Probability Generating Functions

Probability Generating Functions

Info: Board Coverage Edexcel FS1 (Section 5) | AQA Optional Topic | OCR (A) FS1

A probability generating function (PGF) packs the whole probability distribution of a discrete random variable taking non-negative integer values into one expression:

$$G_X(t) = \mathrm{E}(t^X) = \sum_{x=0}^{\infty} \mathrm{P}(X = x)\, t^x$$

The coefficient of $t^x$ in the expansion of $G_X(t)$ is exactly $\mathrm{P}(X = x)$. PGFs are examinable for three reasons: they give expectation and variance quickly, they classify standard distributions, and they handle sums of independent random variables with almost no work.

1. Basic Properties

• $G_X(1) = \sum \mathrm{P}(X = x) = 1$: every PGF equals 1 at $t = 1$. This is the standard check for unknown constants.
• $G_X(0) = \mathrm{P}(X = 0)$.
• The PGF fully determines the distribution of $X$.

2. Expectation and Variance from $G(t)$

Differentiate and evaluate at $t = 1$:

$$\mathrm{E}(X) = G_X'(1), \qquad \mathrm{E}(X(X-1)) = G_X''(1)$$

so

$$\mathrm{Var}(X) = G_X''(1) + G_X'(1) - \bigl(G_X'(1)\bigr)^2$$

Worked example. $G_X(t) = \frac{1}{2}(1 + t)^2$. Then

$$G'(t) = (1+t), \quad G'(1) = 2; \qquad G''(t) = 1, \quad G''(1) = 1$$ $$\mathrm{E}(X) = 2, \qquad \mathrm{Var}(X) = 1 + 2 - 2^2 = 1$$

(Expand to check: this is $X \sim \mathrm{B}(2, \tfrac12)$ — see the standard forms below.)

3. Standard PGFs to Memorise

Distribution	$G_X(t)$	$\mathrm{E}(X)$	$\mathrm{Var}(X)$
$\mathrm{B}(n, p)$	$(q + pt)^n$, $q = 1-p$	$np$	$npq$
$\mathrm{Po}(\lambda)$	$e^{\lambda(t-1)}$	$\lambda$	$\lambda$
Geometric$(p)$ on $\{1, 2, \ldots\}$	$\dfrac{pt}{1 - q t}$	$\dfrac{1}{p}$	$\dfrac{q}{p^2}$

Learn the binomial and Poisson forms derivationally: expand $(q+pt)^n$ by the binomial theorem and read off coefficients; expand $e^{\lambda(t-1)}$ as a Maclaurin series and compare with the Poisson pmf.

4. Sums of Independent Random Variables

For independent $X$ and $Y$:

$$G_{X+Y}(t) = G_X(t)\, G_Y(t)$$

This is the result the exam is built around.

Worked example. $X \sim \mathrm{B}(4, 0.3)$, $Y \sim \mathrm{B}(6, 0.3)$, independent. Then

$$G_{X+Y}(t) = (0.7 + 0.3t)^4 (0.7+0.3t)^6 = (0.7 + 0.3t)^{10}$$

which is the PGF of $\mathrm{B}(10, 0.3)$: the sum of independent binomials with the same $p$ is binomial. The same argument for independent Poissons gives $\mathrm{Po}(\lambda_1) + \mathrm{Po}(\lambda_2) = \mathrm{Po}(\lambda_1 + \lambda_2)$.

A related form: if $X_1, \ldots, X_n$ are i.i.d. then $G_{S_n}(t) = \bigl(G_X(t)\bigr)^n$. For example, the sum of 12 independent geometric variables has PGF $\left(\dfrac{pt}{1-qt}\right)^{12}$ — a negative binomial distribution.

5. Finding the Distribution from a PGF

Given an unfamiliar PGF, either match it to a standard form or expand enough coefficients to identify the probabilities, then name the distribution and state its parameters.

Worked example. $G(t) = e^{2(t - 1)} \cdot (0.6 + 0.4t)^5$ for independent $X \sim \mathrm{Po}(2)$ and $Y \sim \mathrm{B}(5, 0.4)$. Then $X + Y$ has PGF equal to the product, which has no simple closed form — but you can still compute $\mathrm{E}(X+Y) = 2 + 2 = 3$ and $\mathrm{Var}(X+Y) = 2 + 5(0.4)(0.6) = 3.2$ from the product rule applied to derivatives at $t=1$.

6. Common Pitfalls

• Forgetting $G''(1)$ is $\mathrm{E}(X(X-1))$, not $\mathrm{Var}(X)$: the variance formula has three terms.
• Valid PGFs satisfy $G(1) = 1$: if an unknown constant $k$ appears, set $G(1) = 1$ first, then read off probabilities.
• Sums of independent variables only. If $X$ and $Y$ are dependent, the PGF product is wrong; state independence in your working.
• Geometric support: some specifications define the geometric distribution on $\{0, 1, 2, \ldots\}$ (number of failures); its PGF is then $\dfrac{p}{1-qt}$. Check which convention your board uses — the exam paper will match the formula booklet.
• Mixed distributions (constant + random parts) do not have PGFs on their own: write $X = c + Y$ and use $G_Y$.

7. Summary

Task	Method
$\mathrm{P}(X = x)$	Coefficient of $t^x$
$\mathrm{E}(X)$	$G'(1)$
$\mathrm{Var}(X)$	$G''(1) + G'(1) - (G'(1))^2$
Sum of independent RVs	Multiply the PGFs
Unknown constant	Use $G(1) = 1$

Cross-References

• Poisson and Geometric Distributions
• Exponential and Continuous Random Variables
• Chi-Squared Tests