I have a question where it gives us a random variable, that has probability density function as the weighted sum of an inverse gaussian function and a Lomax distribution
$$f_{X}(x) = b\cdot \frac{\alpha}{\sqrt{2\pi\beta}} x^{-\frac{3}{2}} \mathrm{exp}\left(-\frac{(\beta x - \alpha)^2}{2\beta x}\right) + (1-b) \cdot \frac{\gamma \kappa ^\gamma}{(\kappa + x)^{\gamma +1}}$$
where the weights are $b,1-b$. I was wondering what would be some interesting properties of the weighted sum of two distributions? I can't seem to be able to find information online about it.
A silly question (i think) I just thought of was, if I wanted to simulate values of $X$, can I just define $Y,Z$ such that
$Y$ has pdf $f_{Y}(y)=\frac{\alpha}{\sqrt{2\pi\beta}} x^{-\frac{3}{2}} \mathrm{exp}\left(-\frac{(\beta x - \alpha)^2}{2\beta x}\right)$
and Z has pdf $f_{Z}(z) = \frac{\gamma \kappa ^\gamma}{(\kappa + x)^{\gamma +1}}$
and then simulate values of $Y,Z$ and add them together to get $X$?