Which law has the random variable X if it has generating function $G_X(t)=a(3+2t^2)^3$?
First of all, what I did was try to find the value of $a$. We must have that $G_X(1-)=1\iff \lim_{t\rightarrow1^-}G_X(t)=1 \iff a=\frac{1}{125}$.
Now, I have that $G_X(t)=\frac{1}{125}(3+2t^2)^3$. I've tried looking at the classical laws Bernoulli, Binomial, Geometric, Poisson etc and haven't found a generating function similar to this...
$P(X=0)+P(X=1)t+P(X+2)t^{2}...=\frac 1{125} (27+54t^{2}+36t^{4}+8t^{6})$. $X$ is a r.v. taking the values $0,2,4,6$ with probabilites $\frac {27} {125}$, $\frac {54} {125}$, $\frac {36} {125}$ and $\frac {8} {125}$.