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 for studying discrete random variables that take non-negative integer values. It encodes the entire probability distribution in a single algebraic expression, making many calculations much simpler.

概率生成函数（PGF）是研究取非负整数值的离散随机变量的强大工具。它将整个概率分布编码在一个代数表达式中，使许多计算变得简单得多。

1. What is a Probability Generating Function? | 什么是概率生成函数

For a discrete random variable X taking values 0, 1, 2, …, the probability generating function is defined as the expectation of t raised to the power X. This creates a power series in which each probability appears as a coefficient.

对于取值为 0, 1, 2, … 的离散随机变量 X，概率生成函数定义为 t 的 X 次方的期望。这创建了一个幂级数，其中每个概率都作为系数出现。

G_X(t) = E(t^X) = Σ P(X=x) t^x

The sum is taken over all possible values x. This compact form is especially useful because operations on distributions often become simple algebra on their PGFs.

求和针对所有可能的取值 x 进行。这种紧凑形式特别有用，因为对分布的运算往往转化为对其 PGF 的简单代数运算。

2. Definition and Domain | 定义与收敛域

The PGF is defined as a power series. For the series to converge, we usually restrict attention to values of t with |t| ≤ 1. At t = 1 the series is just the sum of all probabilities.

PGF 被定义为一个幂级数。为使级数收敛，我们通常将 t 限制在 |t| ≤ 1 的范围内。在 t = 1 时，级数就是所有概率之和。

G_X(1) = Σ P(X=x) = 1

This is a quick and important check: every valid PGF must satisfy G_X(1) = 1. If you are given a function and it does not satisfy this condition, it cannot be a PGF.

这是一个快速而重要的检验：每个有效的 PGF 都必须满足 G_X(1) = 1。如果给出一个函数不满足这个条件，它就不可能是 PGF。

3. Basic Properties of PGFs | 概率生成函数的基本性质

The value at t = 1 is always 1 because the probabilities sum to 1. This property is often used as a first check in exam questions.

在 t = 1 处的值始终为 1，因为概率之和为 1。这个性质常在考试题中用作第一步检验。

Another fundamental property is uniqueness: if two random variables have the same PGF, then they have exactly the same distribution. This means a PGF completely determines the random variable.

另一个基本性质是唯一性：如果两个随机变量具有相同的 PGF，那么它们的分布完全相同。这意味着 PGF 完全决定了随机变量。

G_X(1) = 1 for all valid PGFs.

所有有效 PGF 都满足 G_X(1) = 1。
Uniqueness: identical PGFs imply identical distributions.

唯一性：相同的 PGF 意味着相同的分布。
PGF coefficients are probabilities, so each coefficient must lie between 0 and 1.

PGF 的系数是概率，因此每个系数必须在 0 和 1 之间。

4. Finding Probabilities from a PGF | 由 PGF 求概率

If a PGF is given as a polynomial or expanded power series, the coefficient of t^x is exactly P(X=x). For example, from G_X(t) = 0.1 + 0.3t + 0.2t² + 0.4t³ you can read off all probabilities directly.

如果 PGF 以多项式或展开的幂级数形式给出，那么 t^x 的系数就是 P(X=x)。例如，从 G_X(t) = 0.1 + 0.3t + 0.2t² + 0.4t³ 中可以直接读出所有概率。

P(X=0)=0.1, P(X=1)=0.3, P(X=2)=0.2, P(X=3)=0.4

For a more complicated PGF, probabilities can be recovered using derivatives at t = 0. The nth probability is the nth derivative of G_X(t) evaluated at 0, divided by n factorial.

对于更复杂的 PGF，可以利用 t = 0 处的导数恢复概率。第 n 个概率等于 G_X(t) 的 n 阶导数在 0 处的值除以 n 的阶乘。

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

5. Mean and Variance Using Derivatives | 用导数求均值与方差

Differentiating the PGF and evaluating at t = 1 gives easy access to the mean and variance. The first derivative at t = 1 gives the expectation of X.

对 PGF 求导并在 t = 1 处取值，可以方便地得到均值和方差。在 t = 1 处的一阶导数给出 X 的期望。

E(X) = G’_X(1)

The second derivative at t = 1 gives E(X(X−1)). Combining these gives the variance. The formula is especially useful when the distribution is defined by a PGF rather than a probability table.

在 t = 1 处的二阶导数给出 E(X(X−1))。结合这些可得到方差。当分布由 PGF 而不是概率表定义时，这个公式特别有用。

Var(X) = G”_X(1) + G’_X(1) − [G’_X(1)]²

Remember that the second derivative gives E(X(X−1)), not E(X²). You must add E(X) to obtain E(X²) before using the usual variance formula.

请记住，二阶导数给出的是 E(X(X−1))，而不是 E(X²)。在使用通常的方差公式之前，必须加上 E(X) 才能得到 E(X²)。

6. PGFs of Common Distributions | 常见分布的 PGF

Several standard distributions have well-known PGFs that you should recognise immediately. The binomial distribution with n trials and success probability p has a PGF that is the nth power of a linear expression.

几个标准分布有你应该立即识别的著名 PGF。具有 n 次试验和成功概率 p 的二项分布的 PGF 是一个线性表达式的 n 次方。

X ~ Binomial(n,p): G_X(t) = (1−p+pt)ⁿ

The Poisson distribution with mean λ has an exponential PGF. The geometric distribution in Edexcel counts the number of trials up to and including the first success, so its PGF has a t in the numerator.

均值为 λ 的泊松分布具有指数形式的 PGF。在 Edexcel 中，几何分布计数直到并包括首次成功的试验次数，因此其 PGF 分子中有一个

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Probability Generating Functions | 概率生成函数