Expectation of Square of Discrete Random Variable

Theorem
Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$.

Then the square of the expectation of $X$ is given by the expression:
 * $E \left({X^2}\right) = \Pi''_X \left({1}\right) + \Pi'_X \left({1}\right)$

where $\Pi''_X \left({1}\right)$ and $\Pi'_X \left({1}\right)$ denote the second and first derivative respectively of the PGF $\Pi_X \left({s}\right)$ evaluated at $1$.

Proof
From Derivatives of Probability Generating Function at One:
 * $\Pi''_X \left({1}\right) = E \left({X \left({X - 1}\right)}\right)$

and from Expectation of Discrete Random Variable from PGF:
 * $\Pi'_X \left({1}\right) = E \left({X}\right)$

So: