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| ≤ 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| ≤ 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+0.3t².

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+0.3t²。

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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+0.3t², then G''(0)=0.6, so P(X=2)=0.6/2=0.3.

例如，若 G(t)=0.2+0.5t+0.3t²，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 ∼ 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 ∼ 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)p²(1-p+pt)^n-2, giving E[X(X-1)] = n(n-1)p². Hence Var(X)=np(1-p).

类似地，G''(t)=n(n-1)p²(1-p+pt)^n-2，得到 E[X(X-1)] = n(n-1)p²。因此 Var(X)=np(1-p)。

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5. Poisson Distribution | 泊松分布

Poisson distribution X ∼ Poisson(λ) 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 ∼ Poisson(λ) 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)=λ e^λ(t-1), so E(X)=λ. Also G''(t)=λ² e^λ(t-1), giving Var(X)=λ.

利用概率生成函数，G'(t)=λ e^λ(t-1)，因此 E(X)=λ。又 G''(t)=λ² e^λ(t-1)，给出 Var(X)=λ。

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 Var(X) = q/p².

二阶导数为 G''(t)=2pq/(1-qt)³。因此 Var(X) = q/p²。

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 pt1-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 pt1-qt。

From the PGF we get E(X)=r/p and Var(X)=r q/p².

由该概率生成函数可得到 E(X)=r/p 与 Var(X)=r q/p²。

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 ∼ Poisson(λ₁) and Y ∼ Poisson(λ₂) independently, then

例如，若 X ∼ Poisson(λ₁) 与 Y ∼ Poisson(λ₂) 独立，则

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

Hence X+Y ∼ Poisson(λ₁+λ₂). Similarly, the sum of independent binomial variables with the same success probability is again binomial.

因此 X+Y ∼ Poisson(λ₁+λ₂)。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)=14(1+t)². Find P(X=2), E(X) and Var(X).

例题：设离散型随机变量 X 的概率生成函数为 G(t)=14(1+t)²。求 P(X=2)、E(X) 和 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)=12+12t, so E(X)=G'(1)=1. The second derivative is G''(t)=12, so E[X(X-1)] = G''(1)=1/2. Hence Var(X) = 1/2 + 1 - 1² = 1/2.

接着求导：G'(t)=12+12t，所以 E(X)=G'(1)=1。二阶导数为 G''(t)=12，因此 E[X(X-1)] = G''(1)=1/2。所以 Var(X) = 1/2 + 1 - 1² = 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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