Probability generating function for negative values of random variables?

Question

What if we have negative integral values for a random variable?Then is it possible to write a probability generating function for it? All definitions I have seen so far is for non negative integer values. I hope someone could assist me. Thanks

Answer 1

Yes, you could define a possion distribution on negative integers as follows. Let support of $X$ be $\{0,-1,-2,...\}$ and let pdf of $X$ be $$f(x)=\frac{e^{-\lambda}\lambda^{-x}}{(-x)!}$$ compared to $Y$ with support $\{0,1,2,...\}$ with possion has pdf $$f(y)=\frac{e^{-\lambda}\lambda^{y}}{(y)!}$$

Answer 2

Using probability generating functions $s\mapsto E(s^X)$ for $s$ in $[0,1]$, for negative valued random variables $X$ would be taking the risk of manipulating divergent series. This is the reason why random variables with negative values (integer or not) are best dealt with using characteristic functions $t\mapsto E(\mathrm e^{\mathrm i tX})$ for every real number $t$, always well defined and allowing every manipulation that probability generating functions would.