If $N \in Fs(p)$ and $X_i \in Ge(\alpha)$ and we define $Y=\sum_{i=1}^N X_i$ then show that the distribution of $Y$ is $Ge(\beta)$.
Clearly the characteristic functions are required, as $\varphi_Y (t) = \varphi_N(\varphi_{X_i}(t))$
$\varphi_N (t)= \frac{p*\exp(i*t)}{1-(1-p)\exp(i*t)}: 0 \le p \le 1$
$\varphi_X (t)= \frac{\alpha}{1-(1-\alpha)\exp(i*t)}: 0 \le \alpha \le 1$
So I need to show that there is SOME $\beta$ such that
$\varphi_Y (t)= \frac{\beta}{1-(1-\beta)\exp(i*t)}: 0 \le \beta \le 1$
I'm getting stuck pretty early with the algebra.... maybe a hint would be greatly appreciated as I want to solve this myself. Thank you all.