I need to find the expectation E$\big[\frac{1}{Y}\big]$ where Y is distributed by Gamma with parameters $\alpha$ and $y$.
Do I need to find the pdf of $\frac{1}{Y}$ by using some transformation technique or is there an easier way? If not could someone direct me? I don't think I can use a Jacobian, as it is not 1 to 1.
Help much appreciated.
No, you don't need to find the pdf of $1/Y$. Use the Law of the Unconscious Statistician:
$$ \mathbb E[1/Y] = \int_0^\infty \dfrac{1}{y} f_Y(y)\; dy$$
where $f_Y$ is the density for $Y$.