I need a little help in solving a homework exercise from uni.
Let X and Y be two independent random variables with distribution given by:
$$ f(x) = \frac{2}{\pi}\frac{1}{e^{x}+e^{-x}} $$
It says to compute the distribution of : $$ Z = X + Y $$
If the density functions of $Z$, $X$, $Y$ are $f\left(z\right)$, $f\left(x\right)$, $f\left(y\right)$, where $z = x + y$, then
$$ f\left(z\right) = \int_{-\infty}^{+\infty} f(x) f(z-x) \text{d}x $$
which means
$$ f\left(z\right) = \int_{-\infty}^{+\infty} -\dfrac{2\mathrm{e}^z\left(\ln\left(\mathrm{e}^{2x}+\mathrm{e}^{2z}\right)-\ln\left(\mathrm{e}^{2x}+1\right)\right)}{{\pi}^2\left(\mathrm{e}^{2z}-1\right)} \text{d}x $$
and them integrate from $-\infty$ to $+\infty$
$$ f\left(z\right) = \frac{4ze^{z}}{\pi^{2}\left(e^{2z}-1\right)} $$
And now, how to find the distribution of Z ?