I want to calculate probability density function (PDF) of $Z=X-Y$, where $X$ and $Y$ are independent. The distribution of $X$ is given by the following equation:
$$f_{X}(x)=\frac{1}{\Gamma{(\alpha_{x})}\beta_{x}^{\alpha_{x}}}\ x^{\alpha_{x}-1}\ e^{\left(-\frac{x}{\beta_{x}}\right)},$$
where $\alpha_{x}>0, \beta_{x}>0 $.
The distribution of $Y$ is given by the following equation:
$$ f_{Y}(y)=\frac{\beta_{y}}{\alpha_{y}\Gamma(1/\beta_{y})}\ y \ e^{-\left(\frac{|y^{2}-\mu|}{\alpha_{y}}\right)^{\beta_{y}}},$$
where $\alpha_{y}>0, \beta_{y}>0 $.
And, random variable $X$ and $Y$ are $ x>y>0 $.
I tried this by using convolution. But, it is so complicated… And also I tried calculation of characteristic function of $Y$. But that is also complicated. How can I get this PDF of $Z$?