Cannot tell when a probability generating function converges for |s|<1

juanma101285

Hi, I have a problem that is already solved... I thought 3 of the 4 functions were probability generating functions, but I got one wrong and don't know why.

The solution says [itex]g(s)=1+s-s^2[/itex] is not a probability generating function. However, g(1)=1 and I think g(s) converges to 1 for |s|<1. Isn't that correct? If so, what is it that invalidates this function as a probability generating function?

The solution says that [itex]g(s)=(1/3)*(1+s+s^4)[/itex] and [itex]g(s)=(2-s^2)^{-1}[/itex] are prob. gen. functions and that [itex]g(s)=1+s-s^2[/itex] and [itex]g(s)=(1/2)(1+s+s^3)[/itex] are not (I know the last one is not because g(1)=3/2).

Thanks a lot for your help! :) And if you could also give me an explanation of other things I need to look for in a function to tell if it is a p.g.f., I would really appreciate that.

awkward

If [itex]1 + s - s^2[/itex] were a probability generating function, then we would have [itex]P(2) = -1[/itex].