Probability Generating Function / Geometric

[spitz]

Homework Statement

a) $P(X=x)=pq^x,\,x\geq 0$

Find the PGF.

b) $P(X=x)=pq^{|x|},\,x\,\epsilon\,\text{Z}$

Find the PGF.

2. The attempt at a solution

a) $G_X(s)=E(s^X)=\displaystyle\sum_{x\geq 0}pq^x s^x=p\displaystyle\sum_{x\geq 0}(qs)^x=\frac{p}{1-qs}$

b) Not sure about this one... Is it: as above for $x\geq0$. And for $x<0$:

$G_X(s)=E(s^X)=\displaystyle\sum_{x>0}pq^{-x} s^{-x}=p\displaystyle\sum_{x\geq 0}(qs)^{-x}=\ldots$

 

[HallsofIvy]

Since |-x|= |x| the negative values of x just double the value.

 

[spitz]

Oh, okay I see. I got it totally mixed up.

Is it: $p+p\displaystyle\sum_{x\geq0}(qs)^{2x}=\ldots$

 

[Ray Vickson]

Homework Statement

a) $P(X=x)=pq^x,\,x\geq 0$

Find the PGF.

b) $P(X=x)=pq^{|x|},\,x\,\epsilon\,\text{Z}$

Find the PGF.

2. The attempt at a solution

a) $G_X(s)=E(s^X)=\displaystyle\sum_{x\geq 0}pq^x s^x=p\displaystyle\sum_{x\geq 0}(qs)^x=\frac{p}{1-qs}$

b) Not sure about this one... Is it: as above for $x\geq0$. And for $x<0$:

$G_X(s)=E(s^X)=\displaystyle\sum_{x>0}pq^{-x} s^{-x}=p\displaystyle\sum_{x\geq 0}(qs)^{-x}=\ldots$

In (b), try writing out a few terms:
$$G = p + pq^1 s^1 + pq^{|-1|} s^{|-1|} + pq^2 s^2 + pq^{|-2|} s^{|-2|} + \cdots .$$

RGV

 

[spitz]

$p+2p\displaystyle\sum_{x>0}(qs)^{x}=p+\frac{2(p+qs-1)}{1-qs}$

C'est correct?