Identifying distributions 
The pgf of a random variable $X$ has pgf 
      $$\frac{1}{7}(1+s+3s^2 +s^3 +s^4 )$$
  What is the distribution of $X$ please? 

I know that a pgf is defines as however the $z$ have just been replaced with $s$ in my example
$$
G(z)=E[z^X]=\sum_{x=0}^\infty p(x)z^x.
$$

As in the definition, the PGF is the sum of probabilities times a variable to some power, where the probability is the probability that the power occurs. (Re-read this several times if that sentence doesn't make sense.)
So, we have a PGF of
$$
\begin{align}
\dfrac{1}{7}(1 + s + 3s^2 + s^3 + s^4) &= \dfrac{1}{7} \cdot 1 + \dfrac{1}{7} \cdot s + \dfrac{1}{7} \cdot 3s^2 + \dfrac{1}{7}\cdot s^3 + \dfrac{1}{7}\cdot s^4 \\
&= \dfrac{1}{7} \cdot s^0 + \dfrac{1}{7} \cdot s^1 + \dfrac{3}{7} \cdot s^2 + \dfrac{1}{7}\cdot s^3 + \dfrac{1}{7}\cdot s^4
\end{align}$$
Thus we have expressed the PGF as a sum of probabilities times a variable ($s$) to some power. Hence, the random variable $X$ is equal to $0$, $1$, $3$, and $4$ each with probability $\dfrac{1}{7}$, and equal to $2$ with probability $\dfrac{3}{7}$.
