Probability Generating Functions | 概率生成函数

📚 Probability Generating Functions | 概率生成函数

A probability generating function (PGF) is a powerful tool used to represent the probability distribution of a discrete random variable that takes non‑negative integer values. It transforms the distribution into a power series, making it easier to derive moments, recognise distributions, and handle sums of independent random variables. In Edexcel Further Mathematics, PGFs appear in Further Statistics 1 and are essential for solving problems involving discrete distributions.

概率生成函数（PGF）是用来表示取非负整数值的离散随机变量概率分布的一种有力工具。它将分布转化为一个幂级数，从而更容易推导矩、识别分布以及处理独立随机变量之和。在 Edexcel 进阶数学中，PGF 出现在 Further Statistics 1 模块，是解决离散分布问题的关键内容。

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1. Definition and Notation | 定义与符号

For a discrete random variable X taking values in {0,1,2,…}, its probability generating function is defined as G_X(t) = E(t^X) = Σ_x=0^∞ t^x P(X = x), for all real t for which the sum converges. Typically we require |t| ≤ 1 to guarantee absolute convergence, but the series may converge on a larger interval.

对于取值于 {0,1,2,…} 的离散随机变量 X，其概率生成函数定义为 G_X(t) = E(t^X) = Σ_x=0^∞ t^x P(X = x)，其中 t 为实数且要求级数收敛。通常 |t| ≤ 1 可保证绝对收敛，但级数可能在更大的区间上收敛。

The notation G_X is clear, we may simply write G(t). The function is always well defined at t = 1, giving G_X(1) = 1.

记号 G_X(t) 强调该 PGF 属于 X。上下文清楚时可以简写为 G(t)。该函数在 t = 1 处总是有定义，且有 G_X(1) = 1。

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2. Basic Properties and PGF Evaluation | 基本性质与求值

Because Σ P(X = x) = 1, we always have G_X = 1. This property serves as a quick check that a candidate function could be a valid PGF.

由于 Σ P(X = x) = 1，我们恒有 G_X(1) = 1。这一性质可用于快速检验某个函数是否可能是一个合法的 PGF。

Evaluating the PGF at t = 0 gives P(X = 0): G_X(0) = P(X = 0). More generally, the coefficient of t^k in the power series expansion of G_X(t) is exactly P(X = k).

在 t = 0 处求值可得 P(X = 0)：G_X(0) = P(X = 0)。更一般地，G_X(t) 的幂级数展开式中 t^k 的系数正是 P(X = k)。

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3. Extracting Probabilities from the PGF | 从 PGF 提取概率

Probabilities can be recovered by differentiation: P(X = k) = G_X^(k)(0) / k!, where G_X^(k) denotes the k‑th derivative. For small k this is straightforward; for example, P(X = 0) = G_X(0), P(X = 1) = G′_X(0), P(X = 2) = G″_X(0)/2.

概率可以通过求导恢复：P(X = k) = G_X^(k)(0) / k!，其中 G_X^(k) 表示 k 阶导数。对于较小的 k 这很直接；例如 P(X = 0) = G_X(0)，P(X = 1) = G′_X(0)，P(X = 2) = G″_X(0)/2。

Alternatively, if the PGF is given as a rational function or in closed form, expanding it as a power series yields the whole distribution.

另一种方法是，若 PGF 以有理函数或封闭形式给出，将其展开为幂级数即可获得整个分布。

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4. Expectation and Variance via Derivatives | 通过导数求期望与方差

The PGF makes it simple to obtain the mean and variance without summing series. Differentiating G_X(t) and setting t = 1 gives the expectation: E(X) = G′_X(1).

利用 PGF 可以轻松得到均值和方差而无需进行级数求和。对 G_X(t) 求导并令 t = 1 可得期望：E(X) = G′_X(1)。

To find the variance, we need the second factorial moment: E[X(X – 1)] = G″_X(1). Then Var(X) = G″_X(1) + G′_X(1) – [G′_X(1)]².

为求方差，我们需要二阶阶乘矩：E[X(X – 1)] = G″_X(1)。于是 Var(X) = G″_X(1) + G′_X(1) – [G′_X(1)]²。

Derivation sketch: G′_X(t) = Σ x t^x–1 P(X = x); evaluating at t = 1 gives Σ x P(X = x) = E(X). G″_X(1) = Σ x(x–1) P(X = x) = E[X(X–1)].

推导概要：G′_X(t) = Σ x t^x–1 P(X = x)；令 t = 1 得 Σ x P(X = x) = E(X)。G″_X(1) = Σ x(x–1) P(X = x) = E[X(X–1)]。

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5. PGF of a Sum of Independent Random Variables | 独立随机变量之和的 PGF

If X and Y are independent non‑negative integer‑valued random variables, then G_X+Y(t) = G_X(t) · G_Y(t). This is because E(t^X+Y) = E(t^X t^Y) = E(t^X) E(t^Y) by independence.

若 X 和 Y 是独立的非负整数值随机变量，则 G_X+Y(t) = G_X(t) · G_Y(t)。这是因为根据独立性有 E(t^X+Y) = E(t^X t^Y) = E(t^X) E(t^Y)。

This result extends to any finite number of independent random variables: the PGF of the sum is the product of the individual PGFs. It is extremely useful for finding the distribution of a sum of independent discrete random variables.

这一结果可推广到任意有限个独立随机变量：和的 PGF 等于各个 PGF 的乘积。这在求独立离散随机变量之和的分布时非常有用。

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6. PGF for the Binomial Distribution | 二项分布的 PGF

Let X ~ B(n, p), with q = 1 – p. Then G_X(t) = Σ_x=0^{n n}C_x (pt)^x q^n–x = (q + pt)ⁿ.

设 X ~ B(n, p)，令 q = 1 – p。则 G_X(t) = Σ_x=0^{n n}C_x (pt)^x q^n–x = (q + pt)ⁿ。

Using this PGF, G′_X(1) = n p (q + p)^n–1 = n p, so E(X) = np, and G″_X(1) = n(n–1)p², giving Var(X) = npq.

利用该 PGF，G′_X(1) = n p (q + p)^n–1 = n p，故 E(X) = np；G″_X(1) = n(n–1)p²，进而 Var(X) = npq。

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7. PGF for the Poisson Distribution | 泊松分布的 PGF

If X ~ Po(λ), then G_X(t) = Σ_x=0^∞ e^–λ (λt)^x / x! = e^–λ e^λt = e^{λ(t – 1)}.

若 X ~ Po(λ)，则 G_X(t) = Σ_x=0^∞ e^–λ (λt)^x / x! = e^–λ e^λt = e^{λ(t – 1)}。

From this, G′_X(t) = λ e^λ(t–1), so E(X) = G′_X(1) = λ. Similarly G″_X(1) = λ², giving Var(X) = λ² + λ – λ² = λ.

由此可得 G′_X(t) = λ e^λ(t–1)，所以 E(X) = G′_X(1) = λ。类似地 G″_X(1) = λ²，得出 Var(X) = λ² + λ – λ² = λ。

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8. PGF for the Geometric Distribution | 几何分布的 PGF

he The Edexcel specification usually defines the geometric distribution as the number of trials up to and including the first success: X ~ Geo(p) with P(X = x) = p q^x–1, x = 1, 2, … . Its PGF is G_X(t) = pt / (1 – qt), for |t| < 1/q.

Edexcel 考纲通常将几何分布定义为直到首次成功为止的试验次数：X ~ Geo(p)，P(X = x) = p q^x–1, x = 1, 2, …。其 PGF 为 G_X(t) = pt / (1 – qt)，|t| < 1/q。

Derivation uses the sum of an infinite geometric series: Σ_x=1^∞ p q^x–1 t^x = p t Σ_k=0^∞ (q t)^k = pt/(1 – qt). From the PGF we obtain E(X) = 1/p and Var(X) = q/p².

推导利用了无穷几何级数求和：Σ_x=1^∞ p q^x–1 t^x = p t Σ_k=0^∞ (q t)^k = pt/(1 – qt)。由该 PGF 可得 E(X) = 1/p，Var(X) = q/p²。

A variant defines Y as the number of failures before the first success (Y = X – 1, Y = 0,1,2,…). Then G_Y(t) = p/(1 – qt). The choice of definition must be clear in any problem.

另一种变体将 Y 定义为首次成功前的失败次数 (Y = X – 1, Y = 0,1,2,…)。则 G_Y(t) = p/(1 – qt)。在任何问题中必须明确使用的是哪种定义。

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9. PGF for the Negative Binomial Distribution | 负二项分布的 PGF

For the negative binomial distribution counting the number of failures before the r‑th success, X ~ NB(r, p), with P(X = x) = ^x+r–1C_r–1 p^r q^x, x = 0,1,2,…, the PGF is G_X(t) = [p/(1 – qt)]^r.

对于计算第 r 次成功前失败次数的负二项分布，X ~ NB(r, p)，P(X = x) = ^x+r–1C_r–1 p^r q^x, x = 0,1,2,…，其 PGF 为 G_X(t) = [p/(1 – qt)]^r。

is defined as the total number of trials needed to obtain r successes (W = X + r), then G_W(t) = [pt/(1 – qt)]^r. Both forms are useful and can be interconverted.

若分布定义为获得 r 次成功所需的总试验次数 (W = X + r)，则 G_W(t) = [pt/(1 – qt)]^r。两种形式都有用且可相互转换。

The mean and variance follow from the PGF: E(X) = r q / p, Var(X) = r q / p².

均值和方差可由 PGF 得出：E(X) = r q / p，Var(X) = r q / p²。

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10. Identifiability and Uniqueness | 可识别性与唯一性

A crucial property is that the PGF uniquely determines the distribution: if two non‑negative integer‑valued random variables have the same PGF (on an interval containing 0), then they have identical probability distributions. This enables us to identify a distribution by comparing its PGF with standard forms.

一个关键性质是 PGF 唯一确定分布：若两个非负整数值随机变量在包含 0 的区间上有相同的 PGF，则它们具有完全相同的概率分布。这使得我们可以通过对比标准形式来识别分布。

For example, if a derived PGF is of the form (0.3 + 0.7t)¹⁰, we immediately recognise X ~ B(10, 0.7). If it is e^5(t–1), X ~ Po(5). This is especially powerful when dealing with sums of independent variables.

例如，若推导出的 PGF 形如 (0.3 + 0.7t)¹⁰，我们马上识别出 X ~ B(10, 0.7)；若是 e^5(t–1)，则 X ~ Po(5)。这在处理独立变量之和时尤其强大。

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11. Problem‑Solving Strategies with PGFs | 使用 PGF 的解题策略

nd the PGF of a sum of independent variables and hence identify the distribution of the sum. (iv) Use the PGF to evaluate probabilities such as P(X = 0) or P(X is even) by evaluating G_X(–1) or using symmetry.

Edexcel 考试常见题型包括：(i) 给定概率分布，写出其 PGF；(ii) 用 PGF 计算 E(X) 和 Var(X)；(iii) 求独立变量之和的 PGF 并由此识别和的分布；(iv) 利用 PGF 求概率，如通过计算 G_X(–1) 等求 P(X = 0) 或 P(X 为偶数)。

A typical trick: P(X is even) = [G_X(–1) + G_X(1)] / 2, because t = –1 signs. Always check the domain of convergence when substituting t values different from 1.

一个典型技巧：P(X 为偶数) = [G_X(–1) + G_X(1)] / 2，因为 t = –1 可翻转符号。代入非 1 的 t 值时务必检查收敛域。

When the PGF is given as a rational function, partial fractions or series expansion can be used to recover the probability mass function.

当 PGF 以有理函数形式给出时，可利用部分分式或级数展开来恢复概率质量函数。

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12. Summary of Key PGFs and Formulae | 关键 PGF 与公式总结

The table below collects the standard PGFs that you must know for Edexcel Further Statistics 1, together with their means and variances.

下表汇总了 Edexcel Further Statistics 1 中必须掌握的标准 PGF 及其均值和方差。

Distribution	PGF GX(t)	E(X)	Var(X)
B(n, p)	(q + p t)n	np	npq
Po(λ)	eλ(t – 1)	λ	λ
Geo(p) (trials to 1st success)	p t / (1 – q t)	1/p	q / p²
Geo₀(p) (failures before 1st success)	p / (1 – q t)	q/p	q/p²
NB(r, p) (failures before r successes)	[p / (1 – q t)]r	r q/p	r q / p²

Always remember the derivative relations: E(X) = G′

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