Negative Binomial Distribution: Concepts and Formulas | 负二项分布的概念与公式

📚 Negative Binomial Distribution: Concepts and Formulas | 负二项分布的概念与公式

The negative binomial distribution is a discrete probability distribution that models the number of trials required until a fixed number of successes occurs in a sequence of independent Bernoulli trials. It generalises the geometric distribution, which is the special case where exactly one success is required.

负二项分布是一种离散概率分布，用于描述在一系列独立的伯努利试验中，直到发生固定次数成功所需的试验次数。它是几何分布的推广，几何分布是需要且仅需要 1 次成功时的特例。

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1. Underlying Assumptions and Definition | 基本假设与定义

The negative binomial distribution is based on the following assumptions.

负二项分布基于以下假设。

• Each trial has exactly two outcomes, which we call success and failure.

  每次试验只有两种结果，我们称之为成功与失败。

• The probability of success, p, is constant from trial to trial.

  成功的概率 p 在每次试验中保持不变。

• The trials are independent of one another.

  各次试验相互独立。

• The experiment continues until exactly r successes have been observed.

  试验持续进行，直到恰好观察到 r 次成功为止。

If X is the number of trials required to obtain r successes, we write X ~ NB(r, p). The possible values of X are x = r, r+1, r+2, …

如果用 X 表示获得 r 次成功所需的试验次数，我们将它记为 X ~ NB(r, p)。X 的可能取值为 x = r, r+1, r+2, …

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2. Probability Mass Function | 概率质量函数

For X ~ NB(r, p), the probability that the r-th success occurs on the x-th trial is given by the following formula.

若 X ~ NB(r, p)，则第 r 次成功恰好出现在第 x 次试验上的概率由以下公式给出。

P(X=x) = C(x−1, r−1) pʳ (1−p)ˣ⁻ʳ, for x = r, r+1, r+2, …

Here C(x−1, r−1) is the binomial coefficient, which counts the ways to choose the positions of the r−1 successes among the first x−1 trials. The term pʳ represents r successes, and (1−p)ˣ⁻ʳ represents x−r failures.

其中 C(x−1, r−1) 是二项系数，表示在前 x−1 次试验中选定位次来放置那 r−1 次成功的方法数。pʳ 对应 r 次成功，(1−p)ˣ⁻ʳ 对应 x−r 次失败。

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3. Why the PMF is Correct: Derivation | 概率质量函数的推导

To have the r-th success occurring exactly on trial x, two conditions must be met.

要让第 r 次成功恰好发生在第 x 次试验上，必须满足两个条件。

First, trial x must be a success. Second, among the first x−1 trials, there must be exactly r−1 successes and x−r failures.

第一，第 x 次试验必须是成功。第二，在前 x−1 次试验中，必须恰好有 r−1 次成功和 x−r 次失败。

There are C(x−1, r−1) different arrangements of these r−1 successes and x−r failures. Each arrangement has probability pʳ(1−p)ˣ⁻ʳ, because the r successes each contribute p and the x−r failures each contribute 1−p.

这 r−1 次成功与 x−r 次失败共有 C(x−1, r−1) 种不同排列。每一种排列的概率都是 pʳ(1−p)ˣ⁻ʳ，因为 r 次成功各自贡献 p，而 x−r 次失败各自贡献 1−p。

Adding the probabilities over all arrangements gives the negative binomial probability mass function.

将所有排列的概率相加，就得到负二项分布的概率质量函数。

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4. Alternative Form: Number of Failures | 另一形式：失败次数

Some textbooks define the negative binomial distribution in terms of the number of failures before the r-th success, rather than the total number of trials.

有些教材用“第 r 次成功之前出现的失败次数”来定义负二项分布，而不是用总试验次数。

Let Y be the number of failures before the r-th success. Since X is the total number of trials, Y = X − r. The probability mass function is then:

设 Y 为第 r 次成功之前出现的失败次数。由于 X 是总试验次数，所以 Y = X − r。此时概率质量函数为：

P(Y=y) = C(y+r−1, y) pʳ (1−p)ʸ, for y = 0, 1, 2, …

The mean and variance in this alternative form become:

在这种形式下，均值与方差变为：

E[Y] = r(1−p)/p, Var(Y) = r(1−p)/p²

Always check the definition used in an exam, because Edexcel generally defines X as the total number of trials up to and including the r-th success.

在考试中一定要先确认所用定义，因为 Edexcel 通常将 X 定义为“直到并包括第 r 次成功在内的总试验次数”。

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5. Mean and Variance | 均值与方差

For X ~ NB(r, p), where X counts the number of trials up to the r-th

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