Is the discrete random variable X has p.g.f Gx(s) what is the p.g.f of mX + n?

I know the p.g.f is given by sum over k (p(k)*s^k) but I'm not sure where to go from here

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- Jan 2nd 2009, 05:39 AMJames0502probability generating function
Is the discrete random variable X has p.g.f Gx(s) what is the p.g.f of mX + n?

I know the p.g.f is given by sum over k (p(k)*s^k) but I'm not sure where to go from here - Jan 2nd 2009, 06:18 AMIsomorphism
Let $\displaystyle Y = mX + n$. Your definition for $\displaystyle G_X(s)$, i.e. sum over k (p(k)*s^k) can alternatively be written as $\displaystyle \mathbb{E}(s^X)$ where $\displaystyle \mathbb{E}(.)$ denotes expectation.

$\displaystyle G_Y(s) = \mathbb{E}(s^Y) = \mathbb{E}(s^{mX+n}) = s^n\mathbb{E}((s^m)^X) = s^n G_X(s^m)$