# Probability mass function from a generating function

I have the generating function $$G_x(\theta) = \frac{\alpha-1}{\alpha-\theta^2}$$ and I am trying to determine the probability mass function.

I believe I need to determine the Taylor series expansion of this but I am stuck at the stage: $$G_x(\theta)=\frac{ 1-(\frac{1}{\alpha})}{1-(\frac{\theta^2}{\alpha})}$$

As you have correctly mentioned, you'll need to express $$G_x(\theta)$$ as a power series.
Recall that $$(\alpha-\theta^{2})^{-1}=\alpha^{-1}(1-\theta^{2}/\alpha)^{-1}=\alpha^{-1}(1+(-1)(-\frac{\theta^2}{\alpha})+\frac{(-1)(-2)}{2}(-\frac{\theta^2}{\alpha})^{2}+...$$=$$\alpha^{-1}(1+\frac{\theta^{2}}{\alpha}+\frac{\theta^{4}}{\alpha^2}+...)=\frac{1}{\alpha}+\frac{\theta^2}{\alpha^2}+\frac{\theta^4}{\alpha^{3}}+...$$
You can now find $$G_x(\theta)$$, as a power series, and you know that $$P(X=k)$$ is the coefficient of $$\theta^k$$.