📚 Probability Generating Functions and Their Applications | 概率生成函数及其应用
Probability generating functions (PGFs) are a fundamental tool in IB Mathematics Higher Level for analysing discrete random variables that take non-negative integer values. They convert a sequence of probabilities into a power series, allowing moments, sums of variables, and entire distributions to be studied through algebraic manipulation.
概率生成函数（PGF）是IB数学高级水平中分析取非负整数值的离散随机变量的核心工具。它将一个概率序列转化为幂级数，从而能够通过代数运算研究矩、变量之和以及整个分布。
1. Definition of Probability Generating Function | 概率生成函数的定义
Let X be a discrete random variable taking values in {0, 1, 2, …}, and let px = P(X = x). The probability generating function of X is defined by the expected value of tX:
设 X 是取值为 {0, 1, 2, …} 的离散随机变量，令 px = P(X = x)。则 X 的概率生成函数定义为 tX 的期望值：
GX(t) = E(tX) = Σx=0∞ px tx
The series converges at least for |t| ≤ 1, and since Σ px = 1, we always have GX(1) = 1.
该级数至少在 |t| ≤ 1 上收敛，且由于 Σ px = 1，我们总有 GX(1) = 1。
2. Basic Properties | 基本性质
Several immediate properties follow from the definition. They are essential for solving IB exam questions involving PGFs.
从定义出发可以直接得到若干基本性质，这些性质在解答IB考试中的PGF问题时至关重要。
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GX(0) = p0 – evaluating at t = 0 isolates the probability that X = 0.
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GX(1) = 1 – the total probability is 1.
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All coefficients are non-negative – each px is a probability.
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GX(t) is monotonically increasing for t ≥ 0 because all terms are non-negative.
In Chinese: 以下性质在解题中经常用到。
GX(0) = p0：在 t = 0 处取值可以直接得到 P(X = 0)。
GX(1) = 1：总概率为 1。
所有系数非负：每个 px 都是概率。
GX(t) 在 t ≥ 0 上单调递增：因为所有项都是非负的。
3. Moments and Variance | 矩与方差
PGFs are particularly useful for computing moments. The k-th derivative at t = 1 gives the k-th factorial moment.
PGF 在计算矩时尤为有用。在 t = 1 处的 k 阶导数给出 k 阶阶乘矩。
GX′(1) = E(X)
GX″(1) = E[X(X − 1)]
Therefore the variance can be expressed as:
因此方差可以表示为：
Var(X) = GX″(1) + GX′(1) − [GX′(1)]²
This avoids summing infinite series in many cases and is a favourite examination technique.
这可以在许多情况下避免求无穷级数之和，是考试中十分青睐的解题技巧。
4. Uniqueness and Inversion | 唯一性与反演
If two discrete random variables have the same PGF, then they have the same probability distribution. This is the uniqueness property.
如果两个离散随机变量具有相同的 PGF，则它们具有相同的概率分布。这就是唯一性。
Conversely, the probability P(X = k) can be recovered from the PGF by the formula:
反过来，可以通过以下公式从 PGF 中恢复概率 P(X = k)：
P(X = k) = GX(k)(0) / k!
This is because the Taylor expansion of GX(t) around 0 has coefficients exactly equal to pk.
这是因为 GX(t) 在 0 附近的泰勒展开式的系数正好等于 pk。
5. Sums of Independent Random Variables | 独立随机变量之和
One of the most powerful applications of PGFs is for sums of independent variables. If X and Y are independent, then:
PGF 最强大的应用之一在于处理独立变量之和。若 X 与 Y 独立，则：
GX+Y(t) = GX(t) GY(t)
This extends to any finite number of independent variables. In IB problems, this result is often used to prove that the sum of two independent Poisson variables is again Poisson.
该结果可推广到任意有限个独立变量。在IB问题中，常利用这一结论证明两个独立泊松变量之和仍为泊松分布。
6. Compound Probability Generating Functions | 复合概率生成函数
Let N be a random number of events, and let X₁, X₂, … be independent identically distributed non-negative integer variables, independent of N. Define S = X₁ + X₂ + … + XN. Then the PGF of S is:
设 N 是随机个数，X₁, X₂, … 是与 N 独立的独立同分布非负整数变量。定义 S = X₁ + X₂ + … + XN。则 S 的 PGF 为：
GS(t) = GN( GX(t) )
This compound PGF is central to branching processes, insurance risk models and queueing theory.
这种复合 PGF 是分支过程、保险风险模型和排队论的核心工具。
7. PGFs of Common Distributions | 常见分布的 PGF
The table below summarises the PGFs for distributions commonly tested in IB mathematics.
下表总结了IB数学中常考分布的 PGF。
| Distribution | PGF G(t) |
| Bernoulli(p) | 1 − p + pt |
| Binomial(n, p) | (1 − p + pt)n |
| Poisson(λ) | eλ(t − 1) |
| Geometric(p) (support k = 1,2,…) | pt / (1 − (1 − p)t) |
| Negative Binomial(r, p) | [p / (1 − (1 − p)t)]r |
These PGFs can be derived directly from the probability mass functions and are often used as starting points in exam questions.
这些 PGF 可以直接由概率质量函数导出，常作为考题的出发点。
8. Applications in Stochastic Processes | 在随机过程中的应用
In branching processes, the number of offspring of each individual has a PGF. The extinction probability is found by solving the fixed-point equation t = G(t).
在分支过程中，每个个体的后代数具有一个 PGF，灭绝概率通过求解不动点方程 t = G(t) 得到。
For example, if each individual has a Poisson(λ) number of offspring, the extinction probability is the smallest non-negative root of t = eλ(t−1).
例如，若每个个体的后代数服从 Poisson(λ) 分布，则灭绝概率是方程 t = eλ(t−1) 的最小非负根。
PGFs also appear in random walks, where the generating function of the step distribution determines hitting probabilities.
PGF 也出现在随机游走中，步分布的生成函数决定了到达概率。
9. Problem-Solving Techniques and Common Exam Questions | 解题技巧与常见题型
When solving PGF problems in IB, always begin by writing down the definition and checking G(1) = 1. Then identify what is being asked: a moment, a probability, or a distribution.
在IB中求解PGF问题时，应首先写出定义并验证 G(1) = 1。然后明确题目要求：是求矩、概率还是分布。
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To find E(X) and Var(X), differentiate G(t) and evaluate at t = 1.
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To find P(X = k), differentiate k times and evaluate at t = 0, then divide by k!.
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To show that a sum of independent variables has a certain distribution, multiply their PGFs and compare with known forms.
中文对应技巧：求 E(X) 和 Var(X) 时对 G(t) 求导并在 t = 1 处取值；求 P(X = k) 时求 k 阶导数并在 t = 0 处取值，再除以 k!；证明独立变量之和服从某分布时，将其 PGF 相乘并与已知形式比较。
10. Summary | 总结
Probability generating functions provide a compact, elegant way to represent discrete distributions. They enable quick computation of moments, simplify sums of independent variables, and unlock solutions to advanced probability problems.
概率生成函数为离散分布提供了一种紧凑而优雅的表示方法。它使我们能够快速计算矩、简化独立变量之和，并为解决高等概率问题提供了通路。
Mastering PGFs is not only essential for IB examinations but also builds a foundation for university-level probability and stochastic processes.
掌握 PGF 不仅对IB考试至关重要，也为大学阶段概率论和随机过程的学习奠定基础。
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