Probability generating functions

Further Maths · Further Probability & Statistics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Further Maths topics

The idea

A probability generating function (PGF) packs a whole discrete distribution into one function. For a random variable X that takes whole number values 0, 1, 2, ..., the PGF is G(t) = E(t^X). Written out, it is a power series in which the coefficient of t^r is P(X = r).

The variable t has no meaning of its own. It is a marker that keeps the probabilities in order. Putting t = 1 adds all the probabilities, so G(1) = 1.

The PGF is useful for two reasons. Differentiating it and putting t = 1 gives the mean and variance quickly. And for independent variables, the PGF of a sum is the product of the separate PGFs.

Rules to remember

G(t) = E(t^X) = Σ P(X = r)t^r; P(X = r) is the coefficient of t^r, and G(1) = 1
Mean: E(X) = G′(1). Variance: Var(X) = G″(1) + G′(1) − [G′(1)]²
Binomial B(n, p) with q = 1 − p: G(t) = (q + pt)ⁿ
Poisson with mean λ: G(t) = e^{λ(t − 1)}
Geometric (number of trials up to and including the first success): G(t) = pt/(1 − qt)
Discrete uniform on 1, 2, ..., n: G(t) = (t + t² + ... + tⁿ)/n = t(1 − tⁿ)/[n(1 − t)]
If X and Y are independent: G_X+Y(t) = G_X(t) × G_Y(t)

Common mistake

Students write Var(X) = G″(1) − [G′(1)]² and leave out the + G′(1) term. G″(1) is E(X(X − 1)), not E(X²), so the mean must be added back.

Worked example

The probability generating function of X is G(t) = (t + 2t² + t³)/4. Find E(X) and Var(X).

G′(t) = (1 + 4t + 3t²)/4, so E(X) = G′(1) = 8/4 = 2.
G″(t) = (4 + 6t)/4, so G″(1) = 10/4 = 2.5.
Var(X) = G″(1) + G′(1) − [G′(1)]² = 2.5 + 2 − 4 = 0.5.

Answer: E(X) = 2 and Var(X) = 0.5

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Keep going

← Non-parametric tests Take a Further Maths test