I want to calculate the MGF of $$ \left(\frac Y X \right)^\alpha, $$ where R.V.'s $Y \in Exp(1)$, $X$ has the Laplace transform $L_X(s)=e^{-s^\alpha}$ and $\alpha \in (0,1)$. $X$ and $Y$ are independent.
I'm thinking that maybe I could compute the MGF as $$ \psi_{\left(\frac Y X\right)^\alpha}=\psi_{(Y^\alpha X^{-\alpha})}=E\left[e^{t(Y^\alpha X^{-\alpha})}\right]= $$ $$ =\int_0^{\infty}\int_0^{\infty} e^{t \cdot y^\alpha \cdot x^{-\alpha}} f_X(x^{-\alpha}) f_Y(y^\alpha) dxdy $$ However, this is where I get unsure of how to continue. Am I on the right path and how should I compute the PDF for Laplacian?