2C Probability Generating Functions | 概率生成函数

📚 2C Probability Generating Functions | 概率生成函数

sentation that encapsulates all the probabilities of a discrete random variable taking non‑ re advanced parts of discrete probability, including branching processes and queueing theory.

概率生成函数（PGF）是一种将取非负整数值的离散随机变量的全部概率“打包”成一个幂级数的工具。借助 PGF，我们可以把独立随机变量求和的问题转化为简单的代数乘法，并通过对 t=1 的求导直接得到期望、方差等矩。这一内容是更高级的离散概率主题（如分支过程、排队论）的基础。

1. Definition of a PGF | 概率生成函数的定义

Let X be a discrete random variable taking values in {0, 1, 2, …}. The probability generating function of X is defined as G_X(t) = E(t^X) = Σ_x=0^∞ t^x P(X = x), provided the sum converges absolutely for |t| ≤ 1.

设 X 是一个取值在 {0, 1, 2, …} 上的离散随机变量。X 的概率生成函数定义为 G_X(t) = E(t^X) = Σ_x=0^∞ t^x P(X = x)，要求当 |t| ≤ 1 时该级数绝对收敛。

and G(1) = Σ P(X = x) = 1. The PGF always exists within the unit disk because the terms t^x are bounded.

t 是一个哑变量；G(0) = P(X = 0)，而 G(1) = Σ P(X = x) = 1。由于各项 t^x 在单位圆内是有界的，这个级数在 |t| ≤ 1 上一定存在。

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2. Basic Properties and Radius of Convergence | 基本性质与收敛半径

For |t| < 1, the power series is smooth (infinitely differentiable), and differentiation can be performed term‑by‑term. The radius of convergence R of the series Σ p_x t^x is always at least 1, because Σ p_x ny practical cases R > 1, which allows us to evaluate derivatives at t = 1 without issues.

当 |t| < 1 时，此幂级数是光滑的（无限次可导），并且可以逐项求导。由于 Σ p_x = 1，级数的收敛半径 R 至少是 1。在许多实际情况下 R > 1，这使我们能够在 t = 1 处安全地计算导数值。

A key property is that G_X(t) uniquely identifies the distribution: if two non‑negative integer‑valued random variables have the same PGF for all t in a neighbourhood of 0, then their probability mass functions are identical.

一个关键性质是：PGF 能够唯一识别分布——如果两个非负整值随机变量在 0 的某邻域内对所有 t 都有相同的 PGF，那么它们的概率质量函数完全相同。

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

ies can be recovered by expanding G(t) as a power series about t = 0. The coefficient of t^x is exactly P(X = x). Alternatively, using Taylor’s formula: P(X = x) = G^(x)(0) / x!, where G^(x)(0) denotes the x‑th derivative evaluated at t = 0.

可以通过将 G(t) 展开为 t=0 处的幂级数来恢复概率。t^x 的系数就是 P(X = x)。或者利用泰勒公式：P(X = x) = G^(x)(0) / x!，其中 G^(x)(0) 表示第 x 阶导数在 t=0 处的取值。

For example, if G(t) = 0.2 + 0.3t + 0.5t² bilities are zero. This immediate reading is one of the most convenient features of a PGF.

例如，如果 G(t) = 0.2 + 0.3t + 0.5t²，那么 P(X=0)=0.2，P(X=1)=0.3，P(X=2)=0.5，更高次项的系数均为零。这种“直接读出”概率的方式是 PGF 最方便的特点之一。

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4. Expectation via First Derivative | 利用一阶导数求期望

Differentiating the definition with respect to t gives G'(t) = Σ x t^x-1 P(X = x). Setting t = 1 yields E(X) = G'(1), provided the sum Σ x P(X = x) is finite.

对定义式关于 t 求导得到 G'(t) = Σ x t^x-1 P(X = x)。令 t = 1 即得 E(X) = G'(1)，前提是 Σ x P(X = x) 有限。

nse, but in typical IB problems the expectation is finite and this formula works smoothly.

如果 E(X) 无限，则 G(t) 在 t=1 处通常不可导，但在典型的 IB 题目中期望都是有限的，所以该公式总能顺利使用。

E(X) = G'(1)

E(X) = G'(1)

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5. Variance via Second Derivative | 利用二阶导数求方差

For the variance we need E(X²). Differentiate twice: G''(t) = Σ x(x-1) t^x-2 P(X = x). At t = 1 we get G''(1) = E(X(X-1)) = E(X²) - E(X).

为求方差我们需要 E(X²)。对 PGF 求二阶导数：G''(t) = Σ x(x-1) t^x-2 P(X = x)。令 t=1 得 G''(1) = E(X(X-1)) = E(X²) - E(X)。

Rearranging gives E(X²) = G''(1) + G'(1). Then Var(X) = E(X²) - [E(X)]² = G''(1) + G'(1) - [G'(1)]².

整理可得 E(X²) = G''(1) + G'(1)。于是 Var(X) = E(X²) - [E(X)]² = G''(1) + G'(1) - [G'(1)]²。

Var(X) = G''(1) + G'(1) - [G'(1)]²

Var(X) = G''(1) + G'(1) - [G'(1)]²

This expression is extremely useful because it avoids summing Σ x² ) directly; once you have the PGF in a closed form, variance follows from simple calculus.

这个表达式极为有用，因为它避免了对 Σ x² P(X=x) 的直接求和；一旦 PGF 表示为封闭形式，方差只需通过简单求导即可得出。

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6. PGF of a Sum of Independent Random Variables | 独立随机变量和的 PGF

If X and Y are independent non‑negative integer random variables, then G_X+Y(t) = E(t^X+Y) = E(t^X) · E(t^Y) = G_X(t) G_Y(t). This multiplicative property extends to any finite number of independent terms.

若 X 和 Y 是独立的非负整值随机变量，则 G_X+Y(t) = E(t^X+Y) = E(t^X) · E(t^Y) = G_X(t) G_Y(t)。这一乘法性质可推广到任意有限个独立变量的和。

This is the single most powerful feature of PGFs: convolutions of probability mass functions, which are messy to compute directly, become simple polynomial multiplication.

这正是 PGF 最强大的特性：原本计算起来很繁琐的概率质量函数卷积，变成了简单的多项式相乘。

For instance, if X ~ Po(λ) and Y ~ Po(μ) are independent, then G_X+Y(t) = e^λ(t-1) · e^μ(t-1) = e^(λ+μ)(t-1), showing immediately that X+Y ~ Po(λ+μ).

例如，若 X ~ Po(λ) 与 Y ~ Po(μ) 独立，则 G_X+Y(t) = e^λ(t-1) · e^μ(t-1) = e^(λ+μ)(t-1)，立即表明 X+Y ~ Po(λ+μ)。

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7. PGFs of Common Discrete Distributions | 常见离散分布的 PGF

The table below summarises the PGF, mean and variance for several standard distributions that are frequently examined.

下表总结了几种常见分布在考试中经常用到的 PGF、均值和方差。

Distribution	PGF G(t)	Mean	Variance
Bernoulli(p)	q + p t,   q = 1-p	p	p q
Binomial B(n, p)	(q + p t)n	n p	n p q
Poisson Po(λ)	eλ(t-1)	λ	λ
Geometric Geo(p) (trials to first success)	p t / (1 - q t),   |t| < 1/q	1/p	q / p2
Negative binomial NB(r, p) (failures before r‑th success)	[p / (1 - q t)]r   (for number of failures)	r q / p	r q / p2

Note: For the geometric distribution defined as "number of trials to first success", X ∈ {1,2,3,… counting failures would have G(t) = p/(1 - q t). Always check the support of the variable.

注意：对于定义为“首次成功所需试验次数”的几何分布，X ∈ {1,2,3,…}，因此 G(0)=0。另一种计数失败的几何分布版本 PGF 为 p/(1 - q t)。使用时务必确认随机变量的取值范围。

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8. Finding the Distribution of a Sum | 求和的分布

Because a PGF is a power series, multiplying two PGFs and expanding the product gives the probability mass function of the sum. For example, suppose X and Y are independent with G_X(t) = 0.3 + 0.7t and G_Y(t) = 0.5 + 0.5t. Then G_X+Y(t) = (0.3+0.7t)(0.5+0.5t) = 0.15 + 0.50t + 0.35t². Hence P(X+Y=0)=0.15, P(X+Y=1)=0.50, P(X+Y=2)=0.35.

因为 PGF 本身就是幂级数，将两个 PGF 相乘并展开即可得到和的概率质量函数。例如，设独立变量 X 和 Y 满足 G_X(t) = 0.3 + 0.7t，G_Y(t) = 0.5 + 0.5t，则 G_X+Y(t) = (0.3+0.7t)(0.5+0.5t) = 0.15 + 0.50t + 0.35t²。因此 P(X+Y=0)=0.15，P(X+Y=1)=0.50，P(X+Y=2)=0.35。

This approach extends elegantly to more than two summands and to random sums, where the number of terms itself is a random variable — a topic that leads to compound distributions.

这一方法可以优雅地推广到多于两个被加数的情况，甚至推广到项数本身为随机变量的“随机和”——这就引出了复合分布的主题。

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9. Using PGFs to Solve Equations | 用 PGF 解方程

has offspring distribution with PGF G(t), the extinction probability ε is the smallest non‑negative root of ε = G(ε).

PGF 在求解随机过程中出现的函数方程时极为有用。例如，若一个分支过程的后代分布具有 PGF G(t)，则其灭绝概率 ε 是方程 ε = G(ε) 的最小非负根。

Even in more elementary IB problems, you may be given an equation like G(t) = (a + b t)^k parameters by using the normalisation condition G(1) = 1, or by matching a given mean to G'(1).

即使在更基础的 IB 题目中，你也可能遇到像 G(t) = (a + b t)^k 这样的方程，需要利用归一化条件 G(1) = 1 或给定的均值与 G'(1) 匹配，来求解未知参数。

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10. Uniqueness of the PGF | PGF 的唯一性

If two PGFs G_X(t) and G_Y(t) are equal as formal power series for all t in some open interval containing 0, then the random variables X and Y have exactly the same distribution. This uniqueness property justifies many ‘coefficient matching’ arguments used in problem solving.

如果两个 PGF G_X(t) 和 G_Y(t) 作为形式幂级数在包含 0 的某个开区间上处处相等，那么随机变量 X 和 Y 具有完全相同的分布。这一唯一性为解题中常用的“比较系数法”提供了理论依据。

In practice, once you recognise the PGF as that of a known distribution (e.g., (q+pt)ⁿ looks like Binomial), uniqueness guarantees that the random variable must follow that distribution, provided the support matches.

在实践中，一旦你识别出一个 PGF 属于某个已知分布（例如 (q+pt)ⁿ 看起来像二项分布），唯一性就保证在取值范围相符的前提下，该随机变量必定服从那个分布。

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