Given a function $G(z)$, what are the sufficient and necessary conditions for being a $G(z)$ to be a probability generating function.

Few necessary condition which I know of are

  1. $G(1) = 1$
  2. All the coefficients of $z^n$ must be positive.

Thanks in advance!


Sufficient and necessary condition for $G$ to be a generating function is that $G$ is smooth on $(-1,1)$ with every derivative $G^{(n)}(0)$ at $0$ nonnegative and $\sum\limits_{n=0}^\infty\frac1{n!}G^{(n)}(0)=1$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.