A question involving the probability generating function I'm stumped by the following exercise: The probability generating function for X is given by $g_{X}(t)$. What is the probability generating function for $X+1$ and $2X$, in terms of $g_{X}(t)$?
For the first part I think I've got an answer down although I can't say I'm 100% sure it's correct.
This is what I got: $g_{X+1}(t)=E(t^{X+1})=E(t^{X}*t)=tE(t^{X})=tg_{X}(t)$
When it comes to the probability generating function for 2X, I don't know what to do though.
I'd be grateful for any answers.

WLOG, consider that $X$ is a discrete random variable, which assumes values in the set $\mathcal{X}$. Using the definitions....
$$ g_{2X}(t) = \mathbb{E}[t^{2X}] = \sum_{x \in \mathcal{X}} p(x)t^{2x} = \sum_{x \in \mathcal{X}} p(x)(t^{2})^x = g_X(t^2).$$
In general:
$$ g_{aX+b}(t) = \mathbb{E}[t^{aX+b}] = \sum_{x \in \mathcal{X}} p(x)t^{ax+b} = \sum_{x \in \mathcal{X}} p(x)(t^{a})^xt^b = t^bg_X(t^a).$$
