The Negative Binomial Distribution | 负二项分布

📚 The Negative Binomial Distribution | 负二项分布

In A-Level Edexcel Statistics, the negative binomial distribution extends the geometric distribution to model the number of trials required to achieve a fixed number of successes in a sequence of independent Bernoulli trials. It is one of the most important discrete probability distributions in the Statistics section of the course, and a firm grasp of its formula, conditions, and properties is essential for exam success. This article provides a complete guide, covering the definition, derivation, mean and variance, worked examples, and common exam pitfalls.

在爱德思A-Level统计学中，负二项分布将几何分布推广到更一般的情形：在重复的独立伯努利试验中，为达成固定次数的成功所需的试验次数。它是课程统计学部分中最重要的一类离散型概率分布之一，熟练掌握其公式、条件与性质，对于考试取得好成绩至关重要。本文将提供完整指南，涵盖定义、推导、均值与方差、例题及常见考试易错点。

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1. Definition and Conditions | 定义与适用条件

The negative binomial distribution applies when a sequence of independent trials satisfies all of the following conditions:

负二项分布适用于满足以下全部条件的一系列独立试验：

• Each trial has exactly two outcomes: success or failure.

  每次试验只有两种结果：成功或失败。

• The probability of success, p, is constant for every trial.

  每次试验成功的概率 p 恒定不变。

• The trials are independent of one another; the outcome of one trial does not affect any other trial.

  各次试验相互独立；一次试验的结果不影响其他任何试验。

• The number of successes required, r, is fixed in advance before the experiment begins.

  所需成功的次数 r 在试验开始前事先固定。

The random variable X, defined as the number of trials needed to obtain r successes, follows a negative binomial distribution. The possible values of X are r, r+1, r+2, and so on, without an upper bound. Intuitively, we keep performing independent trials until we have accumulated exactly r successes, and we record how many total trials this took.

随机变量 X 定义为获得 r 次成功所需的试验次数，它服从负二项分布。X 的可能取值为 r, r+1, r+2, 依此类推，没有上界。直观地看，我们反复进行独立试验，直到累计取得恰好 r 次成功为止，并记录一共进行了多少次试验。

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2. The Probability Mass Function | 概率质量函数

If X denotes the number of trials required to achieve r successes, then the probability that exactly n trials are needed is given by:

若 X 表示达成 r 次成功所需的试验次数，则恰好需要 n 次试验的概率为：

P(X = n) = C(n-1, r-1) × pʳ × (1-p)ⁿ⁻ʳ

for n = r, r+1, r+2, …

其中 n = r, r+1, r+2, …

The binomial coefficient C(n-1, r-1) counts the number of ways to arrange the first n-1 trials so that exactly r-1 successes occur before the final success. The final trial must be a success, which explains why the coefficient uses n-1 and r-1 rather than n and r. The term pʳ accounts for the r successes (r-1 in the first n-1 trials plus one final success), while (1-p)ⁿ⁻ʳ accounts for the n-r failures that occur among the first n-1 trials.

二项式系数 C(n-1, r-1) 统计前 n-1 次试验中恰好出现 r-1 次成功的排列方式数。最后一次试验必定是成功，因此组合数使用 n-1 和 r-1 而非 n 和 r。项 pʳ 对应 r 次成功（前 n-1 次中的 r-1 次加上最后那次成功），而 (1-p)ⁿ⁻ʳ 对应前 n-1 次试验中出现的 n-r 次失败。

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3. Deriving the Formula | 公式推导

To understand the formula, suppose we want r successes and the r-th success occurs on trial n. For this to happen, two requirements must be met simultaneously:

为理解该公式，假设我们需要 r 次成功，且第 r 次成功出现在第 n 次试验。为此，以下两个条件必须同时满足：

• Among the first n-1 trials, there must be exactly r-1 successes and n-r failures.

• 在前 n-1 次试验中，恰好有 r-1 次成功和 n-r 次失败。

• The n-th trial must be a success.

• 第 n 次试验必须是成功。

Consider any specific arrangement of n-1 trials containing exactly r-1 successes. Since the trials are independent, the probability of that particular sequence is p raised to the power r-1 (for the successes) times (1-p) raised to the power n-r (for the failures). Then the n-th trial being a success multiplies the probability by an additional factor of p, giving pʳ(1-p)ⁿ⁻ʳ overall. Because the order of the r-1 successes among the first n-1 positions can be chosen in C(n-1, r-1) different ways, we multiply by this coefficient to obtain the full probability mass function.

考虑任意一个前 n-1 次试验中恰好含 r-1 次成功的特定排列。由于各次试验独立，该特定序列的概率为 p 的 r-1 次幂（对应成功）乘以 (1-p) 的 n-r 次幂（对应失败）。然后第 n 次试验为成功还需再乘一个因子 p，总体上得到 pʳ(1-p)ⁿ⁻ʳ。由于前 n-1 个位置中 r-1 次成功的位置可以在 C(n-1, r-1) 种不同的方式中选择，我们乘以该系数，即得到完整的概率质量函数。

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4. Alternative Formulation | 另一种表述形式

Some textbooks and exam questions define the negative binomial distribution in terms of the number of failures, Y, before the r-th success. In this formulation:

部分教材和考题以第 r 次成功之前所发生的失败次数 Y 来定义负二项分布。在该表述中：

P(Y = y) = C(y+r-1, y) × pʳ × (1-p)ʸ

for y = 0, 1, 2, …

其中 y = 0, 1, 2, …

The two formulations are equivalent: if X is the number of trials and Y is the number of failures, then X = Y + r. The binomial coefficient C(y+r-1, y) is identical to C(y+r-1, r-1); the two forms are just written differently. In the Edexcel syllabus, both forms may appear, so it is essential to read questions carefully to determine which random variable is being used before substituting into the formula.

两种表述等价：若 X 为试验次数、Y 为失败次数，则 X = Y + r。二项式系数 C(y+r-1, y) 与 C(y+r-1, r-1) 完全相同，只是写法不同。在爱德思考纲中，两种形式均可能出现，因此务必仔细审题，确定题目使用的是哪个随机变量，再代入公式计算。

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5. Mean and Variance | 均值与方差

For the number of trials X, the mean and variance of the negative binomial distribution are:

对于试验次数 X，负二项分布的均值与方差为：

E(X) = r/p, Var(X) = r(1-p)/p²

For the number of failures Y, the corresponding results are:

对于失败次数 Y，相应的结果为：

E(Y) = r

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