Let $N$ be a discrete random variable having a probability generating function
$G _ { N } ( z ) = e ^ { - 3 + 2 z + z ^ { 2 } }$
Find $\Pr(N ≤ 2)$
If the PGF is in the form of $\sum z ^ { x } p ( x )$, I can work out the PMF of $N$. But how can I deal with this case?
Break up $G_N(z)=e^{-3}e^{2z}e^{z^2}$ and work out the Cauchy product of the last two factors: $$G_N(z)=e^{-3}(1+2z+2z^2+\cdots)(1+z^2+\cdots)=e^{-3}(1+2z+3z^2+\cdots)$$ where $\cdots$ indicates $z^3$ or higher terms which we don't need. Therefore $P(N\le2)=e^{-3}(1+2+3)=6e^{-3}$.