Graduate Identification of a probability density function

[Ad VanderVen]

TL;DR

Identification of a probability density function.

I have the following probability density function:

$$f(x) = \frac{2^{\frac{1}{2}~\frac{a_{0}-c_{0}}{a_{0}}}~a_{1}^{(-\frac{1}{2}~\frac{a_{0}+c_{0}}{a_{0}})}~c_{1}^{(\frac{1}{2}~\frac{a_{0}+c_{0}}{a_{0}})}~x^{\frac{c_{0}}{a_{0}}}~e^{-\frac{1}{2}~\frac{c_{1}~x^{2}}{a_{1}}}}{\Gamma(\frac{1}{2}~\frac{a_{0}+c_{0}}{a_{0}})}$$

Is this a known probability density function?

 

[Ad VanderVen]

Support: $x \in (0, \infty)$.

 

[mjc123]

I don't know what you call that distribution, but if you make the substitution u = x²/2 then u has a gamma distribution with α = (c₀/a₀+1)/2 and β = a₁/c₁.