma distribution and/or how both versions relate to each other?

Thank you.

Answer 1

Using the shape/rate parameterization on Wikipedia, the pdf of a Gamma($\alpha, \beta$) random variable $X$ is given for $x>0$ by $$ f(x) = \frac{\beta^{\alpha}}{\Gamma (\alpha)}x^{\alpha-1}\mathrm{e}^{-\beta x}, x> 0, $$ and $f(x) = 0$ otherwise. The version in the linked paper reads for $x>0$ $$ H(x;c) = \frac{c}{\Gamma(c)}(cx)^{c-1}\mathrm{e}^{-cx} $$ and 0 otherwise. If you select $\alpha = c$ and $\beta = c$ for some $c > 0$, then the first specification becomes $$ f(x) = \frac{c^{c}}{\Gamma(c)}x^{c-1}\mathrm{e}^{-cx} = \frac{c}{\Gamma(c)}(cx)^{c-1}\mathrm{e}^{-cx}, $$ which is the same as in the paper. So the version used in the paper is a specific case of the Gamma distribution with shape and rate set to be equal to each other.