Probability Generating Functions of Common Distributions | 常见分布的概率生成函数

📚 Probability Generating Functions of Common Distributions | 常见分布的概率生成函数

The probability generating function (PGF) is a powerful tool in statistics, especially for discrete random variables. It encodes the entire probability mass function into a single power series, which makes calculations of moments and sums of independent variables much simpler.

概率生成函数（PGF）是统计学中一个强有力的工具，尤其适用于离散型随机变量。它将整个概率质量函数浓缩为一个幂级数，从而使得矩的计算以及独立变量之和的处理大为简化。

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1. Definition of the Probability Generating Function | 概率生成函数的定义

Let (X) be a discrete random variable taking non-negative integer values 0, 1, 2, … The probability generating function of (X) is defined as

设 (X) 是一个取非负整数值 0, 1, 2, … 的离散型随机变量。(X) 的概率生成函数定义为

G_X(t) = E(t^X) = Σ p(x) t^x

where (p(x) = P(X = x)) and the sum is taken over all possible values of (x). The series converges for (|t| le 1).

其中 (p(x) = P(X = x))，求和遍及 (x) 的所有可能取值。该级数在 (|t| le 1) 时收敛。

For example, if (X) has probability mass function (p(0)=0.2), (p(1)=0.5), (p(2)=0.3), then (G_X(t)=0.2+0.5t+0.3t^2).

例如，若 (X) 的概率质量函数为 (p(0)=0.2), (p(1)=0.5), (p(2)=0.3)，则 (G_X(t)=0.2+0.5t+0.3t^2)。

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2. Key Properties of the PGF | 概率生成函数的基本性质

The PGF has several essential properties that make it useful in solving problems.

概率生成函数具有若干重要性质，使其在解题中非常有用。

Property 1: (G_X(1) = 1), because the total probability is 1.

性质1： (G_X(1) = 1)，因为总概率为 1。

Property 2: The (r)-th derivative at (t=0) gives the factorial moment:

性质2：在 (t=0) 处的 (r) 阶导数给出阶乘矩：

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

This directly recovers the probability mass function from the PGF.

这可以直接从概率生成函数恢复概率质量函数。

Property 3: The mean and variance can be found using derivatives at (t=1):

性质3：均值与方差可通过在 (t=1) 处的导数求得：

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

These properties allow quick computation of moments without summing infinite series.

这些性质允许我们无需计算无穷级数即可快速求矩。

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3. Uniqueness and Inversion | 唯一性与逆变换

The PGF uniquely determines the probability distribution. If two random variables have the same PGF, they have the same probability mass function.

概率生成函数唯一地确定概率分布。若两个随机变量具有相同的概率生成函数，则它们具有相同的概率质量函数。

To recover probabilities from a known PGF, we can use the relation

要从已知的概率生成函数恢复概率，我们可使用关系式

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

This is particularly useful when the PGF is a polynomial or a simple rational function.

当概率生成函数是多项式或简单有理函数时，这一关系尤为有用。

For instance, if (G(t)=0.2+0.5t+0.3t^2), then (G”(0)=0.6), so (P(X=2)=0.6/2=0.3).

例如，若 (G(t)=0.2+0.5t+0.3t^2)，则 (G”(0)=0.6)，因此 (P(X=2)=0.6/2=0.3)。

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4. Binomial Distribution | 二项分布

The binomial distribution (X sim text{Binomial}(n,p)) models the number of successes in (n) independent Bernoulli trials, each with success probability (p). Its probability mass function is

二项分布 (X sim text{Binomial}(n,p)) 描述在 (n) 次独立伯努利试验中成功的次数，每次成功概率为 (p)。其概率质量函数为

P(X = x) = C(n,x) p^x (1−p)^n−x, x = 0,1,…,n

Using the definition of the PGF,

利用概率生成函数的定义，

G_X(t) = Σ C(n,x)(pt)^x(1−p)^n−x = (1 − p + pt)ⁿ

This compact form is easy to differentiate. For example, (G'(t)=n p (1-p+pt)^{n-1}), so (E(X) = G'(1)=np).

这一紧凑形式很容易求导。例如，(G'(t)=n p (1-p+pt)^{n-1})，所以 (E(X) = G'(1)=np)。

Similarly, (G”(t)=n(n-1)p^2(1-p+pt)^{n-2}), giving (E[X(X-1)] = n(n-1)p^2). Hence (text{Var}(X)=np(1-p)).

类似地，(G”(t)=n(n-1)p^2(1-p+pt)^{n-2})，得到 (E[X(X-1)] = n(n-1)p^2)。因此 (text{Var}(X)=np(1-p))。

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5. Poisson Distribution | 泊松分布

Poisson distribution (X sim text{Poisson}(lambda)) is often used to model the number of rare events occurring in a fixed interval. Its probability mass function is

泊松分布 (X sim text{Poisson}(lambda)) 常用于固定区间内稀有事件发生的次数。其概率质量函数为

P(X = x) = e^−λ λ^x / x!, x = 0,1,2,…

The PGF is obtained by

其概率生成函数为

G_X(t) = e^−λ Σ (λt)^x/x! = e^λ(t−1)

Using the PGF, (G'(t)=lambda e^{lambda(t-1)}), so (E(X)=lambda). Also (G”(t)=lambda^2 e^{lambda(t-1)}), giving (text{Var}(X)=lambda).

利用概率生成函数，(G'(t)=lambda e^{lambda(t-1)})，因此 (E(X)=lambda)。又 (G”(t)=lambda^2 e^{lambda(t-1)})，给出 (text{Var}(X)=lambda)。

The Poisson PGF is particularly elegant and often used in deriving the distribution of sums of independent Poisson variables.

泊松分布的概率生成函数特别简洁，常用于推导独立泊松变量之和的分布。

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6. Geometric Distribution | 几何分布

The geometric distribution models the number of trials until the first success. We adopt the convention (X) = number of trials, with (P(X=x)=p(1-p)^{x-1}) for (x=1,2,3,…).

几何分布描述首次成功所需的试验次数。我们采用 (X) 表示试验次数，且 (P(X=x)=p(1-p)^{x-1})，其中 (x=1,2,3,…)。

Its PGF is

其概率生成函数为

G_X(t) = Σ p q^x−1 t^x = pt / (1 − qt), where q = 1−p

To find the mean, differentiate:

为求均值，先求导：

G'(t) = p / (1−qt)², so E(X) = G'(1) = 1/p

The second derivative gives (G”(t)=2pq/(1-qt)^3). Hence (text{Var}(X) = q/p^2).

二阶导数为 (G”(t)=2pq/(1-qt)^3)。因此 (text{Var}(X) = q/p^2)。

Note: Some texts define the geometric distribution as the number of failures before the first success, giving (P(X=x)=p q^x) for (x=0,1,…). In that case the PGF becomes (p/(1-qt)). Always read the question carefully.

注意：有些教材将几何分布定义为首次成功前的失败次数，即 (P(X=x)=p q^x)，其中 (x=0,1,…)。此时概率生成函数变为 (p/(1-qt))。做题时务必仔细审题。

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7. Negative Binomial Distribution | 负二项分布

The negative binomial distribution models the number of trials needed to obtain (r) successes. With convention (X) = number of trials, its probability mass function is

负二项分布描述获得 (r) 次成功所需的试验次数。若 (X) 表示试验次数，其概率质量函数为

P(X=x) = C(x−1, r−1) p^r q^x−r, x = r, r+1, …

The corresponding PGF is

相应的概率生成函数为

G_X(t) = ( pt / (1 − qt) )^r

This result is intuitive: a negative binomial variable is the sum of (r) independent geometric variables, each with PGF (frac{pt}{1-qt}).

这一结果很直观：负二项变量是 (r) 个独立几何变量之和，每个几何变量的概率生成函数均为 (frac{pt}{1-qt})。

From the PGF we get (E(X)=r/p) and (text{Var}(X)=r q/p^2).

由该概率生成函数可得到 (E(X)=r/p) 与 (text{Var}(X)=r q/p^2)。

Some specifications define negative binomial as the number of failures before (r) successes; then the PGF is ((p/(1-qt))^r). Always match the definition used in your exam board.

有些考试大纲将负二项分布定义为在 (r) 次成功之前的失败次数，此时概率生成函数为 ((p/(1-qt))^r)。务必与你的考试局所采用的定义保持一致。

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8. Uniform Discrete Distribution | 离散均匀分布

For a discrete uniform distribution over (1,2,…,n), each value has probability (1/n). Its PGF is

对于取值 (1,2,…,n) 的离散均匀分布，每个取值的概率均为 (1/n)。其概率生成函数为

G_X(t) = (t + t² + … + tⁿ) / n = t(1−tⁿ) / [n(1−t)]

Using this PGF, the mean is

利用该概率生成函数，均值为

E(X) = (n+1)/2

and the variance is

方差为

Var(X) = (n²−1)/12

These results are useful in many combinatorial probability problems.

这些结果在许多组合概率问题中非常有用。

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9. Sums of Independent Random Variables | 独立随机变量之和

One of the most powerful applications of PGFs is finding the distribution of the sum of independent random variables. If (X) and (Y) are independent, then

概率生成函数最强大的应用之一是求独立随机变量之和的分布。若 (X) 与 (Y) 独立，则

G_X+Y(t) = G_X(t) · G_Y(t)

This follows from the fact that (E(t^{X+Y}) = E(t^X t^Y) = E(t^X)E(t^Y)) because of independence.

这由 (E(t^{X+Y}) = E(t^X t^Y) = E(t^X)E(t^Y)) 以及独立性而得。

For example, if (X sim text{Poisson}(lambda_1)) and (Y sim text{Poisson}(lambda_2)) independently, then

例如，若 (X sim text{Poisson}(lambda_1)) 与 (Y sim text{Poisson}(lambda_2)) 独立，则

G_X+Y(t) = e^λ₁(t−1) e^λ₂(t−1) = e^{(λ₁+λ₂)(t−1)}

Hence (X+Y sim text{Poisson}(lambda_1+lambda_2)). Similarly, the sum of independent binomial variables with the same success probability is again binomial.

因此 (X+Y sim text{Poisson}(lambda_1+lambda_2))。类似地，具有相同成功概率的独立二项变量之和仍服从二项分布。

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10. Worked Example | 典型例题

Problem: Let (X) be a discrete random variable whose PGF is (G(t)=frac{1}{4}(1+t)^2). Find (P(X=2)), (E(X)) and (text{Var}(X)).

例题：设离散型随机变量 (X) 的概率生成函数为 (G(t)=frac{1}{4}(1+t)^2)。求 (P(X=2))、(E(X)) 和 (text{Var}(X))。

First expand the PGF:

首先展开概率生成函数：

G(t) = 1/4 + 1/2 t + 1/4 t²

Thus (P(X=0)=1/4), (P(X=1)=1/2), (P(X=2)=1/4). Therefore (P(X=2)=1/4).

因此 (P(X=0)=1/4), (P(X=1)=1/2), (P(X=2)=1/4)。所以 (P(X=2)=1/4)。

Next, differentiate: (G'(t)=frac{1}{2}+frac{1}{2}t), so (E(X)=G'(1)=1). The second derivative is (G”(t)=frac{1}{2}), so (E[X(X-1)] = G”(1)=1/2). Hence (text{Var}(X) = 1/2 + 1 – 1^2 = 1/2).

接着求导：(G'(t)=frac{1}{2}+frac{1}{2}t)，所以 (E(X)=G'(1)=1)。二阶导数为 (G”(t)=frac{1}{2})，因此 (E[X(X-1)] = G”(1)=1/2)。所以 (text{Var}(X) = 1/2 + 1 – 1^2 = 1/2)。

This example illustrates how easily PGFs recover the full distribution and moments.

此例说明概率生成函数如何轻松地恢复完整分布与矩。

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11. Summary and Exam Tips | 总结与考试建议

Probability generating functions provide a unified way to handle discrete distributions. Memorise the PGFs of binomial, Poisson, geometric, negative binomial and uniform, because they are frequently asked in exam questions.

概率生成函数为处理离散分布提供了一种统一的方法。请熟记二项、泊松、几何、负二项和均匀分布的概率生成函数，因为它们在考试中频繁出现。

Always state the definition (G(t)=E(t^X)), and remember the derivative facts for mean and variance. When finding probabilities from a given PGF, differentiate and divide by the appropriate factorial.

始终写出定义 (G(t)=E(t^X))，并牢记求均值与方差的导数公式。当从给定概率生成函数求概率时，进行求导并除以相应的阶乘。

Check whether (G(1)=1); this is a quick and useful verification of your PGF. Also pay attention to the exact convention used for geometric and negative binomial distributions in the question.

检查 (G(1)=1)，这是对概率生成函数快速有效的验证。同时注意题目中几何分布和负二项分布所采用的具体约定。

Finally, remember that for independent variables, PGFs multiply. This can save time in multi-part questions involving sums of random variables.

最后，记住独立变量的概率生成函数相乘。在处理涉及随机变量之和的多步题目时，这可以节省时间。

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