Probability Generating Functions

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Edexcel A Level Further Maths · Further Statistics 1

Probability Generating Functions

🎯 The Big Idea: A PGF is just a clever way of packing an entire probability distribution into a single algebraic expression — where the power of the dummy variable t tells you the value of X, and the coefficient tells you its probability. Once your distribution is "encoded" this way, you can find means, variances, and even combine random variables using ordinary algebra instead of long sums.

Summary — What This Chapter Covers

Constructing a PGF — turning a probability distribution table into a polynomial in t.
G(1) = 1 — a built-in check that always works, because all probabilities sum to 1.
Finding probabilities from a PGF — expanding (using the Binomial Theorem or Maclaurin series) and reading off coefficients.
E(X) = G′(1) — differentiate once, substitute t = 1.
Var(X) = G″(1) + G′(1) − [G′(1)]² — differentiate twice, substitute t = 1.
PGFs of standard distributions — Binomial, Poisson, Geometric, Negative Binomial — and how to derive each from first principles.
PGF of X + Y — multiply the two PGFs together (only valid when X and Y are independent).
PGF of aX + b — replace t with tᵃ, then multiply the whole thing by t^b.

1. Constructing PGFs

1What actually is a PGF?

Imagine you've got a discrete random variable X with this distribution:

x	0	1	4	5
P(X = x)	0.4	0.3	0.2	0.1

A PGF just glues each value of x onto the power of a "dummy" variable t, and each probability becomes the coefficient sitting in front of it:

Definition G_X(t) = 0.4t⁰ + 0.3t¹ + 0.2t⁴ + 0.1t⁵ = ¹⁄₁₀(4 + 3t + 2t⁴ + t⁵) In words: line up every possible value of X as an exponent on t, and every matching probability as the coefficient in front of it.

Why bother? Because once your distribution is written as a single algebraic expression, you can differentiate it, multiply two of them together, or substitute values into it — and all of those algebraic operations turn out to correspond to meaningful probability operations (finding the mean, combining independent variables, and so on). It's the same trick as using logarithms to turn multiplication into addition — you're changing the "space" you work in to make the maths easier.

⚠️ Don't confuse X and t X is your actual random variable — the thing with real-world meaning (a dice score, a number of successes, etc). t is just a "carrier" variable with no meaning of its own — it only exists to build the polynomial structure. Never try to interpret t itself.

2The golden check: G(1) = 1

If you substitute t = 1 into any PGF, every power of t becomes 1^anything = 1, so you're just left with the sum of all the probabilities — which must equal 1.

Rule G_X(1) = 1 — always, for every valid PGF, no exceptions.

This is genuinely useful, not just a curiosity. If a question gives you a PGF containing an unknown constant (like p or k), you can find it instantly by setting t = 1 and solving.

Worked Example

Q: G_X(t) = k(2t + 3t² + 5t³). Find k.

Since G(1) = 1: k(2 + 3 + 5) = 1

10k = 1 → k = 0.1

3The formal definition: G(t) = E(tˣ)

Formal definition (Formulae Booklet) _X ^X

4Going backwards: finding probabilities from a PGF

Practice Question 1

A discrete random variable X has probability distribution:

x

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Also in the full note

2. E(X) and Var(X) from PGFs
3. PGFs of Standard Distributions
4. PGFs of Sums & Transformations
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes
1Finding E(X)
2Finding Var(X)

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Probability Generating Functions