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Math Help - probability generating function

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    probability 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
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  2. #2
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    Quote Originally Posted by James0502 View Post
    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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