Solution
= Solution
For a <probability generating function>, $\mathbb ET=g'(1)$ and $\mathbb E[T(T-1)]=g''(1)$. Here
$$
g'(z)=\frac52(14-5z)^{-3/2},\qquad
g''(z)=\frac{75}4(14-5z)^{-5/2}.
$$
Thus $\mathbb ET=5/54$ and $\mathbb E[T(T-1)]=25/324$. Adding the first moment gives $\mathbb ET^2=55/324$, hence
$$
\boxed{\mathbb ET=\frac5{54},\qquad\operatorname{Var}(T)=\frac{55}{324}-\frac{25}{2916}=\frac{235}{1458}.}
$$