Probability generating function

The probability generating function $G_N(z)$ is used in place of the characteristic function especially when the random variables are non-negative. In this case, for a non-negative, integer-valued RV $N$:

$$G_N(z)=E(z^N)=\sum _{k=0}^{\infty }{p}_N(k)z^k$$

where the sum is the z-transform of the probability mass function. The characteristic function of $N$ is given by:

\Phi_N(\omega)=G_N(e^{j\omega}) $$Working backwards, we can show that the pmf of $N$ is given by:

p_N(k)=\frac 1{k!}\frac{d^k}{dz^k}G_N(z)\bigg|_{z=0}