probability generating function

**James0502** · January 2nd 2009, 05:39 AM

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

**Isomorphism** · January 2nd 2009, 06:18 AM

Originally Posted by James0502

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

Let $Y = mX + n$ . Your definition for $G_X(s)$ , i.e. sum over k (p(k)*s^k) can alternatively be written as $\mathbb{E}(s^X)$ where $\mathbb{E}(.)$ denotes expectation.

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

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