Variance of Discrete Random Variable from PGF

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

Then the variance of $X$ can be easily obtained from the value of the second derivative of $\Pi_X \left({s}\right)$ WRT $s$ at $x=1$:


 * $\operatorname{var} \left({X}\right) = \Pi''_X \left({1}\right) + \mu - \mu^2$

where $\mu = E \left({x}\right)$ is the expectation of $X$.

Proof
From the definition of the probability generating function:
 * $\displaystyle \Pi_X \left({s}\right) = \sum_{x \ge 2} p \left({x}\right) s^x$

Differentiating this twice WRT $s$ gives us:
 * $\displaystyle \Pi''_X \left({s}\right) = \sum_{x \ge 2} x \left({x-1}\right) p \left({x}\right) s^{x-2}$

from Differentiation of Power Series‎.

But it also holds when you include $x = 0$ and $x = 1$ in the sum, as in both cases the term evaluates to zero and therefore vanishes.

So:
 * $\displaystyle \Pi''_X \left({s}\right) = \sum_{x \ge 0} x \left({x-1}\right) p \left({x}\right) s^{x-2}$

Plugging in $s = 1$ gives:

The result follows from the definition of variance:
 * $\operatorname{var} \left({X}\right) = E \left({X^2}\right) - \left({E \left({X}\right)}\right)^2$

after a little algebra.

Comment
So, in order to find the variance of a discrete random variable, then there is no need to go through the tedious process of what might be a complicated and fiddly summation.

All you need to do is differentiate the p.g.f twice, and plug in $1$.

Assuming, of course, you know what the p.g.f is.