I am fairly new to stack exchange, but I need some guidance on the following problem:
Let $X$ and $Y$ be two independent, continuous random variables described by probability density functions $f_{X}$ and $f_{Y}$.
$f_{X}(x) \sim Gamma(n+1,1)$.
$f_{Y}(y) \sim p(1-|y|) + \frac{(1-p)}{\pi(1+y^2)}\: p\in[0,1]\; s.t.\; p+(1-p) = 1$.
Given the above definitions compute the probability density function of $Z = XY$.
The formula for the product of two distributions is the following:
$f_Z(z) = \int^{\infty}_{-\infty} f_X \left( x \right) f_Y \left( z/x \right) \frac{1}{|x|}\, dx.$
The integral is quite involved and I have made approaches using numerical integration with no luck.
To clear up the support confusion:
Gamma has support on $x \in (0, \infty).$
The first half of the mixture has support on $y\in[-1,1]$.
The second half of the mixture has support on $y\in(-\infty,\infty)$.
Feel free to correct any of these definitions if they seem odd. I am an undergraduate doing research on heavy-tailed ICA and am "fairly" new to some of this work.
Edit: I believe I have made some parts of my question a bit more clear.