Let X a random variable with pdf: $$f(y)=\sqrt{\frac{\theta}{y\pi}}e^{-\theta y}, y,\theta>0$$ Find the MGF.
I tried to solve it as follows:
let $u=\sqrt{x}, \frac{1}{\sqrt{y}}dy=2du$
$$M(t)=\int_0^\infty e^{ty}\sqrt{\frac{\theta}{y\pi}}e^{-\theta y}dy=2\sqrt{\frac{\theta}{\pi}}\int_0^\infty e^{-(\theta-t)u^2}du$$
And completing the integral as $N(0,\sigma^2=(2(\theta-t))^{-1})$: $$M(t)=\frac{\sqrt{\theta}}{\sqrt{\theta-t}}$$
It's correct?