So I was trying to find the moment-generating function to the following random variable: $X^{2}+Y^{2}$ such that $X\sim Exp(\lambda)$ and $Y\sim Exp(2\lambda)$, independent to each other. I thought of only finding the distribution of each and using the property:
$M_{X_{1}+X_{2}=M_{X_{1}}.M_{X_{2}}}$ \begin{align} X^{2}&=Z\\ \sqrt{z}&=X\\ f_{Z}&=f_{X}(\sqrt{z})\left| \frac{dx}{dz} \right|\\ &=\lambda e^{-\lambda\sqrt{z}}\frac{1}{2\sqrt{z}} \end{align} So the moment-generating function would be:
\begin{align} E(e^{zt}) &=\int_{0}^{\infty}\lambda e^{-\lambda\sqrt{z}}\frac{1}{2\sqrt{z}}e^{tz}dz\\ &=\frac{\lambda}{2}\int_{0}^{\infty}\frac{1}{\sqrt{z}}e^{-\lambda\sqrt{z}+zt}dz \end{align}