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Math Help - Finding probability generating function of a pdf

  1. #1
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    Finding probability generating function of a pdf

    Would really appreciate help on the following problem. I'm given the pdf of a discrete random variable R given by

    P(R=j) = (j+1)*(1-p)^(2)*p^(j) j=0,1,2,.... and 0<p<1.

    I have to show that the probability generating function of R is:

    G(s)= [(1-p)/(1-ps)]^2

    Apologies for the way I've written it, just tried to use LaTeX but I couldn't get it right.
    Thank you in advance.
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  2. #2
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    Hello,
    Quote Originally Posted by j.matthews View Post
    Would really appreciate help on the following problem. I'm given the pdf of a discrete random variable R given by

    P(R=j) = (j+1)*(1-p)^(2)*p^(j) j=0,1,2,.... and 0<p<1.

    I have to show that the probability generating function of R is:

    G(s)= [(1-p)/(1-ps)]^2

    Apologies for the way I've written it, just tried to use LaTeX but I couldn't get it right.
    Thank you in advance.
    Well your writing is quite clear, so it's not a problem ^^

    G(s)=\sum_{j=0}^\infty s^j P(R=j)=(1-p)^2 \sum_{j=0}^\infty s^j(j+1) p^j=(1-p)^2 \sum_{j=1}^\infty j (sp)^{j-1}


    now, consider the power series \frac{1}{1-x}=\sum_{j=0}^\infty x^j=1+\sum_{j=1}^\infty x^j
    Differentiate :
    \frac{1}{(1-x)^2}=\sum_{j=1}^\infty jx^{j-1}


    hence G(s)=(1-p)^2 \cdot \frac{1}{(1-sp)^2}
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