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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: $$G_x(\theta)=\frac{ 1-(\frac{1}{\alpha})}{1-(\frac{\theta^2}{\alpha})}$$

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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 $\theta^k$.

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