Probability generating functions — common mistakes

Remember the key difference: derivatives at $t=0$ give you probabilities, while derivatives at $t=1$ give you moments (mean and variance). A very common mistake is to mix these up. Always double-check which value of $t$ you should be substituting.

When a question involves the sum of two or more independent random variables and asks for the resulting distribution, using PGFs is almost always the intended method. Calculate the PGF of the sum and then match it to the PGF of a standard distribution (Binomial, Poisson, etc.).

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 $E(X^2) - [E(X)]^2$.