$X$ is a random variable with values in the set of natural numbers and the Generating function G. In Addition: $t(n) = P(X>n)$.
Let $F$ be the generating function of the sequence $\{t(n): n \ge 0\}$, i.e. $F(z) = \sum_{n=0}^\infty t(n)\cdot z^n$.
In this case, the following properties are valid: $F(z) = \frac{(1-G(z))}{(1-z)}$, $E(X) = F(1)$ and $\operatorname{Var}(X) = 2\cdot F'(1) + F(1) - F(1)^2$.
But why is this valid?