Probability generating functions — FAQ

It is called a probability 'generating' function because the function, when written as a power series in $t$, 'generates' the probabilities of the random variable as the coefficients of the terms $t^0, t^1, t^2, \dots$.

No, PGFs are specifically for discrete random variables that take non-negative integer values. The equivalent tool for continuous random variables is the Moment Generating Function (MGF), which is outside the scope of the A-Level Further Mathematics (9231) syllabus.

The formula is $Var(X) = G_X''(1) + G_X'(1) - [G_X'(1)]^2$. The most common mistake is forgetting the middle '$+ G_X'(1)$' term. A good way to remember the structure is to know that $G_X''(1) = E(X(X-1))$. So the formula is equivalent to $Var(X) = E(X(X-1)) + E(X) - [E(X)]^2$, which simplifies to the familiar $E(X^2) - [E(X)]^2$.