Finding probability generating function of a pdf

**j.matthews** · March 11th 2009, 10:24 AM

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.

**Moo** · March 11th 2009, 11:07 AM

Hello,

Originally Posted by j.matthews

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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