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Math Help - Probability Generating Function

  1. #1
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    Probability Generating Function

    A discrete random variable X has a probability mass function given by

    px (x) = (k(p^x))/x for x=1,2...

    where p is a constant such that 0 < p < 1 and k = -1/ln(q) where q = 1-p

    a) Show that the probability generating function of X is given by

    E(t^x) = (ln(1 - pt))/ln(q) for |t| < 1/p

    b) Why must t be restricted to this range?

    any help is greatly appreciated!
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  2. #2
    Moo
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    Hello,

    Well just write the expectation :

    E[t^X]=\sum_{x=1}^\infty \frac{k t^x p^x}{x}=\frac{1}{\ln(1-p)} \sum_{x=1}^\infty -\frac{(pt)^x}{x}

    but the sum is the power series of \ln(1-pt), hence the result.


    Why is it restricted to |t|<1/p ? Because otherwise, the series doesn't converge.
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