Generating Functions are Maclaurin Series

The standard textbook presentation

A typical Further Maths statistics course encounters generating functions through specific distributions in roughly this order:

1. Binomial distribution — G_X(t) = (q + pt)ⁿ. This is a polynomial in t, obtained by applying the binomial theorem.
2. Geometric distribution — G_X(t) = pt / (1 − qt). This is an infinite series, obtained by summing a geometric progression.
3. Poisson distribution — G_X(t) = e^λ(t−1). This involves the exponential function, obtained using the series for e^x.

These look like three unrelated techniques. They are not: in each case G_X(t) is a Maclaurin series, and the Maclaurin coefficient of t^k is P(X = k). The different algebraic forms arise only because the underlying distributions have different probability mass functions.

G_X(t) as a Maclaurin series

The Maclaurin series formula is in the compulsory Further Pure content at all boards. The probability generating function G_X(t) and its application to specific distributions is assessed in the optional Statistics modules listed above.

Suppose G_X(t) can be written as a power series with unknown coefficients:

Set t = 0: immediately a₀ = G_X(0) = P(X = 0). Differentiate once:

Set t = 0: a₁ = G_X′(0) = P(X = 1). Differentiate again:

Set t = 0: a₂ = G_X″(0) / 2 = P(X = 2). After k differentiations:

But aₖ is the coefficient of t^k in G_X(t), which by definition equals P(X = k). So:

This is the Maclaurin formula. There is no separate "PGF technique" for reading off probabilities — it is differentiation at zero, exactly as in every other Maclaurin series.

Binomial distribution — a polynomial PGF

For X ~ B(n, p) with q = 1 − p, the probability mass function is P(X = k) = C(n, k) pᵏ q^n−k. Substituting into the definition:

This is the binomial theorem applied to (q + pt)ⁿ:

This is a polynomial of degree n in t — a Maclaurin series that terminates at n = k because P(X = k) = 0 for k > n. The binomial theorem is not a separate method; it produces the same coefficients that the Maclaurin formula would give.

Verification by the Maclaurin formula

Let X ~ B(3, p) so G_X(t) = (q + pt)³. Extract P(X = 2) by differentiating twice:

Geometric distribution — a geometric series PGF

For X ~ Geo(p) — the number of trials until the first success — P(X = k) = q^k−1p for k = 1, 2, 3, … Substituting:

The sum Σ (qt)^j is a geometric series with ratio qt. Its Maclaurin expansion is:

Therefore:

Reading P(X = k) back from G_X(t)

Expanding G_X(t) = pt · Σ (qt)^j = Σ pq^k−1 t^k, the coefficient of t^k is pq^k−1 = P(X = k) ✓. The closed form pt / (1 − qt) contains the entire geometric distribution in its Maclaurin expansion.

Poisson distribution — an exponential series PGF

For X ~ Po(λ), P(X = k) = e^−λλ^k/k!. Substituting:

The sum Σ (λt)^k/k! is the Maclaurin series for e^λt:

Therefore:

Reading P(X = k) back from G_X(t)

Expanding G_X(t) = e^−λ · e^λt = e^−λ Σ (λt)^k/k!, the coefficient of t^k is e^−λλ^k/k! = P(X = k) ✓. The entire Poisson distribution is encoded in the single expression e^λ(t−1).

Extracting the mean and variance from G_X(t)

Once G_X(t) is known, the mean and variance follow by differentiation. Differentiating G_X(t) = Σ P(X = k) t^k with respect to t:

Setting t = 1 (where the series is guaranteed to converge) gives:

Differentiating a second time:

The variance is then:

Convergence — why t = 1 always works

Every PGF converges for |t| ≤ 1. This is not a coincidence: since all P(X = k) ≥ 0 and Σ P(X = k) = 1, for |t| ≤ 1 we have Σ P(X = k)|t|^k ≤ Σ P(X = k) = 1. The series is dominated by a convergent series of constants. This guarantees that G_X′(1) and G_X″(1) exist (whenever the relevant moments exist), which is why evaluating at t = 1 is the standard route to the mean and variance.

The Binomial PGF is a polynomial — a Maclaurin series that terminates. The Geometric and Poisson PGFs are infinite series. The Maclaurin formula applies to all of them uniformly: the coefficient of t^k is always P(X = k).

The deeper picture — Taylor series

Note on exam boards: Taylor series appears only in the optional Further Pure units. Maclaurin series (expanding around t = 0) is covered in compulsory Further Pure at all four boards; Taylor series (expanding around an arbitrary point a) is the optional extension.

The Maclaurin series is a special case of the Taylor series, which expands G_X around an arbitrary point a rather than zero:

Setting a = 0 recovers the Maclaurin series and the PGF formula P(X = k) = G_X^(k)(0) / k!. Setting a = 1 recovers the moment formulae E[X] = G_X′(1) and Var(X) = G_X″(1) + G_X′(1) − [G_X′(1)]² — these are Taylor coefficients at t = 1, not Maclaurin coefficients. Both views are the same theorem applied at different points.

The PGF also has a clean algebraic property: if X and Y are independent, then G_X+Y(t) = G_X(t) · G_Y(t). Multiplying two power series corresponds to convolving their coefficients — which is exactly the formula for the distribution of a sum of independent random variables. This is the algebra of generating functions: operations on the series mirror operations on the distributions.