📚 Probability Generating Functions | 概率生成函数
Probability generating functions (PGFs) are an elegant tool for studying discrete random variables that take non-negative integer values. By encoding all probabilities into a single power series, they allow us to compute moments, analyse sums of independent variables, and recognise distributions quickly. For Edexcel A-Level Further Mathematics, a solid grasp of PGFs is essential, and this article will guide you through the key concepts, derivations and exam techniques.
概率生成函数（PGF）是研究取非负整数值的离散随机变量的一种精妙工具。它将所有概率编码到一个幂级数中，使我们能够计算矩、分析独立变量之和以及快速识别分布。对于 Edexcel A-Level 进阶数学，扎实掌握 PGF 至关重要，本文将带你理清核心概念、推导过程和考试技巧。
1. What is a Probability Generating Function? | 什么是概率生成函数？
A probability generating function transforms the probability mass function (PMF) of a discrete random variable X into a power series in a dummy variable t. It turns a set of probabilities {px (X=x)} into a compact algebraic object that can be manipulated using calculus and algebra.
概率生成函数将离散随机变量 X 的概率质量函数（PMF）转化为关于虚拟变量 t 的幂级数。它将一组概率 {px = P(X=x)} 变成一个紧凑的代数对象，便于运用微积分和代数进行处理。
The idea originates from the expectation operator: we define GX(t) = E(tX). For any non-negative integer-valued random variable, this expectation expands to a power series whose coefficients are exactly the probabilities. This simple construction underpins powerful methods for finding means, variances, and distributions of sums.
这一思想源于期望算子：我们定义 GX(t) = E(tX)。对于任何非负整数值随机变量，该期望展开为一个幂级数，其系数恰好就是各概率。这一简洁构造为求均值、方差以及和的分布提供了有力方法。
2. Definition and Notation | 定义与符号
Let X be a discrete random variable taking values in {0,1,2,…}. The probability generating function of X, denoted by GX(t), is defined as
GX(t) = E(tX) = Σx=0∞ tx P(X=x)
设 X 为取值于 {0,1,2,…} 的离散随机变量。X 的概率生成函数记作 GX(t)，定义为
GX(t) = E(tX) = Σx=0∞ tx P(X=x)
Here the series is assumed to converge absolutely for |t| ≤ 1. Writing GX(t) = p0 + p1t + p2t2 + p3t3 + … shows that the coefficient of tx e t has no probabilistic meaning but serves as a placeholder to extract the probabilities.
这里假设级数在 |t| ≤ 1 时绝对收敛。将 GX(t) 写成 p0 + p1t + p2t2 + p3t3 + … 可以看出 tx 的系数正是 P(X=x)。虚拟变量 t 没有概率含义，只是一个用于提取概率的占位符。
A fundamental property is GX F is always defined at t = 1, and its behaviour at t = 0 gives the probability that X=0: GX(0) = P(X=0).
一个基本性质是 GX(1) = Σ P(X=x) = 1，因为总概率之和为 1。PGF 在 t = 1 处总有定义，而它在 t = 0 处的值给出 X=0 的概率：GX(0) = P(X=0)。
3. Calculating Probabilities from a PGF | 由概率生成函数计算概率
Once a PGF is known, the individual probabilities can be recovered by expanding GX obability P(X=k) can be found by repeatedly differentiating the PGF and evaluating at t=0:
P(X=k) = GX(k)(0) / k!
一旦已知 PGF，可以通过将 GX(t) 展开为幂级数并读出系数来恢复各个概率。另一种方法是，通过对 PGF 反复求导并在 t=0 处取值得到 P(X=k)：
P(X=k) = GX(k)(0) / k!
where GX(k)(0) denotes the k-th derivative of GX(t) evaluated at t=0. This result follows because the term tk differentiates to k! terms vanish at t=0, while higher-order terms still contain a factor of t and become zero.
其中 GX(k)(0) 表示 GX(t) 的 k 阶导数在 t=0 的值。这一结论成立是因为 tk 项求导后得到 k!，而所有低次项在 t=0 时消去，高次项仍含因子 t 因而为零。
For example, if GX(t) = 0.1 + 0.3t + 0.6t2 probabilities are zero. Calculating the first few derivatives at 0 gives the same numbers.
例如，若 GX(t) = 0.1 + 0.3t + 0.6t2，则直接可得 P(X=0)=0.1, P(X=1)=0.3, P(X=2)=0.6，其余概率为零。求出在 0 处的前几阶导数同样得到这些数值。
4. Mean and Variance using Derivatives | 使用导数求均值和方差
The PGF makes it straightforward to obtain the mean and variance without summing over all x. Differentiate GX(t) term-by-term:
G'X(t) = Σx=1∞ x tx-1 P(X=x)
使用 PGF 可以轻松求均值和方差，无需对所有 x 求和。对 GX(t) 逐项求导：
G'X(t) = Σx=1∞ x tx-1 P(X=x)
Evaluating at t=1 gives the expected value:
E(X) = G'X(1)
在 t=1 处取值即得期望值：
E(X) = G'X(1)
The second derivative yields the factorial moment E(X(X-1)):
G''X(t) = Σx=2∞ x(x-1) tx-2 P(X=x) ⇒ E(X(X-1)) = G''X(1)
二阶导数给出阶乘矩 E(X(X-1))：
G''X(t) = Σx=2∞ x(x-1) tx-2 P(X=x) ⇒ E(X(X-1)) = G''X(1)
Since Var(X) = E(X2) - [E(X)]2 and E(X2) = E(X(X-1)) + E(X), we obtain a very useful formula:
Var(X) = G''X(1) + G'X(1) - [G'X(1)]2
由于 Var(X) = E(X2) - [E(X)]2 且 E(X2) = E(X(X-1)) + E(X)，我们得到一个极其有用的公式：
Var(X) = G''X(1) + G'X(1) - [G'X(1)]2
These relations are heavily tested in exams. When applying them, always check that the series converges and that X is defined on the non-negative integers.
这些关系在考试中考查频繁。应用时务必检查级数收敛且 X 定义在非负整数上。
5. PGF of a Sum of Independent Random Variables | 独立随机变量之和的概率生成函数
One of the most important properties of PGFs is their behaviour under addition of independent random variables. If X and Y are independent discrete random variables on the non-negative integers, then
GX+Y(t) = GX(t) × GY(t)
PGF 最重要的性质之一是其对独立随机变量加法的表现。若 X 与 Y 是独立的非负整数值离散随机变量，则
GX+Y(t) = GX(t) × GY(t)
This follows directly from E(tX+Y) = E(tX tY) = E(tX) E(tY) by independence. The result extends naturally to a sum of n independent random variables: GX₁+…+Xₙ(t) = ∏i=1n GXᵢ(t).
该结论直接来自 E(tX+Y) = E(tX tY) = E(tX) E(tY)（由独立性）。这一结果自然推广到 n 个独立随机变量之和：GX₁+…+Xₙ(t) = ∏i=1n GXᵢ(t)。
This multiplicative property is extremely helpful for identifying the distribution of a sum. For instance, if X~Po(λ) and Y~Po(μ) independently, then GX+Y(t) = eλ(t-1) · eμ(t-1) = e(λ+μ)(t-1), which is the PGF of a Po(λ+μ) distribution. Thus we deduce that the sum is also Poisson.
这一乘法性质对于识别和的分布极为有用。例如，若 X~Po(λ) 与 Y~Po(μ) 独立，则 GX+Y(t) = eλ(t-1) · eμ(t-1) = e(λ+μ)(t-1)，这正是 Po(λ+μ) 分布的 PGF。因此我们推断和也服从泊松分布。
6. PGFs of Standard Discrete Distributions | 标准离散分布的概率生成函数
Memorising the PGFs of common distributions speeds up problem solving. The table below summarises the key distributions examined in Edexcel FS1.
熟记常见分布的 PGF 能加快解题速度。下表总结了 Edexcel FS1 中考查的主要分布。
| Distribution | P(X=x) | PGF G(t) | E(X) | Var(X) |
|---|---|---|---|---|
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