Probability Generating Functions of Standard Distributions | A-Level Edexcel Further Statistics 1 标准分布的概率母函数

📚 Probability Generating Functions of Standard Distributions | A-Level Edexcel Further Statistics 1 标准分布的概率母函数

In Edexcel Further Statistics 1, the probability generating function (PGF) is one of the most rewarding topics on the whole specification. It turns a whole probability distribution into a single algebraic expression, and once you know the PGF of a distribution you can read off its mean, its variance and the distribution of sums of independent variables almost instantly. This article builds the PGF of every standard distribution in the FS1 specification - discrete uniform, binomial, Poisson, geometric and negative binomial - derives each one from first principles, and shows you exactly how examiners test them.

在 Edexcel Further Statistics 1 中，概率母函数（PGF）是整个考纲里性价比最高的知识点之一。它把一整个概率分布浓缩成一个代数表达式；一旦掌握了某个分布的母函数，就能几乎立刻读出它的期望、方差，以及独立随机变量之和的分布。本文将逐一建立 FS1 考纲中所有标准分布的母函数——离散均匀分布、二项分布、泊松分布、几何分布与负二项分布——从基本原理推导，并说明考官究竟怎样出题。

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1. What a Probability Generating Function Actually Is | 概率母函数到底是什么

For a discrete random variable X that takes only the values 0, 1, 2, 3, ..., the probability generating function is defined as the expectation of t to the power X. Because expectation of a function of a discrete variable is just a weighted sum, the definition unpacks into a power series whose coefficients are the probabilities.

对于一个只取 0, 1, 2, 3, ... 的离散随机变量 X，概率母函数定义为 t 的 X 次方的期望。由于离散变量函数的期望就是加权求和，这一定义展开后就是一个幂级数，其系数正是各个概率。

G_X(t) = E(t^X) = Σ P(X = x) tˣ = p₀ + p₁t + p₂t² + p₃t³ + ...

at lets us do algebra with the whole distribution at once. The series always converges for |t| lities sum to 1, and most of the time we only ever evaluate G or its derivatives at t = 1.

这里的 t 只是一个哑变量——它不是概率，也没有实际意义，而是一个"记账工具"，让我们能一次性对整个分布做代数运算。由于所有概率之和为 1，该级数在 |t| ≤ 1 时总是收敛的；而在绝大多数题目中，我们只在 t = 1 处求 G 或其导数的值。

PGFs only exist for random variables whose values are non-negative integers. That is precisely why the specification restricts the technique to the discrete uniform distribution on 1, 2, ..., n, the binomial, the Poisson, the geometric and the negative binomial distributions.

母函数只适用于取值为非负整数的随机变量。正因如此，考纲只把这一方法用于取值 1, 2, ..., n 的离散均匀分布、二项分布、泊松分布、几何分布和负二项分布。

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2. The Four Results You Must Know by Heart | 必须牢记的四个结论

are four standard results, and every FS1 question on this topic uses at least one of them.

母函数的威力在于：对它求导并令 t = 1，就能得到 X 的各阶矩。共有四个标准结论，FS1 中每一道相关题目至少会用到其中一个。

G(1) = 1

G′(1) = E(X)

G″(1) = E(X(X − 1)) = E(X²) − E(X)

Var(X) = G″(1) + G′(1) − [G′(1)]²

lities sum to 1. The second comes from differentiating term by term: the derivative of pₓtˣ is x·pₓt^x−1 s then just the standard identity Var(X) = E(X²) − [E(X)]² rewritten using the first three results.

第一个结论其实就是"概率之和为 1"。第二个来自逐项求导：pₓtˣ 的导数是 x·pₓt^x−1，令 t = 1 后剩下 Σ x·P(X = x)，这正是 E(X) 的定义。第三个结论在求二阶导后同理可得。方差公式则是把标准恒等式 Var(X) = E(X²) − [E(X)]² 用前三个结论改写得到的。

ause the second derivative of a PGF gives the factorial moment E(X(X − 1)), not E(X²). Always add back G′(1) before subtracting the square of the mean.

常见失分点是把方差写成 Var(X) = G″(1) − [G′(1)]²。这是错的，因为母函数的二阶导给出的是阶乘矩 E(X(X − 1))，而不是 E(X²)。务必先加回 G′(1)，再减去期望的平方。

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3. Building a PGF From First Principles | 从基本原理构造母函数

When a question asks you to find a PGF from scratch, never try to recall a formula - go back to the definition and multiply each probability by the corresponding power of t.

当题目要求你"从零求出"母函数时，不要试图回忆公式，而是回到定义：把每个概率乘以对应的 t 的幂次。

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Consider a fair four-sided die whose score X is equally likely to be 1, 2, 3 or 4. Each probability equals ¼, so the PGF is a finite geometric series in t.

假设一枚公平的四面骰，得分 X 等可能地取 1, 2, 3, 4。每个概率都是 ¼，因此母函数是 t 的有限等比级数。

G(t) = ¼(t + t² + t³ + t⁴)

stitute t = 1 to find E(X) and Var(X), and check them against the known results for a discrete uniform distribution.

可以立刻自检：令 t = 1，结果应为 1。这里 ¼(1 + 1 + 1 + 1) = 1，说明母函数至少是一致的。接着求导、代入 t = 1，求出 E(X) 与 Var(X)，再与离散均匀分布的已知结果对照验证。

This "check G(1) = 1" hes most algebraic slips, especially when the PGF contains fractions such as 1/27 or 1/16.

"检验 G(1) = 1" 这个习惯只需五秒，却能抓住绝大多数代数错误，特别是当母函数含有 1/27、1/16 这类分数时。

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4. The Discrete Uniform Distribution | 离散均匀分布

If X is equally likely to take each of the values 1, 2, ..., n, then each probability is 1/n and the PGF is obtained by summing a geometric progression.

若 X 等可能地取 1, 2, ..., n 中的每一个值，则每个概率为 1/n，母函数由一个等比级数求和得到。

G(t) = (1/n)(t + t² + ... + tⁿ) = t(1 − tⁿ) / [n(1 − t)],  t ≠ 1

he limit or use the series form. In practice you never need G(1) from the closed form, because you already know it equals 1.

注意该公式在 t = 1 处没有定义，因此若需要 G(1)，必须取极限或使用级数形式。实际做题时并不会用闭式求 G(1)，因为你早就知道它等于 1。

Differentiating gives the familiar results for this distribution: the mean is (n + 1)/2 and the variance is (n² − 1)/12. If a question gives you a polynomial PGF with all coefficients equal, you should recognise it instantly as a discrete uniform distribution.

求导可得该分布熟悉的结果：期望为 (n + 1)/2，方差为 (n² − 1)/12。若题目给出的母函数是一个各项系数都相等的多项式，你应当立刻认出它是离散均匀分布。

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5. The Binomial Distribution | 二项分布

ss probability p has P(X = 1) = p and P(X = 0) = 1 − p, so its PGF is a linear expression.

得到二项分布母函数最快的路径是通过伯努利分布。单次试验成功概率为 p，则 P(X = 1) = p、P(X = 0) = 1 − p，其母函数是一个一次式。

G_Bernoulli(t) = (1 − p) + pt

If X ~ B(n, p), then X is the sum of n independent Bernoulli variables, and for independent variables PGFs multiply. Raising the Bernoulli PGF to the power n gives the binomial PGF directly.

若 X ~ B(n, p)，则 X 是 n 个独立伯努利变量之和，而独立变量的母函数相乘。把伯努利母函数取 n 次幂，就直接得到二项分布的母函数。

G(t) = (1 − p + pt)ⁿ

Now differentiate. G′(t) = np(1 − p + pt)ⁿ⁻¹ G′(1) = np, the familiar mean. Differentiating again gives G″(t) = n(n − 1)p²(1 − p + pt)ⁿ⁻² and hence G″(1) = n(n − 1)p². Substituting into the variance formula:

接着求导。G′(t) = np(1 − p + pt)ⁿ⁻¹，故 G′(1) = np，即熟悉的期望。再求导得 G″

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