13.4 Exercise 20

If X is a nonnegative integer-valued random variable, express the generating function as a power series.

1. Show that the kth derivative at $s = 1$ is
   $g_{X}^{\left(k\right)} \left(1\right) = E \left[X \left(X - 1\right) \left(X - 2\right) \hdots \left(X - k + 1\right)\right]$
2. Use this to show the $Var \left[X\right] = g_{X}^{' '} \left(1\right) + g_{X}^{'} \left(1\right) - \left[g_{X}^{'} \left(1\right)\right]^{2}$.