Knowing that the laplace transform of a Gamma distribution is given by:
$$F_x(s) = \frac{\beta^a}{(s + \beta)^a}$$
and that for Z = X + Y "Sum of two independent Gamma distribution random variables" $$F_z(s) = (\frac{\beta_x}{s + \beta_x})^a (\frac{\beta_y}{s + \beta_y})^b$$
I am trying to evaluate W= X - Y in the S domain for both X and Y are positive.
1- Is it correct to write: $$F_w(s) = (\frac{\beta_x}{s + \beta_x})^a (-1)( \frac{-\beta_y}{s - \beta_y})^b$$
2- If I am interested in the Inverse Laplace (I know it is not easy) how can I get to the two domains namely "$w>0$" and "$w<0$" from one S function. I don't want to get the actual expression of $f_z(z)$ just how is it transformed into two domains.
3- What if $s=\beta_y$ what will be the value or the meaning of $F_w(s)$