Probability Generating Functions | 概率母函数

📚 Probability Generating Functions | 概率母函数

Probability generating functions (PGFs) are a powerful tool in probability theory that encode the entire probability distribution of a discrete random variable into a single algebraic expression. By transforming problems about probabilities, means, and variances into function manipulations, they make complex calculations remarkably efficient.

概率母函数（PGF）是概率论中的一种强大工具，它将离散型随机变量的整个概率分布编码为一个代数表达式。通过将关于概率、均值和方差的问题转化为函数运算，它们使复杂计算变得极为高效。

1. What is a Probability Generating Function? | 什么是概率母函数？

For a discrete random variable X that takes non-negative integer values, the probability generating function is defined as G_X(t) = E(t^X), which expands to a sum over all possible values of X:

对于取非负整数值的离散型随机变量 X，其概率母函数定义为 G_X(t) = E(t^X)，展开为对所有可能取值的求和：

G_X(t) = Σ P(X = r)t^r = P(X=0) + P(X=1)t + P(X=2)t² + P(X=3)t³ + …

The sum runs over r = 0, 1, 2, 3, … . The notation G_X(t) indicates that this is the PGF of the random variable X. Note that t is a dummy variable — its only purpose is to organise the probabilities into a polynomial or series.

求和遍历 r = 0, 1, 2, 3, …。记号 G_X(t) 表示这是随机变量 X 的概率母函数。注意 t 是一个辅助变量——它的唯一作用是将概率整理为多项式或级数的形式。

2. Recovering Probabilities from the PGF | 从概率母函数恢复概率

Since the coefficient of t^r in the expansion is exactly P(X = r), individual probabilities can be recovered by extracting the relevant coefficient. Alternatively, using calculus, the r-th derivative evaluated at zero gives:

由于展开式中 t^r 的系数恰好是 P(X = r)，因此可以通过提取对应系数来恢复单个概率。另外，利用微积分方法，r 阶导数在零处的值为：

P(X = r) = G_X^(r)(0) / r!

where G_X^(r) denotes the r-th derivative of G_X with respect to t. This is particularly useful when the PGF is given in closed form rather than as a polynomial.

其中 G_X^(r) 表示 G_X 对 t 的 r 阶导数。当概率母函数以封闭形式给出而非多项式时，这一公式尤为有用。

3. Key Property: G(1) = 1 | 关键性质：G(1) = 1

Substituting t = 1 into the definition gives the sum of all probabilities:

将 t = 1 代入定义，得到所有概率之和：

G_X(1) = Σ P(X = r) × 1^r = Σ P(X = r) = 1

This is simply the law of total probability. It serves as a quick check: if a derived PGF does not satisfy G(1) = 1, an error has been made somewhere in the derivation.

这实际上是全概率定律。它可以作为快速检验：如果推导出的概率母函数不满足 G(1) = 1，则推导过程中一定存在错误。

4. Finding the Mean Using the PGF | 用概率母函数求均值

Differentiate the PGF term by term and evaluate at t = 1:

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